![Had the apparatus [of transseries and analyzable functions] been
introduced for the sole purpose of solving Dulac’s “conjecture,” one
might legitimately question the wisdom and cost-effectiveness of
such massive investment in new machinery. However, [these no-
tions] have many more applications, actual or potential, especially
in the study of analytic singularities. But their chief attraction is per-
haps that of giving concrete, if partial, shape to G. H. Hardy’s dream
of an all-inclusive, maximally stable algebra of “totally formalizable
functions.”
— Jean Écalle, Six Lectures on Transseries, Analysable Functions and the
Constructive Proof of Dulac’s Conjecture.
The virtue of model theory is its ability to organize succinctly the
sort of tiresome algebraic details associated with elimination theory.
— Gerald Sacks, The Differential Closure of a Differential Field.
Les analystes p-adiques se fichent tout autant que les géomètres
algébristes . . . , des gammes à plus soif sur les valuations com-
posées, les groupes ordonnés baroques, sous-groupes pleins des-
dits et que sais-je. Ces gammes méritent tout au plus d’enrichir les
exercices de Bourbaki, tant que personne ne s’en sert.
— Alexander Grothendieck, letter to Serre dated October 31, 1961.
I don’t like either writing or reading two-hundred page papers. It’s
not my idea of fun.
— John H. Conway, quoted in Genius at Play: The Curious Mind of John
Horton Conway by Siobhan Roberts.](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-10-320.jpg)
![xiv PREFACE
ACKNOWLEDGMENTS
Part of this work was carried out while some of the authors were in residence at vari-
ous times at the Fields Institute (Toronto), the Institut des Hautes Études Scientifiques
(Bures-sur-Yvette), the Isaac Newton Institute for Mathematical Sciences (Cambridge),
and the Mathematical Sciences Research Institute (Berkeley). The support and hospi-
tality of these institutions is gratefully acknowledged.
Aschenbrenner’s work was partially supported by the National Science Foundation
under grants DMS-0303618, DMS-0556197, and DMS-0969642. Visits by van der
Hoeven to Los Angeles were partially supported by the UCLA Logic Center.
We thank the following copyright holders for permission to reproduce the text in
the epigraphs in the front of this book: Springer Science and Business Media, New
York, for the quote by Jean Écalle from [121]; the American Mathematical Society
for the quote by Gerald Sacks from [376], © 1972 American Mathematical Society;
Professor Jean-Pierre Serre for the quote by Alexander Grothendieck from [88]; and
Siobhan Roberts for the quote by John H. Conway that appears in her book Genius at
Play: The Curious Mind of John Horton Conway [344] © Siobhan Roberts, published
by Bloomsbury Publishing, Inc., 2016.
We thank David Marker and Angus Macintyre for their interest and steadfast moral
support over the years. To Santiago Camacho, Andrei Gabrielov, Tigran Hakobyan,
Elliot Kaplan, Nigel Pynn-Coates, Chieu Minh Tran, and especially to Allen Gehret,
we are indebted for numerous comments on and corrections to the manuscript. We are
also grateful to Philip Ehrlich for setting us right on some historical points, and to the
anonymous reviewers for useful suggestions and for spotting some errors. We are of
course solely responsible for any remaining inadequacies.
Finally, we thank our editor, Vickie Kearn, and the other staff at Princeton Univer-
sity Press, notably Nathan Carr and Glenda Krupa, for helping us to bring this book
into its final form.
Matthias Aschenbrenner, Los Angeles
Lou van den Dries, Urbana
Joris van der Hoeven, Paris
September 2015](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-19-320.jpg)
![Dramatis Personæ
Dramatis Personæ
Dramatis Personæ
We summarize here the definitions of some notions prominent in our work, together
with a list of attributes that apply to them. We include the page number where each
concept is first introduced. We let m, n, r range over N = {0, 1, 2, . . . }. Below K is
a field, possibly equipped with further structure. We let a, b, f, g, y, z range over ele-
ments of K, and we let Y be an indeterminate over K. If K comes equipped with a val-
uation, then we let O be the valuation ring of K, and we freely employ the dominance
relations on K introduced in Section 3.1. If K comes equipped with a derivation ∂,
then we also write f0
, f00
, . . . , f(n)
, . . . for ∂f, ∂2
f, . . . , ∂n
f, . . . , and f†
= f0
/f for
the logarithmic derivative of any f 6= 0; in this case C = {f : f0
= 0} denotes the
constant field of K, and c ranges over C. We let Γ be an ordered abelian group, and let
α, β, γ range over Γ.
VALUED FIELDS
Let K be a valued field, that is, a field equipped with a valuation on it; p. 112.
Complete: every cauchy sequence in K has a limit in K; p. 84.
Spherically complete: every pseudocauchy sequence in K has a pseudolimit in K;
p. 78. “Spherically complete” is equivalent to “maximal” as defined below.
Maximal: there is no proper immediate valued field extension of K; p. 129.
Algebraically maximal: there is no proper immediate algebraic valued field extension
of K; p. 130.
Henselian: for every P ∈ K[Y ] with P 4 1, P(0) ≺ 1, and P0
(0) 1, there exists
y ≺ 1 with P(y) = 0; p. 136.
DIFFERENTIAL FIELDS
Let K be a differential field, that is, a field of characteristic zero equipped with a deri-
vation on it; p. 200.
Linearly surjective: for all a0, . . . , ar ∈ K with ar 6= 0 there exists y such that
a0y + a1y0
+ · · · + ary(r)
= 1; p. 253.](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-24-320.jpg)
![xx DRAMATIS PERSONÆ
Linearly closed: for all r ⩾ 1 and a0, . . . , ar ∈ K there are b0, . . . , br−1, b ∈ K with
a0Y +a1Y 0
+· · ·+arY (r)
= b0(Y 0
+bY )+b1(Y 0
+bY )0
+· · ·+br−1(Y 0
+bY )(r−1)
;
p. 252.
Picard-Vessiot closed (or pv-closed): for all r ⩾ 1 and a0, . . . , ar ∈ K with ar 6= 0
there are C-linearly independent y1, . . . , yr such that a0yi + a1y0
i + · · · + ary
(r)
i = 0
for i = 1, . . . , r; p. 254.
Differentially closed: for all P ∈ K[Y, . . . , Y (r)
]6=
and Q ∈ K[Y, . . . , Y (r−1)
]6=
such
that ∂P
∂Y (r) 6= 0 there is y with P(y, y0
, . . . , y(r)
) = 0 and Q(y, y0
, . . . , y(r−1)
) 6= 0;
p. 237.
VALUED DIFFERENTIAL FIELDS
Let K be a valued differential field, that is, a differential field equipped with a valuation
on it; p. 221.
Small derivation: f ≺ 1 ⇒ f0
≺ 1; p. 222.
Monotone: f ≺ 1 ⇒ f0
4 f; p. 226.
Few constants: c 4 1 for all c; p. 226.
Many constants: for every f there exists c with f c; p. 226.
Differential-henselian (or d-henselian): K has small derivation and:
(DH1) for all a0, . . . , ar 4 1 in K with ar 1 there exists y 1 such that
a0y + a1y0
+ · · · + ary(r)
∼ 1;
(DH2) for every P ∈ K[Y, Y 0
, . . . , Y (r)
] with P 4 1, P(0) ≺ 1, and ∂P
∂Y (n) (0) 1
for some n, there exists y ≺ 1 such that P(y, y0
, . . . , y(r)
) = 0; p. 340.
ASYMPTOTIC FIELDS
Let K be an asymptotic field, that is, a valued differential field such that for all nonzero
f, g ≺ 1: f ≺ g ⇐⇒ f0
≺ g0
; p. 379.
H-asymptotic (or of H-type): 0 6= f ≺ g ≺ 1 ⇒ f†
g†
; p. 379.
Differential-valued (or d-valued): for all f 1 there exists c with f ∼ c; p. 379.
Grounded: there exists nonzero f 6 1 such that g†
f†
for all nonzero g 6 1; p. 384.
Asymptotic integration: for all f 6= 0 there exists g 6 1 with g0
f; p. 383.
Asymptotically maximal: K has no proper immediate asymptotic field extension;
p. 380.
Asymptotically d-algebraically maximal: K has no proper immediate differential-alge-
braic asymptotic field extension; p. 380.](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-25-320.jpg)
![DRAMATIS PERSONÆ xxi
λ-free: H-asymptotic, ungrounded, and for all f there exists g 1 with f − g††
g†
;
p. 506.
ω-free: H-asymptotic, ungrounded, and for all f there is g 1 with f − ω(g††
) g†
,
where ω(z) := −(2z0
+ z2
); p. 515.
Newtonian: H-asymptotic, ungrounded, and every P ∈ K[Y, Y 0
, . . . , Y (r)
]6=
of New-
ton degree 1 has a zero in O; p. 640. (See p. 480 for Newton degree.)
ORDERED VALUED DIFFERENTIAL FIELDS
Let K be an ordered valued differential field, that is, a valued differential field equipped
with an ordering in the usual sense of ordered field; p. 378.
Pre-H-field: O is convex in the ordered field K, and for all f:
f O =⇒ f0
0; p. 452.
H-field: O is the convex hull of C in the ordered field K, and for all f:
f C =⇒ f0
0, f 1 =⇒ there exists c with f ∼ c; p. 451.
Liouville closed: K is a real closed H-field and for all f, g there exists y 6= 0 such that
y0
+ fy = g; p. 460.
ASYMPTOTIC COUPLES
Let (Γ, ψ) be an asymptotic couple, that is, the ordered abelian group Γ is equipped
with a map ψ: Γ6=
→ Γ such that for all α, β 6= 0:
(AC1) α + β 6= 0 ⇒ ψ(α + β) ⩾ min ψ(α), ψ(β)
;
(AC2) ψ(kα) = ψ(α) for all k ∈ Z6=
;
(AC3) α 0 ⇒ α0
:= α + ψ(α) ψ(β);
p. 322. For γ 6= 0 we set γ0
:= γ + ψ(γ).
H-asymptotic (or of H-type): 0 α β ⇒ ψ(α) ⩾ ψ(β); p. 323.
Grounded: Ψ := {ψ(α) : α 6= 0} has a largest element; p. 388.
Small derivation: γ 0 ⇒ γ0
0; p. 388.
Asymptotic integration: for all α there exists β 6= 0 with α = β0
; p. 383.](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-26-320.jpg)
![Introduction and Overview
A DIFFERENTIAL FIELD WITH NO ESCAPE
Our principal object of interest is the differential field T of transseries. Transseries are
formal series in an indeterminate x R, such as
ϕ(x) = −3 eex
+ e
ex
log x + ex
log2 x
+ ex
log3 x
+ · · ·
−x11
+ 7
(1)
+
π
x
+
1
x log x
+
1
x log2
x
+
1
x log3
x
+ · · ·
+
2
x2
+
6
x3
+
24
x4
+
120
x5
+
720
x6
+ · · ·
+ e−x
+2 e−x2
+3 e−x3
+4 e−x4
+ · · · ,
where log2
x := (log x)2
, etc. As in this example, each transseries is a (possibly
transfinite) sum, with terms written from left to right, in asymptotically decreasing
order. Each term is the product of a real coefficient and a transmonomial. Appendix A
contains the inductive construction of T, including the definition of “transmonomial”
and other notions about transseries that occur in this introduction. For expositions of T
with proofs, see [112, 122, 194]. In [112], T is denoted by R((x−1
))LE
, and its elements
are called logarithmic-exponential series. At this point we just mention that transseries
can be added and multiplied in the natural way, and that with these operations, T is a
field containing R as a subfield. Transseries can also be differentiated term by term,
subject to r0
= 0 for each r ∈ R and x0
= 1. In this way T acquires the structure of a
differential field.
Why transseries?
Transseries naturally arise in solving differential equations at infinity and studying the
asymptotic behavior of their solutions, where ordinary power series, Laurent series, or
even Puiseux series in x−1
are inadequate. Indeed, functions as simple as ex
or log x
cannot be expanded with respect to the asymptotic scale xR
of real powers of x at +∞.
For merely solving algebraic equations, no exponentials or logarithms are needed: it is
classical that the fields of Puiseux series over R and C are real closed and algebraically
closed, respectively.
One approach to asymptotics with respect to more general scales was initiated by
Hardy [163, 165], inspired by earlier work of du Bois-Reymond [51] in the late 19th](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-30-320.jpg)
![2 INTRODUCTION AND OVERVIEW
century. Hardy considered logarithmico-exponential functions: real-valued functions
built up from constants and the variable x using addition, multiplication, division, ex-
ponentiation and taking logarithms. He showed that such a function, when defined on
some interval (a, +∞), has eventually constant sign (no oscillation!), and so the germs
at +∞ of these functions form an ordered field H with derivation d
dx . Thus H is what
Bourbaki [62] calls a Hardy field: a subfield K of the ring of germs at +∞ of dif-
ferentiable functions f : (a, +∞) → R with a ∈ R, closed under taking derivatives;
for more precision, see Section 9.1. Each Hardy field is naturally an ordered differ-
ential field. The Hardy field H is rather special: every f ∈ H satisfies an algebraic
differential equation over R. But H lacks some closure properties that are desirable
for a comprehensive theory. For instance, H has no antiderivative of ex2
(by Liouville;
see [361]), and the functional inverse of (log x)(log log x) doesn’t lie in H, and is not
even asymptotic to any element of H: [111, 190]; see also [333].
With T and transseries we go beyond H and logarithmico-exponential functions
by admitting infinite sums. It is important to be aware, however, that by virtue of its
inductive construction, T does not contain, for example, the series
x + log x + log log x + log log log x + · · · ,
which does make sense in a suitable extension of T. Thus T allows only certain kinds
of infinite sums. Nevertheless, it turns out that the differential field T enjoys many
remarkable closure properties that H lacks. For instance, T is closed under natural
operations of exponentiation, integration, composition, compositional inversion, and
the resolution of feasible algebraic differential equations (where the meaning of feasible
can be made explicit). This makes T of interest for different areas of mathematics:
Analysis
In connection with the Dulac Problem, T is sufficiently rich for modeling the asymp-
totic behavior of so-called Poincaré return maps. This analytically deep result is a
crucial part of Écalle’s solution of the Dulac Problem [119, 120, 121]. (At the end of
this introduction we discuss this in more detail.)
Computer algebra
Many transseries are concrete enough to compute with them, in the sense of computer
algebra [190, 402]. Moreover, many of the closure properties mentioned above can
be made effective. This allows for the automation of an important part of asymptotic
calculus for functions of one variable.
Logic
Given an o-minimal expansion of the real field, the germs at +∞ of its definable one-
variable functions form a Hardy field, which in many cases can be embedded into T.
This gives useful information about the possible asymptotic behavior of these definable
functions; see [21, 292] for more about this connection.](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-31-320.jpg)
![INTRODUCTION AND OVERVIEW 3
Soon after the introduction of T in the 1980s it was suspected that T might well be a
kind of universal domain for the differential algebra of Hardy fields and similar ordered
differential fields, analogous to the role of the algebraically closed field C as a universal
domain for algebraic geometry of characteristic 0 (Weil [461, Chapter X, §2]), and
of R, Qp, and C((t)) in related ordered and valued settings. This is corroborated by the
strong closure properties enjoyed by T. See in particular p. 148 of Écalle’s book [120]
for eloquent expressions of this idea. The present volume and the next substantiate
the universal domain nature of the differential field T, using the language of model
theory. The model-theoretic properties of the classical fields C, R, Qp and C((t)) are
well established thanks to Tarski, Seidenberg, Robinson, Ax Kochen, Eršov, Cohen,
Macintyre, Denef, and others; see [443, 395, 350, 28, 29, 131, 84, 275, 100]. Our goal
is to analyze likewise the differential field T, which comes with a definable ordering
and valuation, and in this book we achieve this goal.
The ordered and valued differential field T
For what follows, it will be convenient to quickly survey some of the most distinctive
features of T. Appendix A contains precise definitions and further details.
Each transseries f = f(x) can be uniquely decomposed as a sum
f = f + f + f≺,
where f is the infinite part of f, f is its constant term (a real number), and f≺ is its
infinitesimal part. In the example (1) above,
ϕ = −3 eex
+ e
ex
log x + ex
log2 x
+ ex
log3 x
+ · · ·
−x11
,
ϕ = 7,
ϕ≺ =
π
x
+
1
x log x
+ · · · .
In this example, ϕ happens to be a finite sum, but this is not a necessary feature of
transseries: take for example f := ex
log x + ex
log2 x
+ ex
log3 x
+· · · , with f = f. Declaring
a transseries to be positive iff its dominant (= leftmost) coefficient is positive turns T
into an ordered field extension of R with x R. In our example (1), the dominant
transmonomial of ϕ(x) is eex
and its dominant coefficient is −3, whence ϕ(x) is neg-
ative; in fact, ϕ(x) R.
The inductive definition of T involves constructing a certain exponential operation
exp: T → T×
, with exp(f) also written as ef
, and
exp(f) = exp(f) · exp(f) · exp(f≺) = exp(f) · ef
·
∞
X
n=0
fn
≺
n!
where the first factor exp(f) is a transmonomial, the second factor ef
is the real
number obtained by exponentiating the real number f in the usual way, and the third](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-32-320.jpg)
![4 INTRODUCTION AND OVERVIEW
factor exp(f≺) =
P∞
n=0
fn
≺
n! is expanded as a series in the usual way. Conversely,
each transmonomial is of the form exp(f) for some transseries f. Viewed as an
exponential field, T is an elementary extension of the exponential field of real numbers;
see [111]. In particular, T is real closed, and so its ordering is existentially definable
(and universally definable) from its ring operations:
(2) f ⩾ 0 ⇐⇒ f = g2
for some g.
However, as emphasized above, our main interest is in T as a differential field, with
derivation f 7→ f0
on T defined termwise, with r0
= 0 for r ∈ R, x0
= 1, (ef
)0
= f0
ef
,
and (log f)0
= f0
/f for f 0. Let us fix here some notation and terminology in force
throughout this volume: a differential field is a field K of characteristic 0 together with
a single derivation ∂: K → K; if ∂ is clear from the context we often write a0
instead
of ∂(a), for a ∈ K. The constant field of a differential field K is the subfield
CK := {a ∈ K : a0
= 0}
of K, also denoted by C if K is clear from the context. The constant field of T turns
out to be R, that is,
R = {f ∈ T : f0
= 0}.
By an ordered differential field we mean a differential field equipped with a total order-
ing on its underlying set making it an ordered field in the usual sense of that expression.
So T is an ordered differential field. More important than the ordering is the valuation
on T with valuation ring
OT :=
f ∈ T : |f| ⩽ r for some r ∈ R = {f ∈ T : f = 0},
a convex subring of T. The unique maximal ideal of OT is
OT :=
f ∈ T : |f| ⩽ r for all r 0 in R = {f ∈ T : f = f≺}
and thus OT = R+OT. Its very definition shows that OT is existentially definable in the
differential field T. However, OT is not universally definable in the differential field T:
Corollary 16.2.6. In light of the model completeness conjecture discussed below, it is
therefore advisable to add the valuation as an extra primitive, and so in the rest of this
introduction we construe T as an ordered and valued differential field, with valuation
given by OT. By a valued differential field we mean throughout a differential field K
equipped with a valuation ring of K that contains the prime subfield Q of K.
Grid-based transseries
When referring to transseries we have in mind the well-based transseries of finite log-
arithmic and exponential depth of [190], also called logarithmic-exponential series
in [112]. The construction of the field T in Appendix A allows variants, and we briefly
comment on one of them.](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-33-320.jpg)
![INTRODUCTION AND OVERVIEW 5
Each transseries f is an infinite sum f =
P
m fmm where each m is a transmono-
mial and fm ∈ R. The support of such a transseries f is the set supp(f) of transmono-
mials m for which the coefficient fm is nonzero. For instance, the transmonomials in
the support of the transseries ϕ of example (1) are
eex
, e
ex
log x + ex
(log x)2 + ex
(log x)3 + · · ·
, x11
,
1,
1
x
,
1
x log x
, . . . ,
1
x2
,
1
x3
, . . . , e−x
, e−x2
, . . . .
By imposing various restrictions on the kinds of permissible supports, the construction
from Appendix A yields various interesting differential subfields of T.
To define multiplication on T, supports should be well-based: every nonempty sub-
set of the support of a transseries f should contain an asymptotically dominant element.
So well-basedness is a minimal requirement on supports. A much stronger condition
on supp(f) is as follows: there are transmonomials m and n1, . . . , nk ∈ OT (k ∈ N)
such that
supp f ⊆
m ni1
1 · · · nik
k : i1, . . . , ik ∈ N .
Supports of this kind are called grid-based. Imposing this constraint all along, the
construction from Appendix A builds the differential subfield Tg of grid-based trans-
series of T. Other suitable restrictions on the support yield other interesting differential
subfields of T.
The differential field Tg of grid-based transseries has been studied in detail in [194].
In particular, that book contains a kind of algorithm for solving algebraic differential
equations over Tg. These equations are of the form
(3) P y, . . . , y(r)
= 0,
where P ∈ Tg[Y, . . . , Y (r)
] is a nonzero polynomial in Y and a finite number of its
formal derivatives Y 0
, . . . , Y (r)
. We note here that by combining results from [194] and
the present volume, any solution y ∈ T to (3) is actually grid-based. Thus transseries
outside Tg such as ϕ(x) from (1) or ζ(x) = 1 + 2−x
+ 3−x
+ · · · are differentially
transcendental over Tg; see the Notes and comments to Section 16.2 for more details,
and Grigor0
ev-Singer [155] for an earlier result of this kind.
Model completeness
One reason that “geometric” fields like C, R, Qp are more manageable than “arith-
metic” fields like Q is that the former are model complete; see Appendix B for this and
other basic model-theoretic notions used in this volume. A consequence of the model
completeness of R is that any finite system of polynomial equations over R (in any
number of unknowns) with a solution in an ordered field extension of R, has a solution
in R itself. By the R-version of (2) we can also allow polynomial inequalities in such
a system. (A related fact: if such a system has real algebraic coefficients, then it has a
real algebraic solution.)](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-34-320.jpg)
![6 INTRODUCTION AND OVERVIEW
For a more geometric view of model completeness we first specify an algebraic
subset of Rn
to be the set of common zeros,
y = (y1, . . . , yn) ∈ Rn
: P1(y) = · · · = Pk(y) = 0 ,
of finitely many polynomials P1, . . . , Pk ∈ R[Y1, . . . , Yn]. Define a subset of Rm
to
be subalgebraic if it is the image of an algebraic set in Rn
for some n ⩾ m under the
projection map
(y1, . . . , yn) 7→ (y1, . . . , ym) : Rn
→ Rm
.
Then a consequence of the model completeness of R is that the complement in Rm
of
any subalgebraic set is again subalgebraic. Model completeness of R is a little stronger
in that only polynomials with integer coefficients should be involved.
A nice analogy between R and T is the following intermediate value property, an-
nounced in [193] and established for Tg in [194]: Let P(Y ) = p(Y, . . . , Y (r)
) be a
differential polynomial over T, that is, with coefficients in T, and let f, h be trans-
series with f h; then P(g) takes on all values strictly between P(f) and P(h)
for transseries g with f g h. Underlying this opulence of T is a more ro-
bust property that we call newtonianity, which is analogous to henselianity for valued
fields. The fact that T is newtonian implies, for instance, that any differential equation
y0
= Q(y, y0
, . . . , y(r)
) with Q ∈ x−2
OT[Y, Y 0
, . . . , Y (r)
] has an infinitesimal solu-
tion y ∈ OT. The definition of “newtonian” is rather subtle, and is discussed later in
this introduction.
Another way that R and T are similar concerns the factorization of linear differ-
ential operators: any linear differential operator A = ∂r
+ a1∂r−1
· · · + ar of or-
der r ⩾ 1 with coefficients a1, . . . , ar ∈ T, is a product of such operators of order
one and order two, with coefficients in T. Moreover, any linear differential equation
y(r)
+ a1y(r−1)
+ · · · + ary = b (a1, . . . , ar, b ∈ T) has a solution y ∈ T (possibly
y = 0). In particular, every transseries f has a transseries integral g, that is, f = g0
. (It
is noteworthy that a convergent transseries can very well have a divergent transseries as
an integral; for example, the transmonomial ex
x has as an integral the divergent trans-
series
P∞
n=0 n! ex
xn+1 . The analytic aspects of transseries are addressed by Écalle’s the-
ory of analyzable functions [120], where genuine functions are associated to transseries
such as
P∞
n=0 n! ex
xn+1 , using the process of accelero-summation, a far reaching gen-
eralization of Borel summation; these analytic issues are not addressed in the present
volume.)
These strong closure properties make it plausible to conjecture that T is model
complete, as a valued differential field. This and some other conjectures to be men-
tioned in this introduction go back some 20 years, and are proved in the present vol-
ume. To state model completeness of T geometrically we use the terms d-algebraic
and d-polynomial to abbreviate differential-algebraic and differential polynomial and
we define a d-algebraic set in Tn
to be the set of common zeros,
f = (f1, . . . , fn) ∈ Tn
: P1(f) = · · · = Pk(f) = 0
of some d-polynomials P1, . . . , Pk in differential indeterminates Y1, . . . , Yn,
Pi(Y1, . . . , Yn) = pi Y1, . . . , Yn, Y 0
1, . . . , Y 0
n, Y 00
1 , . . . , Y 00
n , Y 000
1 , . . . , Y 000
n , . . .](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-35-320.jpg)
![INTRODUCTION AND OVERVIEW 7
over T. We also define an H-algebraic set to be the intersection of a d-algebraic set
with a set of the form
y = (y1, . . . , yn) ∈ Tn
: yi ∈ OT for all i ∈ I where I ⊆ {1, . . . , n},
and we finally define a subset of Tm
to be sub-H-algebraic if it is the image of an
H-algebraic set in Tn
for some n ⩾ m under the projection map
(f1, . . . , fn) 7→ (f1, . . . , fm): Tn
→ Tm
.
It follows from the model completeness of T that the complement in Tm
of any sub-
H-algebraic set is again sub-H-algebraic, in analogy with Gabrielov’s “theorem of
the complement” for real subanalytic sets [145]. (The model completeness of T is a
little stronger: it is equivalent to this “complement” formulation where the defining d-
polynomials of the d-algebraic sets involved have integer coefficients.) A consequence
is that for subsets of Tm
,
sub-H-algebraic = definable in T.
The usual model-theoretic approach to establishing that a given structure is model com-
plete consists of two steps. (There is also a preliminary choice to be made of primitives;
our choice for T: its ring operations, its derivation, its ordering, and its valuation.) The
first step is to record the basic compatibilities between primitives; “basic” here means in
practice that they are also satisfied by the substructures of the structure of interest. For
the more familiar structure of the ordered field R of real numbers, these basic compati-
bilities are the ordered field axioms. The second and harder step is to find some closure
properties satisfied by our structure that together with these basic compatibilities can
be shown to imply all its elementary properties. In the model-theoretic treatment of R,
it turns out that this job is done by the closure properties defining real closed fields:
every positive element has a square root, and every odd degree polynomial has a zero.
H-fields
For T we try to capture the first step of the axiomatization by the notion of an H-field.
We chose the prefix H in honor of E. Borel, H. Hahn, G. H. Hardy, and F. Hausdorff,
who pioneered our subject about a century ago [55, 162, 164, 171], and who share the
initial H, except for Borel. To define H-fields, let K be an ordered differential field
(with constant field C) and set
O :=
a ∈ K : |a| ⩽ c for some c 0 in C (a convex subring of K),
O :=
a ∈ K : |a| c for all c 0 in C .
These notations should remind the reader of Landau’s big O and small o. The elements
of O are thought of as infinitesimal, the elements of O as bounded, and those of K O
as infinite. Note that O is definable in the ordered differential field K, and is a valuation
ring of K with (unique) maximal ideal O. We define K to be an H-field if it satisfies
the two conditions below:](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-36-320.jpg)
![8 INTRODUCTION AND OVERVIEW
(H1) for all a ∈ K, if a C, then a0
0,
(H2) O = C + O.
By (H2) the constant field C can be identified canonically with the residue field O/O
of O. As we did with T we construe an H-field K as an ordered valued differential
field. An H-field K is said to have small derivation if ∂O ⊆ O (and thus ∂O ⊆ O). If K
is an H-field and a ∈ K, a 0, then K with its derivation ∂ replaced by a∂ is also an
H-field. Such changes of derivation play a major role in our work.
Among H-fields with small derivation are T and its ordered differential subfields
containing R, and any Hardy field containing R. Thus R(x), R(x, ex
, log x) as well as
Hardy’s larger field of logarithmico-exponential functions are H-fields.
Closure properties
Let Th(M) be the first-order theory of an L-structure M, that is, Th(M) is the set of
L-sentences that are true in M; see Appendix B for details. In terms of H-fields, we
can now make the model completeness conjecture more precise, as was done in [19]:
Th(T) = model companion of the theory of H-fields with small derivation,
where T is construed as an ordered and valued differential field. This amounts to adding
to the earlier model completeness of T the claim that any H-field with small derivation
can be embedded as an ordered valued differential field into some ultrapower of T.
Among the consequences of this conjecture is that any finite system of algebraic differ-
ential equations over T (in several unknowns) has a solution in T whenever it has one
in some H-field extension of T. It means that the concept of “H-field” is intrinsic to
the differential field T. It also suggests studying systematically the extension theory of
H-fields: A. Robinson taught us that for a theory to have a model companion at all—a
rare phenomenon—is equivalent to certain embedding and extension properties of its
class of models. Here it helps to know that H-fields fall under the so-called differential-
valued fields (abbreviated as d-valued fields below) of Rosenlicht, who began a study
of these valued differential fields and their extensions in the early 1980s; see [364]. (A
d-valued field is defined to be a valued differential field such that O = C + O, and
a0
b ∈ b0
O for all a, b ∈ O; here O is the valuation ring with maximal ideal O, and C
is the constant field.) Most of our work is actually in the setting of valued differential
fields where no field ordering is given, since even for H-fields the valuation is a more
robust and useful feature than its field ordering.
Besides developing the extension theory of H-fields we need to isolate the relevant
closure properties of T. First, T is real closed, but that property does not involve the
derivation. Next, T is closed under integration and, by its very construction, also under
exponentiation. In terms of the derivation this gives two natural closure properties of T:
∀a∃b (a = b0
), ∀a∃b (b 6= 0 ab = b0
).
An H-field K is said to be Liouville closed if it is real closed and satisfies these two
sentences; cf. Liouville [260, 261]. So T is Liouville closed. It was shown in [19]](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-37-320.jpg)
![INTRODUCTION AND OVERVIEW 9
that any H-field has a Liouville closure, that is, a minimal Liouville closed H-field
extension. If K is a Hardy field containing R as a subfield, then it has a unique Hardy
field extension that is also a Liouville closure of K, but it can happen that an H-field K
has two Liouville closures that are not isomorphic over K; it cannot have more than
two. Understanding this “fork in the road” and dealing with it is fundamental in our
work. Useful notions in this connection are comparability classes, groundedness, and
asymptotic integration. We discuss this briefly below for H-fields. (Parts of Chapters 9
and 11 treat these notions for a much larger class of valued differential fields.) Later in
this introduction we encounter an important but rather hidden closure property, called
ω-freeness, which rules over the fork in the road. Finally, there is the very powerful
closure property of newtonianity that we already mentioned earlier.
Valuations and asymptotic relations
Let K be an H-field, let a, b range over K, and let v: K → Γ∞ be the (Krull) valuation
on K associated to O, with value group Γ = v(K×
) and Γ∞ := Γ∪{∞} with Γ ∞.
Recall that Γ is an ordered abelian group, additively written as is customary in valuation
theory. Then
va vb ⇐⇒ |a| c|b| for all c 0 in C.
Thinking of elements of K as germs of functions at +∞, we also adopt Hardy’s nota-
tions from asymptotic analysis:
a b, a b, a ≺ b, a 4 b, a b, a ∼ b
are defined to mean, respectively,
va vb, va ⩽ vb, va vb, va ⩾ vb, va = vb, v(a − b) va.
(Some of these notations from [165] actually go back to du Bois-Reymond [48].) Note
that a 1 means that a is infinite, that is, |a| C, and a ≺ 1 means that a is
infinitesimal, that is, a ∈ O. It is crucial that the asymptotic relations above can be
differentiated, provided we restrict to nonzero a, b with a 6 1, b 6 1:
a b ⇐⇒ a0
b0
, a b ⇐⇒ a0
b0
, a ∼ b ⇐⇒ a0
∼ b0
.
For a 6= 0 we let a†
:= a0
/a be its logarithmic derivative, so (ab)†
= a†
+ b†
for a, b 6= 0. Elements a, b 1 are said to be comparable if a†
b†
; if K is a Hardy
field containing R as subfield, or K = T, this is equivalent to the existence of an n ⩾ 1
such that |a| ⩽ |b|n
and |b| ⩽ |a|n
. Comparability is an equivalence relation on the set
of infinite elements of K, and the comparability classes Cl(a) of K are totally ordered
by Cl(a) ⩽ Cl(b) :⇐⇒ a†
4 b†
.
EXAMPLE. For K = T, set e0 = x and en+1 = exp(en). Then the sequence (Cl(en))
is strictly increasing and cofinal in the set of comparability classes. More important are
the `n defined recursively by `0 = x, and `n+1 = log `n. Then the sequence Cl(`0)](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-38-320.jpg)
![10 INTRODUCTION AND OVERVIEW
Cl(`1) Cl(`2) · · · Cl(`n) · · · is coinitial in the set of comparability classes
of T. For later use it is worth noting at this point that
`†
n =
1
`0 · · · `n
, −`††
n =
1
`0
+
1
`0`1
+ · · · +
1
`0`1 · · · `n
.
We call K grounded if K has a smallest comparability class. Thus T is not grounded.
If Γ
contains an element α such that for every γ ∈ Γ
we have nγ ⩾ α for
some n ⩾ 1, then K is grounded; this condition on Γ is in particular satisfied if Γ 6= {0}
and Γ has finite archimedean rank. If K is grounded, then K has only one Liouville
closure (up to isomorphism over K).
The H-field K is said to have asymptotic integration if K satisfies ∀a∃b(a b0
),
equivalently, {vb0
: b ∈ K} = Γ∞. It is obvious that every Liouville closed H-field
has asymptotic integration; in particular, T has asymptotic integration. In general, at
most one γ ∈ Γ lies outside {vb0
: b ∈ K}; if K is grounded, then such a γ exists, by
results in Section 9.2, and so K cannot have asymptotic integration.
STRATEGY AND MAIN RESULTS
Model completeness of T concerns finite systems of algebraic differential equations
over T with asymptotic side conditions in several differential indeterminates.
Robinson’s strategy for establishing model completeness applied to T requires us
to move beyond T to consider H-fields and their extensions. If we are lucky—as we
are in this case—it will suffice to consider extensions of H-fields by one element y at
a time. This leads to equations P(y) = 0 with an asymptotic side condition y ≺ g.
Here P ∈ K{Y } is a univariate differential polynomial with coefficients in an H-
field K with g ∈ K×
, and K{Y } = K[Y, Y 0
, Y 00
, . . . ] is the differential domain of
d-polynomials in the differential indeterminate Y over K. The key issue: when is there
a solution in some H-field extension of K? A detailed study of such equations in the
special case K = Tg and where we only look for solutions in Tg itself was undertaken
in [194], using an assortment of techniques (for instance, various fixpoint theorems)
heavily based on the particular structure of Tg. Generalizing these results to suitable
H-fields is an important guideline in our work.
Differential Newton diagrams
Let K be an H-field, and consider a d-algebraic equation with asymptotic side condi-
tion,
(4) P(y) = 0, y ≺ g,
where P ∈ K{Y }, P 6= 0, and g ∈ K×
; we look for nonzero solutions in H-field
extensions of K. For the sake of concreteness we take K = Tg and look for nonzero
solutions in Tg, focusing on the example below:
(5) e− ex
y2
y00
+ y2
− 2xyy0
− 7 e−x
y0
− 4 + 1
log x = 0, y ≺ x.](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-39-320.jpg)
![INTRODUCTION AND OVERVIEW 11
We sketch briefly how [194] goes about solving (5). First of all, we need to find the
possible dominant terms of solutions y. This is done by considering possible can-
cellations. For example, y2
and −4 might be the terms of least valuation in the left
side of (5), with all other terms having greater valuation, so negligible compared to y2
and −4. This yields a cancellation y2
∼ 4, so y ∼ 2 or y ∼ −2, giving 2 and −2 as
potential dominant terms of a solution y.
Another case: e− ex
y2
y00
and y2
are the terms of least valuation. Then we get a
cancellation e− ex
y2
y00
∼ −y2
, that is, y00
∼ − eex
, which leads to y ∼ − eex
/ e2x
.
But this possibility is discarded, since (5) also requires y ≺ x. (On the other hand, if
the asymptotic condition in (5) had been y ≺ eex
, we would have kept − eex
/ e2x
as a
potential dominant term of a solution y.)
What makes things work in these two cases is that the cancellations arise from terms
of different degrees in y, y0
, y00
, . . . . Such cancellations are reminiscent of the more
familiar setting of algebraic equations where the dominant monomials of solutions can
be read off from a Newton diagram and the corresponding dominant coefficients are
zeros of the corresponding Newton polynomials; see Section 3.7. This method still
works in our d-algebraic setting, for cancellations among terms of different degrees,
but requires the construction of so-called equalizers.
A different situation arises for cancellations between terms of the same degree. Con-
sider for example the case that y2
and −2xyy0
have least valuation among the terms
in the left side of (5), with all other terms of higher valuation. Then y2
∼ 2xyy0
, so
y†
∼ 1
2x . Now y†
= 1
2x gives y = cx1/2
with c ∈ R×
, but the weaker condition
y†
∼ 1
2x only gives y = ux1/2
with u 6= 0, u†
≺ x−1
, that is, |v(u)| |v(x)|/n
for all n ⩾ 1. Substituting ux1/2
for y in (5) and considering u as the new unknown,
the condition on v(u) forces u 1, so after all we do get y ∼ cx1/2
with c ∈ R×
,
giving cx1/2
as a potential dominant term of a solution y. It is important to note that
here an integration constant c gets introduced.
Manipulations as we just did are similar to rewriting an equation H(y) = 0 with ho-
mogeneous nonzero H ∈ K{Y } of positive degree as a (Riccati) equation R(y†
) = 0
with R of lower order than H.
This technique can be shown to work in general for cancellations among terms of
the same degree, provided we are also allowed to transform the equation to an equiv-
alent one by applying a suitable iteration of the upward shift f(x) 7→ f(ex
). (For
reasonable H-fields K one can apply instead compositional conjugation by positive
active elements; see below for compositional conjugation and active.)
Having determined a possible dominant term f = cm of a solution of (4), where
c ∈ R×
and m is a transmonomial, we next perform a so-called refinement
(6) P(f + y) = 0, y ≺ f
of (4). For instance, taking f = 2, the equation (5) transforms into
e− ex
y2
y00
+ y2
− 2xyy0
+ 4 e− ex
yy00
+ 4y − (4x + 7 e−x
)y0
+ 4 e− ex
y00
+ 1
log x = 0, y ≺ 2.
Now apply the same procedure to this refinement, to find the “next” term.](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-40-320.jpg)
![12 INTRODUCTION AND OVERVIEW
Roughly speaking, this yields an infinite process to obtain all possible asymptotic ex-
pansions of solutions to any asymptotic equation. How do we make this into a finite
process? For this, it is useful to introduce the Newton degree of (4). This notion is
similar to the Weierstrass degree of a multivariate power series and corresponds to the
degree of the asymptotically significant part of the equation. If the Newton degree is 0,
then (4) has no solution. The Newton degree of (5) turns out to be 2: this has to do
with the fact that e− ex
y2
y00
≺ y2
whenever y ≺ x. We shall return soon to the precise
definition of Newton degree for differential polynomials over rather general H-fields.
As to the resolution of asymptotic equations over K = Tg, the following key facts
were established in [194]:
• The Newton degree stays the same or decreases under refinement.
• If the Newton degree of the refinement (6) equals that of (4), we employ so-called
unravelings; these resemble the Tschirnhaus transformations that overcome sim-
ilar obstacles in the algebraic setting. Combining unravelings with refinements
as described above, we arrive after finitely many steps at an asymptotic equation
of Newton degree 0 or 1.
• The H-field Tg is newtonian, that is, any asymptotic equation over Tg of Newton
degree 1 has a solution in Tg.
All in all, we have for any given asymptotic equation over Tg a more or less finite
procedure for gaining an overview of the entire space of solutions in Tg.
To define the Newton degree of an asymptotic equation (4) over rather general H-fields,
we first need to introduce the dominant part of P and then, based on a process called
compositional conjugation, the Newton polynomial of P.
The dominant part
Let K be an H-field. We extend the valuation v of K to the integral domain K{Y } by
setting
vP = min{va : a is a coefficient of P},
and we extend the binary relations and ∼ on K to K{Y } accordingly. It is also
convenient to fix a monomial set M in K, that is, a subset M of K
that is mapped
bijectively by v onto the value group Γ of K. This allows us to define the dominant
part DP (Y ) of a nonzero d-polynomial P(Y ) over K to be the unique element of
C{Y } ⊆ K{Y } with P ∼ dP DP , where dP ∈ M is the dominant monomial of P
determined by P dP . (Another choice of monomial set would just multiply DP by
some positive constant.) For K = T we always take the set of transmonomials as our
monomial set.
EXAMPLE 1. Let K = T. For P = x5
+ (2 + ex
)Y + (3 ex
+ log x)(Y 0
)2
, we have
dP = ex
and DP = Y + 3(Y 0
)2
. For Q = Y 2
− 2xY Y 0
we have DQ = −2Y Y 0
.](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-41-320.jpg)
![INTRODUCTION AND OVERVIEW 13
For K with small derivation we can use DP to get near the zeros a 1 of P: if
P(a) = 0, a 1, then DP (c) = 0 where c is the unique constant with a ∼ c. We
need to understand, however, the behavior of P(a) not only for a 1, that is, va = 0,
but also for “sufficiently flat” elements a ∈ K, that is, for va approaching 0 ∈ Γ. For
instance, in T, the iterated logarithms
`0 = x, `1 = log x, `2 = log log x, . . .
satisfy v(`n) → 0 in ΓT and likewise v(1/`n) → 0. The dominant term dP DP of P
often provides a good approximation for P when evaluating at sufficiently flat elements,
but not always: for K = T and Q as in Example 1 we note that for y = `2 we have:
y2
= `2
2 2xyy0
= 2`2/`1, so Q(y) ∼ y2
6 (dQDQ)(y).
In order to approximate P(y) by (dP DP )(y) for sufficiently flat y, we need one
more ingredient: compositional conjugation. For K = T and Q as in Example 1, this
amounts to a change of variables x = eee
x
, so that Q(y) = y2
− 2y(dy/de
x) e−e
x
for
y ∈ T. With respect to this new variable e
x, the dominant term Y 2
of the adjusted d-
polynomial Y 2
−2Y Y 0
e−e
x
is then an adequate approximation of Q when evaluating at
sufficiently flat elements of T. Such changes of variable do not make sense for general
H-fields, but as it turns out, compositional conjugation is a good substitute.
Compositional conjugation
We define this for an arbitrary differential field K. For φ ∈ K×
we let Kφ
be the differ-
ential field obtained from K by replacing its derivation ∂ by the multiple φ−1
∂. Then
a differential polynomial P(Y ) ∈ K{Y } defines the same function on the common
underlying set of K and Kφ
as a certain differential polynomial Pφ
(Y ) ∈ Kφ
{Y }:
for P = Y 0
, we have Pφ
(Y ) = φY 0
(since over Kφ
we evaluate Y 0
according to the
derivation φ−1
∂), for P = Y 00
we have Pφ
(Y ) = φ0
Y 0
+ φ2
Y 00
(with φ0
= ∂φ), and so
on. This yields a ring isomorphism
P 7→ Pφ
: K{Y } → Kφ
{Y }
that is the identity on the common subring K[Y ]. It is also an automorphism of the
common underlying K-algebra of K{Y } and Kφ
{Y }, and studied as such in Chap-
ter 12. We call Kφ
the compositional conjugate of K by φ, and Pφ
the compositional
conjugate of P by φ. Note that K and Kφ
have the same constant field C. If K is
an H-field and φ ∈ K
, then so is Kφ
. It pays to note how things change under
compositional conjugation, and what remains invariant.
The Newton polynomial
Suppose now that K is an H-field with asymptotic integration. For φ ∈ K
we say
that φ is active (in K) if φ a†
for some nonzero a 6 1 in K; equivalently, the
derivation φ−1
∂ of Kφ
is small. Let φ ∈ K
range over the active elements of K
in what follows, fix a monomial set M ⊆ K
of K, and let P ∈ K{Y }, P 6= 0.
The dominant part DP φ of Pφ
lies in C{Y }, and we show in Section 13.1 that it](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-42-320.jpg)
![14 INTRODUCTION AND OVERVIEW
eventually stabilizes as φ varies: there is a differential polynomial NP ∈ C{Y } and an
active φ0 ∈ K
such that for all φ 4 φ0,
DP φ = cφNP , cφ ∈ C
.
We call NP the Newton polynomial of P. It is of course only determined up to a factor
from C
, but this ambiguity is harmless. The (total) degree of NP is called the Newton
degree of P.
EXAMPLE 2. Let K = T. Then f ∈ K
is active iff f `†
n = 1
`0`1···`n
for some n.
If P is as in Example 1, then for each φ,
Pφ
= x5
+ (2 + ex
)Y + φ2
(3 ex
+ log x)(Y 0
)2
,
so DP φ = Y if φ ≺ 1. This yields NP = Y , so P has Newton degree 1. It is an easy
exercise to show that for Q = Y 2
− 2xY Y 0
we have NQ = Y 2
.
A crucial result in [194] (Theorem 8.6) says that if K = Tg, then NP ∈ R[Y ](Y 0
)N
.
A major step in our work was to isolate a robust class of H-fields K with asymptotic
integration for which likewise NP ∈ C[Y ](Y 0
)N
for all nonzero P ∈ K{Y }. This
required several completely new tools to be discussed below.
The special cuts γ, λ and ω
Recall that `n denotes the nth iterated logarithm of x in T, so `0 = x and `n+1 =
log `n. We introduce the elements
γn = `†
n =
1
`0 · · · `n
λn = −γ†
n =
1
`0
+
1
`0`1
+ · · · +
1
`0`1 · · · `n
ωn = −2λ0
n − λ2
n =
1
`2
0
+
1
`2
0`2
1
+ · · · +
1
`2
0`2
1 · · · `2
n
of T. As n → ∞ these elements approach their formal limits
γT =
1
`0`1`2 · · ·
λT =
1
`0
+
1
`0`1
+
1
`0`1`2
+ · · ·
ωT =
1
`2
0
+
1
`2
0`2
1
+
1
`2
0`2
1`2
2
+ · · · ,
which for now are just suggestive expressions. Indeed, our field T of transseries of
finite logarithmic and exponential depth does not contain any pseudolimit of the pseu-
docauchy sequence (λn), nor of the pseudocauchy sequence (ωn). There are, however,
immediate H-field extensions of T where such pseudolimits exist, and if we let λT be](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-43-320.jpg)
![16 INTRODUCTION AND OVERVIEW
There are also K with K 6= C (even Hardy fields) that have a gap. Not having a gap is
equivalent to being grounded or having asymptotic integration.
We already mentioned the result from [19] that K may have two Liouville closures
that are not isomorphic over K (but fortunately not more than two). Indeed, if K has
a gap γ, then in one Liouville closure all primitives of γ are infinitely large, whereas in
the other γ has an infinitesimal primitive. Even if K has no gap, the above fork in the
road can arise more indirectly: Assume that K has asymptotic integration and λ ∈ K
is such that for all a ∈ K×
with a ≺ 1,
a0†
−λ a††
.
Then K has no element γ 6= 0 with λ = −γ†
, but K has an H-field extension Khγi
generated by an element γ with λ = −γ†
, and any such γ is a gap in Khγi. It follows
again that K has two Liouville closures that are not K-isomorphic.
For real closed K with asymptotic integration, the existence of such an element λ
corresponds to the informal statement that γK /
∈ K λK ∈ K. We define K to be
λ-free if K has asymptotic integration and satisfies the sentence
∀a∃b
b 1 a − b††
b†
.
It can be shown that for real closed K with asymptotic integration, λ-freeness is equiv-
alent to the nonexistence of an element λ as above. More generally, K is λ-free iff K
has asymptotic integration and (λρ) has no pseudolimit in K.
The property of ω-freeness
Even λ-freeness might not prevent a fork in the road for some d-algebraic extension.
Let K be an H-field, and define
ω = ωK : K → K, ω(z) := −2z0
− z2
.
Assume that K is λ-free and ω ∈ K is such that for all b 1 in K,
ω − ω(b††
) ≺ (b†
)2
.
Then the first-order differential equation ω(z) = ω admits no solution in K, but K has
an H-field extension Khλi generated by a solution z = λ to ω(z) = ω such that Khλi
is no longer λ-free (and with a fork in its road towards Liouville closure).
For λ-free K the existence of an element ω as above corresponds to the informal
statement that λK /
∈ K ωK ∈ K. We say that K is ω-free if no such ω exists, more
precisely, K has asymptotic integration and satisfies the sentence
∀a∃b
b 1 a − ω(b††
) (b†
)2
.
(It is easy to show that if K is ω-free, then it is λ-free.) For K with asymptotic in-
tegration, ω-freeness is equivalent to the pseudocauchy sequence (ωρ) not having a
pseudolimit in K. Thus T is ω-free. More generally, if K has asymptotic integration
and is a union of grounded H-subfields, then K is ω-free by Corollary 11.7.15.](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-45-320.jpg)
![INTRODUCTION AND OVERVIEW 17
Much deeper and very useful is that if K is an ω-free H-field and L is a d-algebraic
H-field extension of K, then L is also ω-free and has no comparability class smaller
than all those of K; this is part of Theorem 13.6.1. Thus the property of ω-freeness is
very robust: if K is ω-free, then forks in the road towards Liouville closure no longer
occur, even for d-algebraic H-field extensions of K (Corollary 13.6.2). There are,
however, Liouville closed H-fields that are not ω-free; see [22].
Another important consequence of ω-freeness is that Newton polynomials of dif-
ferential polynomials then take the same simple shape as those over Tg:
THEOREM 1. If K is ω-free and P ∈ K{Y }, P 6= 0, then NP ∈ C[Y ](Y 0
)N
.
The proof in Chapter 13 depends heavily on Chapter 12, where we determine the in-
variants of certain automorphism groups of polynomial algebras in infinitely many vari-
ables Y0, Y1, Y2, . . . over a field of characteristic zero.
The function ω and the notion of ω-freeness are closely related to second order lin-
ear differential equations over K. More precisely (Riccati), for y ∈ K×
, 4y00
+ fy = 0
is equivalent to ω(z) = f with z := 2y†
; so the second-order linear differential equa-
tion 4y00
+fy = 0 reduces in a way to a first-order (but non-linear) differential equation
ω(z) = f. (The factor 4 is just for convenience, to get simpler expressions below.)
EXAMPLE. The differential equation y00
= −y has no solution y ∈ T×
, whereas the
Airy equation y00
= xy has two R-linearly independent solutions in T [308, Chapter 11,
(1.07)]. Indeed, in Sections 11.7 and 11.8 we show that for f ∈ T, the differential
equation 4y00
+fy = 0 has a solution y ∈ T×
if and only if f ωT, that is, f ωn =
1
`2
0
+ 1
`2
0`2
1
+ · · · + 1
`2
0`2
1···`2
n
for some n. This fact reflects classical results [167, 184] on
the question: for which logarithmico-exponential functions f (in Hardy’s sense) does
the equation 4y00
+ fy = 0 have a non-oscillating real-valued solution (more precisely,
a nonzero solution in a Hardy field)?
Newtonianity
This is the most consequential elementary property of T. An ω-free H-field K is said
to be newtonian if every d-polynomial P(Y ) over K of Newton degree 1 has a zero
in O. This turns out to be the correct analogue for valued differential fields like T of the
property of being henselian for a valued field. We chose the adjective newtonian since
it is this property that allows us to develop in Chapter 13 a Newton diagram method for
differential polynomials. It is good to keep in mind that the role of newtonianity in the
results of Chapters 14, 15, and 16 is more or less analogous to that of henselianity in
the theory of valued fields and as the key condition in the Ax-Kochen-Eršov results.
We already mentioned the result from [194] that Tg is newtonian. That T is newto-
nian is a consequence of the following analogue in Chapter 15 of the familiar valuation-
theoretic fact that spherically complete valued fields are henselian:
THEOREM 2. If K is an H-field, ∂K = K, and K is a directed union of spherically
complete grounded H-subfields, then K is (ω-free and) newtonian.
EXAMPLE. Let K = T and consider for α ∈ R the differential polynomial
P(Y ) = Y 00
− 2Y 3
− xY − α ∈ T{Y }.](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-46-320.jpg)
![18 INTRODUCTION AND OVERVIEW
For φ ∈ T×
we have (Y 00
)φ
= φ2
Y 00
+ φ0
Y 0
for φ ∈ T×
, so
Pφ
= φ2
Y 00
+ φ0
Y 0
− 2Y 3
− xY − α.
Now φ2
, φ0
≺ 1 ≺ x for active φ ≺ 1 in T
. Hence NP ∈ R×
Y , so P has Newton
degree 1. Thus the Painlevé II equation y00
= 2y3
+ xy + α has a solution y ∈ OT. (It
is known that P has a zero y 4 1 in the differential subfield R(x) of T iff α ∈ Z; see
for example [156, Theorem 20.2].)
The main results of Chapter 14 amount for H-fields to the following:
THEOREM 3. If K is a newtonian ω-free H-field with divisible value group, then K
has no proper immediate d-algebraic H-field extension.
COROLLARY 1. Let K be a real closed newtonian ω-free H-field, and let Ka
= K[i]
(where i2
= −1) be its algebraic closure. Then:
(i) each d-polynomial in Ka
{Y } of positive degree has a zero in Ka
;
(ii) each linear differential operator in Ka
[∂] of positive order is a composition of
such operators of order 1;
(iii) each d-polynomial in K{Y } of odd degree has a zero in K; and
(iv) each linear differential operator in K[∂] of positive order is a composition of
such operators of order 1 and order 2.
THEOREM 4. If K is an ω-free H-field with divisible value group, then K has an
immediate d-algebraic newtonian H-field extension, and any such extension embeds
over K into every ω-free newtonian H-field extension of K.
An extension of K as in Theorem 4 is minimal over K and thus unique up to isomor-
phism over K. We call such an extension a newtonization of K.
THEOREM 5. If K is an ω-free H-field, then K has a d-algebraic newtonian Liouville
closed H-field extension that embeds over K into every ω-free newtonian Liouville
closed H-field extension of K.
An extension of K as in Theorem 5 is minimal over K and thus unique up to isomor-
phism over K. We call such an extension a Newton-Liouville closure of K.
The main theorems
We now come to the results in Chapter 16, which in our view justify this volume. First,
the various elementary conditions we have discussed axiomatize a model complete
theory. To be precise, construe H-fields in the natural way as L-structures where L :=
{0, 1, +, −, · , ∂, ⩽, 4}, and let Tnl
be the L-theory whose models are the newtonian
ω-free Liouville closed H-fields.
THEOREM 6. Tnl
is model complete.](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-47-320.jpg)
![INTRODUCTION AND OVERVIEW 19
The theory Tnl
is not complete and has exactly two completions, namely Tnl
small (small
derivation) and Tnl
large (large derivation). Thus newtonian ω-free Liouville closed H-
fields with small derivation have the same elementary properties as T.
Every H-field with small derivation can be embedded into a model of Tnl
small; thus
Theorem 6 yields the strong version of the model completeness conjecture from [19]
stated earlier in this introduction. As Tnl
small is complete and effectively axiomatized,
it is decidable. In particular, there is an algorithm which, for any given d-polynomials
P1, . . . , Pm in indeterminates Y1, . . . , Yn with coefficients from Z[x], decides whether
there is a tuple y ∈ Tn
such that P1(y) = · · · = Pm(y) = 0. Such an algorithm with T
replaced by its differential subring R[[x−1
]] is due to Denef and Lipshitz [101], but no
such algorithm can exist with T replaced by R((x−1
)) or by any of various other natural
H-subfields of T [20, 155].
Theorem 6 is the main step towards an elimination of quantifiers, in a slightly extended
language: Let Lι
Λ,Ω be L augmented by the unary function symbol ι and the unary
predicates Λ, Ω, and extend Tnl
to the Lι
Λ,Ω-theory Tnl,ι
Λ,Ω by adding as defining axioms
for these new symbols the universal closures of
a 6= 0 −→ a · ι(a) = 1
a = 0 −→ ι(a) = 0
,
Λ(a) ←→ ∃y
y 1 a = −y††
,
Ω(a) ←→ ∃y
y 6= 0 4y00
+ ay = 0
.
For a model K of Tnl
this makes the sets Λ(K) and Ω(K) downward closed with
respect to the ordering of K. For example, for f ∈ T,
f ∈ Λ(T) ⇐⇒ f λn =
1
`0
+
1
`0`1
+ · · · +
1
`0`1 · · · `n
for some n,
f ∈ Ω(T) ⇐⇒ f ωn =
1
`2
0
+
1
`2
0`2
1
+ · · · +
1
`2
0`2
1 · · · `2
n
for some n,
that is, Λ(T) and Ω(T) are the cuts in T determined by λT, ωT introduced earlier. We
can now state what we view as the main result of this volume:
THEOREM 7. The theory Tnl,ι
Λ,Ω admits elimination of quantifiers.
We cannot omit here either Λ or Ω. In Chapter 16 we do include for convenience one
more unary predicate I in Lι
Λ,Ω: for a model K of Tnl
and a ∈ K,
I(a) ←→ ∃y
a 4 y0
y 4 1
←→ a = 0 ∨
a 6= 0 ¬Λ(−a†
)
,
where the first equivalence is the defining axiom for I, and the second shows that I is
superfluous in Theorem 7. We note here that this predicate I governs the solvability of
first-order linear differential equations with asymptotic side condition. More precisely,
for K as above and f ∈ K, g, h ∈ K×
, the following are equivalent:
(a) there exists y ∈ K with y0
= fy + g and y ≺ h;
(b)
(f −h†
) ∈ I(K) and (g/h) ∈ I(K)
or
(f −h†
) /
∈ I(K) and (g/h) ≺ f −h†
.](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-48-320.jpg)
![20 INTRODUCTION AND OVERVIEW
This equivalence is part of Corollary 11.8.12 and exemplifies Theorem 7 (but is not
derived from that theorem, nor used in its proof).
In the proof of Theorem 7, and throughout the construction of suitable H-field
extensions, the predicates I, Λ and Ω act as switchmen. Whenever a fork in the road
occurs due to the presence of a gap γ, then I(γ) tells us to take the branch where
R
γ 4 1,
while ¬I(γ) forces
R
γ 1. Likewise, the predicates Λ and Ω control what happens
when adjoining elements γ and λ with γ†
= −λ and ω(λ) = ω.
From the above defining axioms for Λ and Ω it is clear that these predicates are
(uniformly) existentially definable in models of Tnl
. By model completeness of Tnl
they are also uniformly universally definable in these models; Section 16.5 deals with
such algebraic-linguistic issues.
Next we list some more intrinsic consequences of our elimination theory.
COROLLARY 2. Let K be a newtonian ω-free Liouville closed H-field, and suppose
the set X ⊆ Kn
is definable. Then X has empty interior in Kn
(with respect to the
order topology on K and the product topology on Kn
) if and only if for some nonzero
P ∈ K{Y1, . . . , Yn} we have X ⊆
y ∈ Kn
: P(y) = 0 .
In (i) below the intervals are in the sense of the ordered field K.
COROLLARY 3. Let K be a newtonian ω-free Liouville closed H-field. Then:
(i) K is o-minimal at infinity: if X ⊆ K is definable in K, then for some a ∈ K,
either (a, +∞) ⊆ X, or (a, +∞) ∩ X = ∅;
(ii) if X ⊆ Kn
is definable in K, then X ∩ Cn
is semialgebraic in the sense of the
real closed constant field C of K;
(iii) K has NIP. (See Appendix B for this very robust property.)
It is hard to imagine obtaining these results for K = T without Theorem 7. Item (i)
relates to classical bounds on solutions of algebraic differential equations over Hardy
fields; see [20, Section 3]. To illustrate item (ii) of Corollary 3, we note that the set of
real parameters (λ0, . . . , λn) ∈ Rn+1
for which the system
λ0y + λ1y0
+ · · · + λny(n)
= 0, 0 6= y ≺ 1
has a solution in T is a semialgebraic subset of Rn+1
; in fact, it agrees with the set of all
(λ0, . . . , λn) ∈ Rn+1
such that the polynomial λ0 + λ1Y + · · · + λnY n
∈ R[Y ] has a
negative zero in R; see Corollary 11.8.26. To illustrate item (iii), let Y = (Y1, . . . , Yn)
be a tuple of distinct differential indeterminates; for an m-tuple σ = (σ1, . . . , σm) of
elements of {≺, , } we say that P1, . . . , Pm ∈ T{Y } realize σ if there exists a ∈
Tn
such that Pi(a) σi 1 holds for i = 1, . . . , m. Then a special case of (iii) says
that for fixed d, n, r ∈ N, the number of tuples σ ∈ {≺, , }m
realized by some
P1, . . . , Pm ∈ T{Y } of degree at most d and order at most r grows only polynomially
with m, even though the total number of tuples is 3m
. These manifestations of (ii)
and (iii), though instructive, are perhaps a bit misleading, since they can be obtained
without appealing to (ii) and (iii).
In the course of proving Theorem 6 we also get:](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-49-320.jpg)
![22 INTRODUCTION AND OVERVIEW
Asymptotic couples
A useful gadget is the asymptotic couple of an H-field K. This is the value group Γ
of K equipped with the map γ 7→ γ†
: Γ6=
→ Γ defined by: if γ = vf, f ∈ K×
,
then γ†
= v(f†
). This map is a valuation on Γ, and we extend it to a map Γ → Γ∞ by
setting 0†
:= ∞. Two key facts are that α†
β+β†
for all α, β 0 in Γ, and α†
⩾ β†
whenever 0 α ⩽ β in Γ. The condition on an H-field of having small derivation can
be expressed in terms of its asymptotic couple; the same holds for having a gap, for
being grounded, and for having asymptotic integration, but not for being ω-free.
Asymptotic couples were introduced by Rosenlicht [364] for d-valued fields. In
Chapter 6 we assign to any valued differential field with small derivation an asymptotic
couple, with good effect. Asymptotic couples play also an important role in Chapters 9,
10, 11, 13, and 16.
Differential-henselian fields
Valued differential fields with small derivation include the so-called monotone dif-
ferential fields defined by the condition a0
4 a. In analogy with the notion of a
henselian valued field, Scanlon [382] introduced differential-henselian monotone dif-
ferential fields. Using the Equalizer Theorem we extend this notion and basic facts
about it to arbitrary valued differential fields with small derivation in Chapter 7. (We
abbreviate differential-henselian to d-henselian.) This material plays a role in Chap-
ter 14, using the following relation between d-henselian and newtonian: an ω-free
H-field K is newtonian iff for every active φ ∈ K
the compositional conjugate Kφ
is d-henselian, with the valuation v on Kφ
replaced by the coarser valuation π ◦ v
where π: v(K×
) = Γ → Γ/∆ is the canonical map to the quotient of Γ by its convex
subgroup
∆ := {γ ∈ Γ : γ†
vφ}.
We pay particular attention to two special cases: v(C×
) = {0} (few constants), and
v(C×
) = Γ (many constants). The first case is relevant for newtonianity, the second
case is considered in a short Chapter 8, where we present Scanlon’s extension of the
Ax-Kochen-Eršov theorems to d-henselian valued fields with many constants, and add
some things on definability.
While d-henselianity is defined in terms of solving differential equations in one
unknown, it implies the solvability of suitably non-singular systems of n differential
equations in n unknowns: this is shown at the end of Chapter 7, and has a nice conse-
quence for newtonianity: Proposition 14.5.7.
Asymptotic differential fields
To keep things simple we confined most of the exposition above to H-fields, but this
setting is a bit too narrow for various technical reasons. For example, a differential
subfield of an H-field with the induced ordering is not always an H-field, and passing
to an algebraic closure like T[i] destroys the ordering, though T[i] is still a d-valued
field. On occasion we also wish to change the valuation of an H-field or d-valued field](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-51-320.jpg)
![INTRODUCTION AND OVERVIEW 23
by coarsening. For all these reasons we introduce in Chapter 9 the class of asymptotic
differential fields, which is larger and more flexible than Rosenlicht’s class of d-valued
fields. Many basic facts about H-fields and d-valued fields do have good analogues
for asymptotic differential fields. This is shown in Chapter 9, which also contains a
lot of basic material on asymptotic couples. Chapter 10 deals more specifically with
H-fields.
Immediate extensions
Indispensable for attaining our main results is the fact that every H-field with divisible
value group and with asymptotic integration has a spherically complete immediate H-
field extension. This is part of Theorem 11.4.1, and proving it about five years ago
removed a bottleneck. It provides the only way known to us of extending every H-field
to an ω-free H-field. Possibly more important than Theorem 11.4.1 itself are the tools
involved in its proof. In view of the theorem’s content, it is ironic that models of Tnl
are never spherically complete, in contrast to all prior positive results on elementary
theories of valued fields with or without extra structure, cf. [28, 29, 41, 131, 382].
The differential Newton diagram method
Chapters 13 and 14 present the differential Newton diagram method in the general con-
text of asymptotic fields that satisfy suitable technical conditions, such as ω-freeness.
Before tackling these chapters, the reader may profit from first studying our exposition
of the Newton diagram method for ordinary one-variable polynomials over henselian
valued fields of equicharacteristic zero in Section 3.7. Some of the issues encountered
there (for example, the unraveling technique) appear again, albeit in more intricate
form, in the differential context of these chapters. In the proofs of a few crucial facts
about the special cuts λ and ω in Chapter 13 we use some results from the preceding
Chapter 12 on triangular automorphisms. Chapter 12 is a bit special in being essentially
independent of the earlier chapters.
Proving newtonianity
Chapter 15 contains the proof of Theorem 2, and thus establishes that T is a model of
our theory Tnl
small. This theorem is also useful in other contexts: In [43], Berarducci
and Mantova construct a derivation on Conway’s field No of surreal numbers [92, 150]
turning it into a Liouville closed H-field with constant field R. From Theorem 2 and
the completeness of Tnl
it follows that No with this derivation and T are elementarily
equivalent, as we show in [24].
Quantifier elimination
In Chapter 16 we first prove Theorem 6 on model completeness, next we consider
H-fields equipped with a ΛΩ-structure, and then deduce Theorem 7 about quantifier
elimination with various interesting consequences, such as Corollaries 2 and 3. The](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-52-320.jpg)
![24 INTRODUCTION AND OVERVIEW
introduction to this chapter gives an overview of the proof and the role of various em-
bedding and extension results in it.
THE NEXT VOLUME
The present volume focuses on achieving quantifier elimination (Theorem 7), and so
we left out various things we did since 1995 that were not needed for that. In a second
volume we intend to cover these things as required for developing our work further. Let
us briefly survey some highlights of what is to come.
Linear differential equations
We plan to consider linear differential equations in much greater detail, comprising
the corresponding differential Galois theory, in connection with constructing the linear
surjective closure of a differential field, factoring linear differential operators over suit-
able algebraically closed d-valued fields, and explicitly constructing the Picard-Vessiot
extension of such an operator. Concerning the latter, the complexification T[i] of T
is no longer closed under exponential integration, since oscillatory “transmonomials”
such as eix
are not in T [i]. Adjoining these oscillatory transmonomials to T[i] yields a
d-valued field that contains a Picard-Vessiot extension of T for each operator in T[∂].
Hardy fields
We also wish to pay more attention to Hardy fields, and this will bring up analytic
issues. For example, every Hardy field containing R can be shown to extend to an
ω-free Hardy field. Using methods from [195], we also hope to prove that it always
extends to a newtonian ω-free Hardy field. Indeed, that paper proves among other
things the following pertinent result (formulated here with our present terminology):
Let Tda
g consist of the grid-based transseries that are d-algebraic over R. Then Tda
g is
a newtonian ω-free Liouville closed H-subfield of Tg and is isomorphic over R to a
Hardy field containing R.
Embedding into fields of transseries
Another natural question we expect to deal with is whether every H-field can be given
some kind of transserial structure. This can be made more precise in terms of the
axiomatic definition of a field of transseries in terms of a transmonomial group M in
Schmeling’s thesis [388]. For instance, one axiom there is that for all m ∈ M we
have supp log m ⊆ M
. We hope that any H-field can be embedded into such a
field of transseries. This would be a natural counterpart of Kaplansky’s theorem [209]
embedding certain valued fields into Hahn fields, and would make it possible to think
of H-field elements as generalized transseries.](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-53-320.jpg)
![INTRODUCTION AND OVERVIEW 25
More on the model theory of T
In the second volume we hope to deal with further issues around T of a model-theoretic
nature: for example, identifying the induced structure on its value group (conjectured
to be given by its H-couple, as specified in [18]); and determining the definable closure
of a subset of a model of Tnl
, in order to get a handle on what functions are definable
in T.
A by-product of the present volume is a full description of several important 1-
types over a given model of Tnl
, but the entire space of such 1-types remains to be
surveyed. Theorem 8 suggests that the model-theoretic notions of non-orthogonality
to C or C-internality may be significant for models of Tnl
; see also [25].
FUTURE CHALLENGES
We now discuss a few more open-ended avenues of inquiry.
Differentiation and exponentiation
The restriction to OT of the exponential function on T is easily seen to be definable
in T, but by part (ii) of Corollary 3, the restriction to R of this exponential function is
not definable in T. This raises the question whether our results can be extended to the
differential field T with exponentiation, or with some other extra o-minimal structure
on it.
Logarithmic transseries
A transseries is logarithmic if all transmonomials in it are of the form `r0
0 · · · `rn
n with
r0, . . . , rn ∈ R. (See Appendix A.) The logarithmic transseries make up an ω-free
newtonian H-subfield Tlog of T that is not Liouville closed. We conjecture that Tlog
as a valued differential field is model complete. The asymptotic couple of Tlog has
been successfully analyzed by Gehret [146], and turns out to be model-theoretically
tame, in particular, has NIP [147]. (There is also the notion of a transseries being expo-
nential. The exponential transseries form a real closed H-subfield Texp of T in which
the set Z is existentially definable, see [20]. It follows that the differential field Texp
does not have a reasonable model theory: it is as complicated as so-called second-order
arithmetic.)
Accelero-summable transseries
The paper [195] on transserial Hardy fields yields on the one hand a method to as-
sociate a genuine function to a suitable formal transseries, and in the other direction
also provides means to associate concrete asymptotic expansions to elements of Hardy
fields. We expect that more can be done in this direction.
Écalle’s theory of analyzable functions has a more canonical procedure that asso-
ciates a function to an accelero-summable transseries. These transseries make up an
H-subfield Tas of T. This procedure has the advantage that it does not only preserve](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-54-320.jpg)
![This ebook is for the use of anyone anywhere in the United States and most
other parts of the world at no cost and with almost no restrictions
whatsoever. You may copy it, give it away or re-use it under the terms of the
Project Gutenberg License included with this ebook or online at
www.gutenberg.org. If you are not located in the United States, you will have
to check the laws of the country where you are located before using this
eBook.
Title: Happy House
Author: Freifrau von Betsey Riddle Hutten zum Stolzenberg
Release date: May 22, 2013 [eBook #42771]
Most recently updated: October 23, 2024
Language: English
Credits: E-text prepared by Annie McGuire from page images generously
made available by the Google Books Library Project
(http://books.google.com)
*** START OF THE PROJECT GUTENBERG EBOOK HAPPY HOUSE ***](https://image.slidesharecdn.com/21243820-250520090512-5fc286ca/85/Asymptotic-Differential-Algebra-And-Model-Theory-Of-Transseries-Matthias-Aschenbrenner-60-320.jpg)
Asymptotic Differential Algebra And Model Theory Of Transseries Matthias Aschenbrenner
- 1. Asymptotic Differential Algebra And Model Theory Of Transseries Matthias Aschenbrenner download https://ebookbell.com/product/asymptotic-differential-algebra- and-model-theory-of-transseries-matthias-aschenbrenner-42487640 Explore and download more ebooks at ebookbell.com
- 2. Here are some recommended products that we believe you will be interested in. You can click the link to download. Asymptotic Differential Algebra And Model Theory Of Transseries Ams195 Matthias Aschenbrenner Lou Van Den Dries Joris Van Der Hoeven https://ebookbell.com/product/asymptotic-differential-algebra-and- model-theory-of-transseries-ams195-matthias-aschenbrenner-lou-van-den- dries-joris-van-der-hoeven-51946076 Asymptotic Analysis Of Differential Equations Revised White Rb https://ebookbell.com/product/asymptotic-analysis-of-differential- equations-revised-white-rb-4990868 Asymptotic Integration Of Differential And Difference Equations 2015th Edition Sigrun Bodine https://ebookbell.com/product/asymptotic-integration-of-differential- and-difference-equations-2015th-edition-sigrun-bodine-5102506 Nonlinear Partial Differential Equations Asymptotic Behavior Of Solutions And Selfsimilar Solutions 1st Edition Miho Giga https://ebookbell.com/product/nonlinear-partial-differential- equations-asymptotic-behavior-of-solutions-and-selfsimilar- solutions-1st-edition-miho-giga-4259962
- 3. Differential Equations Asymptotic Theory In Mathematical Physics Wuhan University Hubei China 20 29 October 2003 Series In Analysis Chen Hua https://ebookbell.com/product/differential-equations-asymptotic- theory-in-mathematical-physics-wuhan-university-hubei- china-20-29-october-2003-series-in-analysis-chen-hua-1296250 Qualitative And Asymptotic Analysis Of Differential Equations With Random Perturbations World Scientific Series On Nonlinear Science Anatoliy M Samoilenko https://ebookbell.com/product/qualitative-and-asymptotic-analysis-of- differential-equations-with-random-perturbations-world-scientific- series-on-nonlinear-science-anatoliy-m-samoilenko-2377872 Progress In Partial Differential Equations Asymptotic Profiles Regularity And Wellposedness Michael Reissig M Ruzhansky Eds https://ebookbell.com/product/progress-in-partial-differential- equations-asymptotic-profiles-regularity-and-wellposedness-michael- reissig-m-ruzhansky-eds-4158386 Asymptotic Perturbation Methods For Nonlinear Differential Equations In Physics Attilio Maccari https://ebookbell.com/product/asymptotic-perturbation-methods-for- nonlinear-differential-equations-in-physics-attilio-maccari-47832556 Asymptotic Perturbation Methods For Nonlinear Differential Equations In Physics Attilio Maccari https://ebookbell.com/product/asymptotic-perturbation-methods-for- nonlinear-differential-equations-in-physics-attilio-maccari-48946094
- 6. Annals of Mathematics Studies Number 195
- 8. Asymptotic Differential Algebra and Model Theory of Transseries Matthias Aschenbrenner Lou van den Dries Joris van der Hoeven PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2017
- 9. Copyright © 2017 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TR press.princeton.edu All Rights Reserved Library of Congress Cataloging-in-Publication Data Names: Aschenbrenner, Matthias, 1972– | van den Dries, Lou | Hoeven, J. van der (Joris) Title: Asymptotic differential algebra and model theory of transseries / Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven. Description: Princeton : Princeton University Press, 2017. | Series: Annals of mathe- matics studies ; number 195 | Includes bibliographical references and index. Identifiers: LCCN 2017005899 | ISBN 9780691175423 (hardcover : alk. paper) | ISBN 9780691175430 (pbk. : alk. paper) Subjects: LCSH: Series, Arithmetic. | Divergent series. | Asymptotic expansions. | Differential algebra. Classification: LCC QA295 .A87 2017 | DDC 512/.56–dc23 LC record available at https://lccn.loc.gov/2017005899 British Library Cataloging-in-Publication Data is available The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed. This book has been composed in L A TEX. Printed on acid-free paper. ∞ 10 9 8 7 6 5 4 3 2 1
- 10. Had the apparatus [of transseries and analyzable functions] been introduced for the sole purpose of solving Dulac’s “conjecture,” one might legitimately question the wisdom and cost-effectiveness of such massive investment in new machinery. However, [these no- tions] have many more applications, actual or potential, especially in the study of analytic singularities. But their chief attraction is per- haps that of giving concrete, if partial, shape to G. H. Hardy’s dream of an all-inclusive, maximally stable algebra of “totally formalizable functions.” — Jean Écalle, Six Lectures on Transseries, Analysable Functions and the Constructive Proof of Dulac’s Conjecture. The virtue of model theory is its ability to organize succinctly the sort of tiresome algebraic details associated with elimination theory. — Gerald Sacks, The Differential Closure of a Differential Field. Les analystes p-adiques se fichent tout autant que les géomètres algébristes . . . , des gammes à plus soif sur les valuations com- posées, les groupes ordonnés baroques, sous-groupes pleins des- dits et que sais-je. Ces gammes méritent tout au plus d’enrichir les exercices de Bourbaki, tant que personne ne s’en sert. — Alexander Grothendieck, letter to Serre dated October 31, 1961. I don’t like either writing or reading two-hundred page papers. It’s not my idea of fun. — John H. Conway, quoted in Genius at Play: The Curious Mind of John Horton Conway by Siobhan Roberts.
- 12. Contents Preface xiii Conventions and Notations xv Leitfaden xvii Dramatis Personæ xix Introduction and Overview 1 A Differential Field with No Escape 1 Strategy and Main Results 10 Organization 21 The Next Volume 24 Future Challenges 25 A Historical Note on Transseries 26 1 Some Commutative Algebra 29 1.1 The Zariski Topology and Noetherianity 29 1.2 Rings and Modules of Finite Length 36 1.3 Integral Extensions and Integrally Closed Domains 39 1.4 Local Rings 43 1.5 Krull’s Principal Ideal Theorem 50 1.6 Regular Local Rings 52 1.7 Modules and Derivations 55 1.8 Differentials 59 1.9 Derivations on Field Extensions 67 2 Valued Abelian Groups 70 2.1 Ordered Sets 70 2.2 Valued Abelian Groups 73 2.3 Valued Vector Spaces 89 2.4 Ordered Abelian Groups 98
- 13. viii CONTENTS 3 Valued Fields 110 3.1 Valuations on Fields 110 3.2 Pseudoconvergence in Valued Fields 126 3.3 Henselian Valued Fields 136 3.4 Decomposing Valuations 157 3.5 Valued Ordered Fields 171 3.6 Some Model Theory of Valued Fields 179 3.7 The Newton Tree of a Polynomial over a Valued Field 186 4 Differential Polynomials 199 4.1 Differential Fields and Differential Polynomials 199 4.2 Decompositions of Differential Polynomials 209 4.3 Operations on Differential Polynomials 214 4.4 Valued Differential Fields and Continuity 221 4.5 The Gaussian Valuation 227 4.6 Differential Rings 231 4.7 Differentially Closed Fields 237 5 Linear Differential Polynomials 241 5.1 Linear Differential Operators 241 5.2 Second-Order Linear Differential Operators 258 5.3 Diagonalization of Matrices 264 5.4 Systems of Linear Differential Equations 270 5.5 Differential Modules 276 5.6 Linear Differential Operators in the Presence of a Valuation 285 5.7 Compositional Conjugation 290 5.8 The Riccati Transform 298 5.9 Johnson’s Theorem 303 6 Valued Differential Fields 310 6.1 Asymptotic Behavior of vP 311 6.2 Algebraic Extensions 314 6.3 Residue Extensions 316 6.4 The Valuation Induced on the Value Group 320 6.5 Asymptotic Couples 322 6.6 Dominant Part 325 6.7 The Equalizer Theorem 329 6.8 Evaluation at Pseudocauchy Sequences 334 6.9 Constructing Canonical Immediate Extensions 335 7 Differential-Henselian Fields 340 7.1 Preliminaries on Differential-Henselianity 341 7.2 Maximality and Differential-Henselianity 345 7.3 Differential-Hensel Configurations 351 7.4 Maximal Immediate Extensions in the Monotone Case 353
- 14. CONTENTS ix 7.5 The Case of Few Constants 356 7.6 Differential-Henselianity in Several Variables 359 8 Differential-Henselian Fields with Many Constants 365 8.1 Angular Components 367 8.2 Equivalence over Substructures 369 8.3 Relative Quantifier Elimination 374 8.4 A Model Companion 377 9 Asymptotic Fields and Asymptotic Couples 378 9.1 Asymptotic Fields and Their Asymptotic Couples 379 9.2 H-Asymptotic Couples 387 9.3 Application to Differential Polynomials 398 9.4 Basic Facts about Asymptotic Fields 402 9.5 Algebraic Extensions of Asymptotic Fields 409 9.6 Immediate Extensions of Asymptotic Fields 413 9.7 Differential Polynomials of Order One 416 9.8 Extending H-Asymptotic Couples 421 9.9 Closed H-Asymptotic Couples 425 10 H-Fields 433 10.1 Pre-Differential-Valued Fields 433 10.2 Adjoining Integrals 439 10.3 The Differential-Valued Hull 443 10.4 Adjoining Exponential Integrals 445 10.5 H-Fields and Pre-H-Fields 451 10.6 Liouville Closed H-Fields 460 10.7 Miscellaneous Facts about Asymptotic Fields 468 11 Eventual Quantities, Immediate Extensions, and Special Cuts 474 11.1 Eventual Behavior 474 11.2 Newton Degree and Newton Multiplicity 482 11.3 Using Newton Multiplicity and Newton Weight 487 11.4 Constructing Immediate Extensions 492 11.5 Special Cuts in H-Asymptotic Fields 499 11.6 The Property of λ-Freeness 505 11.7 Behavior of the Function ω 511 11.8 Some Special Definable Sets 519 12 Triangular Automorphisms 532 12.1 Filtered Modules and Algebras 532 12.2 Triangular Linear Maps 541 12.3 The Lie Algebra of an Algebraic Unitriangular Group 545 12.4 Derivations on the Ring of Column-Finite Matrices 548 12.5 Iteration Matrices 552
- 15. x CONTENTS 12.6 Riordan Matrices 563 12.7 Derivations on Polynomial Rings 568 12.8 Application to Differential Polynomials 579 13 The Newton Polynomial 585 13.1 Revisiting the Dominant Part 586 13.2 Elementary Properties of the Newton Polynomial 593 13.3 The Shape of the Newton Polynomial 598 13.4 Realizing Cuts in the Value Group 606 13.5 Eventual Equalizers 610 13.6 Further Consequences of ω-Freeness 615 13.7 Further Consequences of λ-Freeness 622 13.8 Asymptotic Equations 628 13.9 Some Special H-Fields 635 14 Newtonian Differential Fields 640 14.1 Relation to Differential-Henselianity 641 14.2 Cases of Low Complexity 645 14.3 Solving Quasilinear Equations 651 14.4 Unravelers 657 14.5 Newtonization 665 15 Newtonianity of Directed Unions 671 15.1 Finitely Many Exceptional Values 671 15.2 Integration and the Extension K(x) 672 15.3 Approximating Zeros of Differential Polynomials 673 15.4 Proof of Newtonianity 676 16 Quantifier Elimination 678 16.1 Extensions Controlled by Asymptotic Couples 680 16.2 Model Completeness 685 16.3 ΛΩ-Cuts and ΛΩ-Fields 688 16.4 Embedding Pre-ΛΩ-Fields into ω-Free ΛΩ-Fields 697 16.5 The Language of ΛΩ-Fields 701 16.6 Elimination of Quantifiers with Applications 704 A Transseries 712 B Basic Model Theory 724 B.1 Structures and Their Definable Sets 724 B.2 Languages 729 B.3 Variables and Terms 734 B.4 Formulas 738 B.5 Elementary Equivalence and Elementary Substructures 744 B.6 Models and the Compactness Theorem 749
- 16. CONTENTS xi B.7 Ultraproducts and Proof of the Compactness Theorem 755 B.8 Some Uses of Compactness 759 B.9 Types and Saturated Structures 763 B.10 Model Completeness 767 B.11 Quantifier Elimination 771 B.12 Application to Algebraically Closed and Real Closed Fields 776 B.13 Structures without the Independence Property 782 Bibliography 787 List of Symbols 817 Index 833
- 18. Preface We develop here the algebra and model theory of the differential field of transseries, a fascinating mathematical structure obtained by iterating a construction going back more than a century to Levi-Civita and Hahn. It was introduced about thirty years ago as an exponential ordered field by Dahn and Göring in connection with Tarski’s problem on the real field with exponentiation, and independently by Écalle in his proof of the Dulac Conjecture on plane analytic vector fields. The analytic aspects of transseries have a precursor in Borel’s summation of di- vergent series. Indeed, Écalle’s theory of accelero-summation vastly extends Borel summation, and associates to each accelero-summable transseries an analyzable func- tion. In this way many non-oscillating real-valued functions that arise naturally (for example, as solutions of algebraic differential equations) can be represented faithfully by transseries. For about twenty years we have studied the differential field of transseries within the broader program of developing asymptotic differential algebra. We have recently obtained decisive positive results on its model theory, and we describe these results in an Introduction and Overview. That introduction assumes some rudimentary knowl- edge of differential fields, valued fields, and model theory, but no acquaintance with transseries. It is intended to familiarize readers with the main issues in this book and with the terminology that we frequently use. Initially, Joris van der Hoeven in Paris and Matthias Aschenbrenner and Lou van den Dries in Urbana on the other side of the Atlantic worked independently, but around 2000 we decided to join forces. In 2011 we arrived at a rough outline for proving some precise conjectures: see our programmatic survey Toward a model theory for transseries. All the conjectures stated in that paper (with one minor change) did turn out to be true, even though some seemed to us at the time rather optimistic. Why is this book so long? For one, several problems we faced had no short solu- tions. Also, we have chosen to work in a setting that is sufficiently flexible for further developments, as we plan to show in a later volume. Finally, we have tried to be reason- ably self-contained by assuming only a working knowledge of basic algebra: groups, rings, modules, fields. Occasionally we refer to Lang’s Algebra. After the Introduction and Overview this book consists of 16 chapters and 2 ap- pendices. Each chapter has an introduction and is divided into sections. Each section has subsections, the last one often consisting of (partly historical) notes and comments. Many chapters state in the beginning some assumptions—sometimes just notational in nature— that are in force throughout that chapter, and of course the reader should be aware of those in studying a particular chapter, since we do not repeat these assump- tions when stating theorems, etc. The same holds for many sections and subsections. The end of the volume has a list of symbols and an index.
- 19. xiv PREFACE ACKNOWLEDGMENTS Part of this work was carried out while some of the authors were in residence at vari- ous times at the Fields Institute (Toronto), the Institut des Hautes Études Scientifiques (Bures-sur-Yvette), the Isaac Newton Institute for Mathematical Sciences (Cambridge), and the Mathematical Sciences Research Institute (Berkeley). The support and hospi- tality of these institutions is gratefully acknowledged. Aschenbrenner’s work was partially supported by the National Science Foundation under grants DMS-0303618, DMS-0556197, and DMS-0969642. Visits by van der Hoeven to Los Angeles were partially supported by the UCLA Logic Center. We thank the following copyright holders for permission to reproduce the text in the epigraphs in the front of this book: Springer Science and Business Media, New York, for the quote by Jean Écalle from [121]; the American Mathematical Society for the quote by Gerald Sacks from [376], © 1972 American Mathematical Society; Professor Jean-Pierre Serre for the quote by Alexander Grothendieck from [88]; and Siobhan Roberts for the quote by John H. Conway that appears in her book Genius at Play: The Curious Mind of John Horton Conway [344] © Siobhan Roberts, published by Bloomsbury Publishing, Inc., 2016. We thank David Marker and Angus Macintyre for their interest and steadfast moral support over the years. To Santiago Camacho, Andrei Gabrielov, Tigran Hakobyan, Elliot Kaplan, Nigel Pynn-Coates, Chieu Minh Tran, and especially to Allen Gehret, we are indebted for numerous comments on and corrections to the manuscript. We are also grateful to Philip Ehrlich for setting us right on some historical points, and to the anonymous reviewers for useful suggestions and for spotting some errors. We are of course solely responsible for any remaining inadequacies. Finally, we thank our editor, Vickie Kearn, and the other staff at Princeton Univer- sity Press, notably Nathan Carr and Glenda Krupa, for helping us to bring this book into its final form. Matthias Aschenbrenner, Los Angeles Lou van den Dries, Urbana Joris van der Hoeven, Paris September 2015
- 20. Conventions and Notations Throughout, m and n range over the set N = {0, 1, 2, . . . } of natural numbers. For sets X, Y we distinguish between X ⊆ Y , meaning that X is a subset of Y , and X ⊂ Y , meaning that X is a proper subset of Y . For an (additively written) abelian group A we set A6= := A {0}. By ring we mean an associative but possibly non-commutative ring with identity 1. Let R be a ring. A unit of R is a u ∈ R with a right-inverse (an x ∈ R with ux = 1) and a left- inverse (an x ∈ R with xu = 1). If u is a unit of R, then u has only one right-inverse and only one left-inverse, and these coincide. With respect to multiplication the units of R form a group R× with identity 1. Thus the multiplicative group of a field K is K× = K {0} = K6= . Subrings and ring morphisms preserve 1. A domain is a ring with 1 6= 0 such that for all x, y in the ring, if xy = 0, then x = 0 or y = 0. Usually domains are commutative, but not always. However, an integral domain is always commutative, that is, a subring of a field. Let R be a ring. An R-module is a left R-module unless specified otherwise, and the scalar 1 ∈ R acts as the identity on any R-module. Let M be an R-module and (xi)i∈I a family in M. A family (ri)i∈I in R is admissible if ri = 0 for all but finitely many i ∈ I. An R-linear combination of (xi) is an x ∈ M such that x = P i rixi of M for some admissible family (ri) in R. We say that (xi) generates M if every element of M is an R-linear combination of (xi). We say that (xi) is R-dependent (or linearly dependent over R) if P i rixi = 0 for some admissible family (ri)i∈I in R with ri 6= 0 for some i ∈ I; for I = {1, . . . , n} we also abuse language by expressing this as: x1, . . . , xn are R-dependent. We say that (xi) is R-independent (or linearly independent over R) if (xi) is not R-dependent. We call M free on (xi) (or (xi) a basis of M) if (xi) generates M and (xi) is R-independent. Sometimes we use this terminology for sets X ⊆ M to mean that for some (equivalently, for every) index set I and bijection i 7→ xi : I → X the family (xi) has the corresponding property. Let K be a commutative ring. A K-algebra is defined to be a ring A together with a ring morphism φ: K → A that takes its values in the center of A; we then refer to φ as the structural morphism of the K-algebra A, and construe A as a K-module by λa := φ(λ)a for λ ∈ K and a ∈ A. Given a field extension F of a field K and a family (xi) in F we use the expres- sions (xi) is algebraically (in)dependent over K and (xi) is a transcendence basis of F over K in a way similar to the above linear analogues; likewise, a set X ⊆ F can be referred to as being a transcendence basis of F over K. When a vector space V over a field C is given, then subspace of V means vector subspace of V .
- 22. Leitfaden 2. Valued Abelian Groups 1. Some Commutative Algebra 3. Valued Fields 4. Differential Polynomials 5. Linear Differential Polynomials 6. Valued Differential Fields 7. Differential-Henselian Fields 8. Differential-Henselian Fields with Many Constants 9. Asymptotic Fields and Asymptotic Couples 10. H-Fields 11. Eventual Quantities, Immediate Extensions, and Special Cuts 12. Triangular Automorphisms 13. The Newton Polynomial 14. Newtonian Differential Fields 16. Quantifier Elimination 15. Newtonianity of Directed Unions
- 24. Dramatis Personæ Dramatis Personæ Dramatis Personæ We summarize here the definitions of some notions prominent in our work, together with a list of attributes that apply to them. We include the page number where each concept is first introduced. We let m, n, r range over N = {0, 1, 2, . . . }. Below K is a field, possibly equipped with further structure. We let a, b, f, g, y, z range over ele- ments of K, and we let Y be an indeterminate over K. If K comes equipped with a val- uation, then we let O be the valuation ring of K, and we freely employ the dominance relations on K introduced in Section 3.1. If K comes equipped with a derivation ∂, then we also write f0 , f00 , . . . , f(n) , . . . for ∂f, ∂2 f, . . . , ∂n f, . . . , and f† = f0 /f for the logarithmic derivative of any f 6= 0; in this case C = {f : f0 = 0} denotes the constant field of K, and c ranges over C. We let Γ be an ordered abelian group, and let α, β, γ range over Γ. VALUED FIELDS Let K be a valued field, that is, a field equipped with a valuation on it; p. 112. Complete: every cauchy sequence in K has a limit in K; p. 84. Spherically complete: every pseudocauchy sequence in K has a pseudolimit in K; p. 78. “Spherically complete” is equivalent to “maximal” as defined below. Maximal: there is no proper immediate valued field extension of K; p. 129. Algebraically maximal: there is no proper immediate algebraic valued field extension of K; p. 130. Henselian: for every P ∈ K[Y ] with P 4 1, P(0) ≺ 1, and P0 (0) 1, there exists y ≺ 1 with P(y) = 0; p. 136. DIFFERENTIAL FIELDS Let K be a differential field, that is, a field of characteristic zero equipped with a deri- vation on it; p. 200. Linearly surjective: for all a0, . . . , ar ∈ K with ar 6= 0 there exists y such that a0y + a1y0 + · · · + ary(r) = 1; p. 253.
- 25. xx DRAMATIS PERSONÆ Linearly closed: for all r ⩾ 1 and a0, . . . , ar ∈ K there are b0, . . . , br−1, b ∈ K with a0Y +a1Y 0 +· · ·+arY (r) = b0(Y 0 +bY )+b1(Y 0 +bY )0 +· · ·+br−1(Y 0 +bY )(r−1) ; p. 252. Picard-Vessiot closed (or pv-closed): for all r ⩾ 1 and a0, . . . , ar ∈ K with ar 6= 0 there are C-linearly independent y1, . . . , yr such that a0yi + a1y0 i + · · · + ary (r) i = 0 for i = 1, . . . , r; p. 254. Differentially closed: for all P ∈ K[Y, . . . , Y (r) ]6= and Q ∈ K[Y, . . . , Y (r−1) ]6= such that ∂P ∂Y (r) 6= 0 there is y with P(y, y0 , . . . , y(r) ) = 0 and Q(y, y0 , . . . , y(r−1) ) 6= 0; p. 237. VALUED DIFFERENTIAL FIELDS Let K be a valued differential field, that is, a differential field equipped with a valuation on it; p. 221. Small derivation: f ≺ 1 ⇒ f0 ≺ 1; p. 222. Monotone: f ≺ 1 ⇒ f0 4 f; p. 226. Few constants: c 4 1 for all c; p. 226. Many constants: for every f there exists c with f c; p. 226. Differential-henselian (or d-henselian): K has small derivation and: (DH1) for all a0, . . . , ar 4 1 in K with ar 1 there exists y 1 such that a0y + a1y0 + · · · + ary(r) ∼ 1; (DH2) for every P ∈ K[Y, Y 0 , . . . , Y (r) ] with P 4 1, P(0) ≺ 1, and ∂P ∂Y (n) (0) 1 for some n, there exists y ≺ 1 such that P(y, y0 , . . . , y(r) ) = 0; p. 340. ASYMPTOTIC FIELDS Let K be an asymptotic field, that is, a valued differential field such that for all nonzero f, g ≺ 1: f ≺ g ⇐⇒ f0 ≺ g0 ; p. 379. H-asymptotic (or of H-type): 0 6= f ≺ g ≺ 1 ⇒ f† g† ; p. 379. Differential-valued (or d-valued): for all f 1 there exists c with f ∼ c; p. 379. Grounded: there exists nonzero f 6 1 such that g† f† for all nonzero g 6 1; p. 384. Asymptotic integration: for all f 6= 0 there exists g 6 1 with g0 f; p. 383. Asymptotically maximal: K has no proper immediate asymptotic field extension; p. 380. Asymptotically d-algebraically maximal: K has no proper immediate differential-alge- braic asymptotic field extension; p. 380.
- 26. DRAMATIS PERSONÆ xxi λ-free: H-asymptotic, ungrounded, and for all f there exists g 1 with f − g†† g† ; p. 506. ω-free: H-asymptotic, ungrounded, and for all f there is g 1 with f − ω(g†† ) g† , where ω(z) := −(2z0 + z2 ); p. 515. Newtonian: H-asymptotic, ungrounded, and every P ∈ K[Y, Y 0 , . . . , Y (r) ]6= of New- ton degree 1 has a zero in O; p. 640. (See p. 480 for Newton degree.) ORDERED VALUED DIFFERENTIAL FIELDS Let K be an ordered valued differential field, that is, a valued differential field equipped with an ordering in the usual sense of ordered field; p. 378. Pre-H-field: O is convex in the ordered field K, and for all f: f O =⇒ f0 0; p. 452. H-field: O is the convex hull of C in the ordered field K, and for all f: f C =⇒ f0 0, f 1 =⇒ there exists c with f ∼ c; p. 451. Liouville closed: K is a real closed H-field and for all f, g there exists y 6= 0 such that y0 + fy = g; p. 460. ASYMPTOTIC COUPLES Let (Γ, ψ) be an asymptotic couple, that is, the ordered abelian group Γ is equipped with a map ψ: Γ6= → Γ such that for all α, β 6= 0: (AC1) α + β 6= 0 ⇒ ψ(α + β) ⩾ min ψ(α), ψ(β) ; (AC2) ψ(kα) = ψ(α) for all k ∈ Z6= ; (AC3) α 0 ⇒ α0 := α + ψ(α) ψ(β); p. 322. For γ 6= 0 we set γ0 := γ + ψ(γ). H-asymptotic (or of H-type): 0 α β ⇒ ψ(α) ⩾ ψ(β); p. 323. Grounded: Ψ := {ψ(α) : α 6= 0} has a largest element; p. 388. Small derivation: γ 0 ⇒ γ0 0; p. 388. Asymptotic integration: for all α there exists β 6= 0 with α = β0 ; p. 383.
- 28. Asymptotic Differential Algebra and Model Theory of Transseries
- 30. Introduction and Overview A DIFFERENTIAL FIELD WITH NO ESCAPE Our principal object of interest is the differential field T of transseries. Transseries are formal series in an indeterminate x R, such as ϕ(x) = −3 eex + e ex log x + ex log2 x + ex log3 x + · · · −x11 + 7 (1) + π x + 1 x log x + 1 x log2 x + 1 x log3 x + · · · + 2 x2 + 6 x3 + 24 x4 + 120 x5 + 720 x6 + · · · + e−x +2 e−x2 +3 e−x3 +4 e−x4 + · · · , where log2 x := (log x)2 , etc. As in this example, each transseries is a (possibly transfinite) sum, with terms written from left to right, in asymptotically decreasing order. Each term is the product of a real coefficient and a transmonomial. Appendix A contains the inductive construction of T, including the definition of “transmonomial” and other notions about transseries that occur in this introduction. For expositions of T with proofs, see [112, 122, 194]. In [112], T is denoted by R((x−1 ))LE , and its elements are called logarithmic-exponential series. At this point we just mention that transseries can be added and multiplied in the natural way, and that with these operations, T is a field containing R as a subfield. Transseries can also be differentiated term by term, subject to r0 = 0 for each r ∈ R and x0 = 1. In this way T acquires the structure of a differential field. Why transseries? Transseries naturally arise in solving differential equations at infinity and studying the asymptotic behavior of their solutions, where ordinary power series, Laurent series, or even Puiseux series in x−1 are inadequate. Indeed, functions as simple as ex or log x cannot be expanded with respect to the asymptotic scale xR of real powers of x at +∞. For merely solving algebraic equations, no exponentials or logarithms are needed: it is classical that the fields of Puiseux series over R and C are real closed and algebraically closed, respectively. One approach to asymptotics with respect to more general scales was initiated by Hardy [163, 165], inspired by earlier work of du Bois-Reymond [51] in the late 19th
- 31. 2 INTRODUCTION AND OVERVIEW century. Hardy considered logarithmico-exponential functions: real-valued functions built up from constants and the variable x using addition, multiplication, division, ex- ponentiation and taking logarithms. He showed that such a function, when defined on some interval (a, +∞), has eventually constant sign (no oscillation!), and so the germs at +∞ of these functions form an ordered field H with derivation d dx . Thus H is what Bourbaki [62] calls a Hardy field: a subfield K of the ring of germs at +∞ of dif- ferentiable functions f : (a, +∞) → R with a ∈ R, closed under taking derivatives; for more precision, see Section 9.1. Each Hardy field is naturally an ordered differ- ential field. The Hardy field H is rather special: every f ∈ H satisfies an algebraic differential equation over R. But H lacks some closure properties that are desirable for a comprehensive theory. For instance, H has no antiderivative of ex2 (by Liouville; see [361]), and the functional inverse of (log x)(log log x) doesn’t lie in H, and is not even asymptotic to any element of H: [111, 190]; see also [333]. With T and transseries we go beyond H and logarithmico-exponential functions by admitting infinite sums. It is important to be aware, however, that by virtue of its inductive construction, T does not contain, for example, the series x + log x + log log x + log log log x + · · · , which does make sense in a suitable extension of T. Thus T allows only certain kinds of infinite sums. Nevertheless, it turns out that the differential field T enjoys many remarkable closure properties that H lacks. For instance, T is closed under natural operations of exponentiation, integration, composition, compositional inversion, and the resolution of feasible algebraic differential equations (where the meaning of feasible can be made explicit). This makes T of interest for different areas of mathematics: Analysis In connection with the Dulac Problem, T is sufficiently rich for modeling the asymp- totic behavior of so-called Poincaré return maps. This analytically deep result is a crucial part of Écalle’s solution of the Dulac Problem [119, 120, 121]. (At the end of this introduction we discuss this in more detail.) Computer algebra Many transseries are concrete enough to compute with them, in the sense of computer algebra [190, 402]. Moreover, many of the closure properties mentioned above can be made effective. This allows for the automation of an important part of asymptotic calculus for functions of one variable. Logic Given an o-minimal expansion of the real field, the germs at +∞ of its definable one- variable functions form a Hardy field, which in many cases can be embedded into T. This gives useful information about the possible asymptotic behavior of these definable functions; see [21, 292] for more about this connection.
- 32. INTRODUCTION AND OVERVIEW 3 Soon after the introduction of T in the 1980s it was suspected that T might well be a kind of universal domain for the differential algebra of Hardy fields and similar ordered differential fields, analogous to the role of the algebraically closed field C as a universal domain for algebraic geometry of characteristic 0 (Weil [461, Chapter X, §2]), and of R, Qp, and C((t)) in related ordered and valued settings. This is corroborated by the strong closure properties enjoyed by T. See in particular p. 148 of Écalle’s book [120] for eloquent expressions of this idea. The present volume and the next substantiate the universal domain nature of the differential field T, using the language of model theory. The model-theoretic properties of the classical fields C, R, Qp and C((t)) are well established thanks to Tarski, Seidenberg, Robinson, Ax Kochen, Eršov, Cohen, Macintyre, Denef, and others; see [443, 395, 350, 28, 29, 131, 84, 275, 100]. Our goal is to analyze likewise the differential field T, which comes with a definable ordering and valuation, and in this book we achieve this goal. The ordered and valued differential field T For what follows, it will be convenient to quickly survey some of the most distinctive features of T. Appendix A contains precise definitions and further details. Each transseries f = f(x) can be uniquely decomposed as a sum f = f + f + f≺, where f is the infinite part of f, f is its constant term (a real number), and f≺ is its infinitesimal part. In the example (1) above, ϕ = −3 eex + e ex log x + ex log2 x + ex log3 x + · · · −x11 , ϕ = 7, ϕ≺ = π x + 1 x log x + · · · . In this example, ϕ happens to be a finite sum, but this is not a necessary feature of transseries: take for example f := ex log x + ex log2 x + ex log3 x +· · · , with f = f. Declaring a transseries to be positive iff its dominant (= leftmost) coefficient is positive turns T into an ordered field extension of R with x R. In our example (1), the dominant transmonomial of ϕ(x) is eex and its dominant coefficient is −3, whence ϕ(x) is neg- ative; in fact, ϕ(x) R. The inductive definition of T involves constructing a certain exponential operation exp: T → T× , with exp(f) also written as ef , and exp(f) = exp(f) · exp(f) · exp(f≺) = exp(f) · ef · ∞ X n=0 fn ≺ n! where the first factor exp(f) is a transmonomial, the second factor ef is the real number obtained by exponentiating the real number f in the usual way, and the third
- 33. 4 INTRODUCTION AND OVERVIEW factor exp(f≺) = P∞ n=0 fn ≺ n! is expanded as a series in the usual way. Conversely, each transmonomial is of the form exp(f) for some transseries f. Viewed as an exponential field, T is an elementary extension of the exponential field of real numbers; see [111]. In particular, T is real closed, and so its ordering is existentially definable (and universally definable) from its ring operations: (2) f ⩾ 0 ⇐⇒ f = g2 for some g. However, as emphasized above, our main interest is in T as a differential field, with derivation f 7→ f0 on T defined termwise, with r0 = 0 for r ∈ R, x0 = 1, (ef )0 = f0 ef , and (log f)0 = f0 /f for f 0. Let us fix here some notation and terminology in force throughout this volume: a differential field is a field K of characteristic 0 together with a single derivation ∂: K → K; if ∂ is clear from the context we often write a0 instead of ∂(a), for a ∈ K. The constant field of a differential field K is the subfield CK := {a ∈ K : a0 = 0} of K, also denoted by C if K is clear from the context. The constant field of T turns out to be R, that is, R = {f ∈ T : f0 = 0}. By an ordered differential field we mean a differential field equipped with a total order- ing on its underlying set making it an ordered field in the usual sense of that expression. So T is an ordered differential field. More important than the ordering is the valuation on T with valuation ring OT := f ∈ T : |f| ⩽ r for some r ∈ R = {f ∈ T : f = 0}, a convex subring of T. The unique maximal ideal of OT is OT := f ∈ T : |f| ⩽ r for all r 0 in R = {f ∈ T : f = f≺} and thus OT = R+OT. Its very definition shows that OT is existentially definable in the differential field T. However, OT is not universally definable in the differential field T: Corollary 16.2.6. In light of the model completeness conjecture discussed below, it is therefore advisable to add the valuation as an extra primitive, and so in the rest of this introduction we construe T as an ordered and valued differential field, with valuation given by OT. By a valued differential field we mean throughout a differential field K equipped with a valuation ring of K that contains the prime subfield Q of K. Grid-based transseries When referring to transseries we have in mind the well-based transseries of finite log- arithmic and exponential depth of [190], also called logarithmic-exponential series in [112]. The construction of the field T in Appendix A allows variants, and we briefly comment on one of them.
- 34. INTRODUCTION AND OVERVIEW 5 Each transseries f is an infinite sum f = P m fmm where each m is a transmono- mial and fm ∈ R. The support of such a transseries f is the set supp(f) of transmono- mials m for which the coefficient fm is nonzero. For instance, the transmonomials in the support of the transseries ϕ of example (1) are eex , e ex log x + ex (log x)2 + ex (log x)3 + · · · , x11 , 1, 1 x , 1 x log x , . . . , 1 x2 , 1 x3 , . . . , e−x , e−x2 , . . . . By imposing various restrictions on the kinds of permissible supports, the construction from Appendix A yields various interesting differential subfields of T. To define multiplication on T, supports should be well-based: every nonempty sub- set of the support of a transseries f should contain an asymptotically dominant element. So well-basedness is a minimal requirement on supports. A much stronger condition on supp(f) is as follows: there are transmonomials m and n1, . . . , nk ∈ OT (k ∈ N) such that supp f ⊆ m ni1 1 · · · nik k : i1, . . . , ik ∈ N . Supports of this kind are called grid-based. Imposing this constraint all along, the construction from Appendix A builds the differential subfield Tg of grid-based trans- series of T. Other suitable restrictions on the support yield other interesting differential subfields of T. The differential field Tg of grid-based transseries has been studied in detail in [194]. In particular, that book contains a kind of algorithm for solving algebraic differential equations over Tg. These equations are of the form (3) P y, . . . , y(r) = 0, where P ∈ Tg[Y, . . . , Y (r) ] is a nonzero polynomial in Y and a finite number of its formal derivatives Y 0 , . . . , Y (r) . We note here that by combining results from [194] and the present volume, any solution y ∈ T to (3) is actually grid-based. Thus transseries outside Tg such as ϕ(x) from (1) or ζ(x) = 1 + 2−x + 3−x + · · · are differentially transcendental over Tg; see the Notes and comments to Section 16.2 for more details, and Grigor0 ev-Singer [155] for an earlier result of this kind. Model completeness One reason that “geometric” fields like C, R, Qp are more manageable than “arith- metic” fields like Q is that the former are model complete; see Appendix B for this and other basic model-theoretic notions used in this volume. A consequence of the model completeness of R is that any finite system of polynomial equations over R (in any number of unknowns) with a solution in an ordered field extension of R, has a solution in R itself. By the R-version of (2) we can also allow polynomial inequalities in such a system. (A related fact: if such a system has real algebraic coefficients, then it has a real algebraic solution.)
- 35. 6 INTRODUCTION AND OVERVIEW For a more geometric view of model completeness we first specify an algebraic subset of Rn to be the set of common zeros, y = (y1, . . . , yn) ∈ Rn : P1(y) = · · · = Pk(y) = 0 , of finitely many polynomials P1, . . . , Pk ∈ R[Y1, . . . , Yn]. Define a subset of Rm to be subalgebraic if it is the image of an algebraic set in Rn for some n ⩾ m under the projection map (y1, . . . , yn) 7→ (y1, . . . , ym) : Rn → Rm . Then a consequence of the model completeness of R is that the complement in Rm of any subalgebraic set is again subalgebraic. Model completeness of R is a little stronger in that only polynomials with integer coefficients should be involved. A nice analogy between R and T is the following intermediate value property, an- nounced in [193] and established for Tg in [194]: Let P(Y ) = p(Y, . . . , Y (r) ) be a differential polynomial over T, that is, with coefficients in T, and let f, h be trans- series with f h; then P(g) takes on all values strictly between P(f) and P(h) for transseries g with f g h. Underlying this opulence of T is a more ro- bust property that we call newtonianity, which is analogous to henselianity for valued fields. The fact that T is newtonian implies, for instance, that any differential equation y0 = Q(y, y0 , . . . , y(r) ) with Q ∈ x−2 OT[Y, Y 0 , . . . , Y (r) ] has an infinitesimal solu- tion y ∈ OT. The definition of “newtonian” is rather subtle, and is discussed later in this introduction. Another way that R and T are similar concerns the factorization of linear differ- ential operators: any linear differential operator A = ∂r + a1∂r−1 · · · + ar of or- der r ⩾ 1 with coefficients a1, . . . , ar ∈ T, is a product of such operators of order one and order two, with coefficients in T. Moreover, any linear differential equation y(r) + a1y(r−1) + · · · + ary = b (a1, . . . , ar, b ∈ T) has a solution y ∈ T (possibly y = 0). In particular, every transseries f has a transseries integral g, that is, f = g0 . (It is noteworthy that a convergent transseries can very well have a divergent transseries as an integral; for example, the transmonomial ex x has as an integral the divergent trans- series P∞ n=0 n! ex xn+1 . The analytic aspects of transseries are addressed by Écalle’s the- ory of analyzable functions [120], where genuine functions are associated to transseries such as P∞ n=0 n! ex xn+1 , using the process of accelero-summation, a far reaching gen- eralization of Borel summation; these analytic issues are not addressed in the present volume.) These strong closure properties make it plausible to conjecture that T is model complete, as a valued differential field. This and some other conjectures to be men- tioned in this introduction go back some 20 years, and are proved in the present vol- ume. To state model completeness of T geometrically we use the terms d-algebraic and d-polynomial to abbreviate differential-algebraic and differential polynomial and we define a d-algebraic set in Tn to be the set of common zeros, f = (f1, . . . , fn) ∈ Tn : P1(f) = · · · = Pk(f) = 0 of some d-polynomials P1, . . . , Pk in differential indeterminates Y1, . . . , Yn, Pi(Y1, . . . , Yn) = pi Y1, . . . , Yn, Y 0 1, . . . , Y 0 n, Y 00 1 , . . . , Y 00 n , Y 000 1 , . . . , Y 000 n , . . .
- 36. INTRODUCTION AND OVERVIEW 7 over T. We also define an H-algebraic set to be the intersection of a d-algebraic set with a set of the form y = (y1, . . . , yn) ∈ Tn : yi ∈ OT for all i ∈ I where I ⊆ {1, . . . , n}, and we finally define a subset of Tm to be sub-H-algebraic if it is the image of an H-algebraic set in Tn for some n ⩾ m under the projection map (f1, . . . , fn) 7→ (f1, . . . , fm): Tn → Tm . It follows from the model completeness of T that the complement in Tm of any sub- H-algebraic set is again sub-H-algebraic, in analogy with Gabrielov’s “theorem of the complement” for real subanalytic sets [145]. (The model completeness of T is a little stronger: it is equivalent to this “complement” formulation where the defining d- polynomials of the d-algebraic sets involved have integer coefficients.) A consequence is that for subsets of Tm , sub-H-algebraic = definable in T. The usual model-theoretic approach to establishing that a given structure is model com- plete consists of two steps. (There is also a preliminary choice to be made of primitives; our choice for T: its ring operations, its derivation, its ordering, and its valuation.) The first step is to record the basic compatibilities between primitives; “basic” here means in practice that they are also satisfied by the substructures of the structure of interest. For the more familiar structure of the ordered field R of real numbers, these basic compati- bilities are the ordered field axioms. The second and harder step is to find some closure properties satisfied by our structure that together with these basic compatibilities can be shown to imply all its elementary properties. In the model-theoretic treatment of R, it turns out that this job is done by the closure properties defining real closed fields: every positive element has a square root, and every odd degree polynomial has a zero. H-fields For T we try to capture the first step of the axiomatization by the notion of an H-field. We chose the prefix H in honor of E. Borel, H. Hahn, G. H. Hardy, and F. Hausdorff, who pioneered our subject about a century ago [55, 162, 164, 171], and who share the initial H, except for Borel. To define H-fields, let K be an ordered differential field (with constant field C) and set O := a ∈ K : |a| ⩽ c for some c 0 in C (a convex subring of K), O := a ∈ K : |a| c for all c 0 in C . These notations should remind the reader of Landau’s big O and small o. The elements of O are thought of as infinitesimal, the elements of O as bounded, and those of K O as infinite. Note that O is definable in the ordered differential field K, and is a valuation ring of K with (unique) maximal ideal O. We define K to be an H-field if it satisfies the two conditions below:
- 37. 8 INTRODUCTION AND OVERVIEW (H1) for all a ∈ K, if a C, then a0 0, (H2) O = C + O. By (H2) the constant field C can be identified canonically with the residue field O/O of O. As we did with T we construe an H-field K as an ordered valued differential field. An H-field K is said to have small derivation if ∂O ⊆ O (and thus ∂O ⊆ O). If K is an H-field and a ∈ K, a 0, then K with its derivation ∂ replaced by a∂ is also an H-field. Such changes of derivation play a major role in our work. Among H-fields with small derivation are T and its ordered differential subfields containing R, and any Hardy field containing R. Thus R(x), R(x, ex , log x) as well as Hardy’s larger field of logarithmico-exponential functions are H-fields. Closure properties Let Th(M) be the first-order theory of an L-structure M, that is, Th(M) is the set of L-sentences that are true in M; see Appendix B for details. In terms of H-fields, we can now make the model completeness conjecture more precise, as was done in [19]: Th(T) = model companion of the theory of H-fields with small derivation, where T is construed as an ordered and valued differential field. This amounts to adding to the earlier model completeness of T the claim that any H-field with small derivation can be embedded as an ordered valued differential field into some ultrapower of T. Among the consequences of this conjecture is that any finite system of algebraic differ- ential equations over T (in several unknowns) has a solution in T whenever it has one in some H-field extension of T. It means that the concept of “H-field” is intrinsic to the differential field T. It also suggests studying systematically the extension theory of H-fields: A. Robinson taught us that for a theory to have a model companion at all—a rare phenomenon—is equivalent to certain embedding and extension properties of its class of models. Here it helps to know that H-fields fall under the so-called differential- valued fields (abbreviated as d-valued fields below) of Rosenlicht, who began a study of these valued differential fields and their extensions in the early 1980s; see [364]. (A d-valued field is defined to be a valued differential field such that O = C + O, and a0 b ∈ b0 O for all a, b ∈ O; here O is the valuation ring with maximal ideal O, and C is the constant field.) Most of our work is actually in the setting of valued differential fields where no field ordering is given, since even for H-fields the valuation is a more robust and useful feature than its field ordering. Besides developing the extension theory of H-fields we need to isolate the relevant closure properties of T. First, T is real closed, but that property does not involve the derivation. Next, T is closed under integration and, by its very construction, also under exponentiation. In terms of the derivation this gives two natural closure properties of T: ∀a∃b (a = b0 ), ∀a∃b (b 6= 0 ab = b0 ). An H-field K is said to be Liouville closed if it is real closed and satisfies these two sentences; cf. Liouville [260, 261]. So T is Liouville closed. It was shown in [19]
- 38. INTRODUCTION AND OVERVIEW 9 that any H-field has a Liouville closure, that is, a minimal Liouville closed H-field extension. If K is a Hardy field containing R as a subfield, then it has a unique Hardy field extension that is also a Liouville closure of K, but it can happen that an H-field K has two Liouville closures that are not isomorphic over K; it cannot have more than two. Understanding this “fork in the road” and dealing with it is fundamental in our work. Useful notions in this connection are comparability classes, groundedness, and asymptotic integration. We discuss this briefly below for H-fields. (Parts of Chapters 9 and 11 treat these notions for a much larger class of valued differential fields.) Later in this introduction we encounter an important but rather hidden closure property, called ω-freeness, which rules over the fork in the road. Finally, there is the very powerful closure property of newtonianity that we already mentioned earlier. Valuations and asymptotic relations Let K be an H-field, let a, b range over K, and let v: K → Γ∞ be the (Krull) valuation on K associated to O, with value group Γ = v(K× ) and Γ∞ := Γ∪{∞} with Γ ∞. Recall that Γ is an ordered abelian group, additively written as is customary in valuation theory. Then va vb ⇐⇒ |a| c|b| for all c 0 in C. Thinking of elements of K as germs of functions at +∞, we also adopt Hardy’s nota- tions from asymptotic analysis: a b, a b, a ≺ b, a 4 b, a b, a ∼ b are defined to mean, respectively, va vb, va ⩽ vb, va vb, va ⩾ vb, va = vb, v(a − b) va. (Some of these notations from [165] actually go back to du Bois-Reymond [48].) Note that a 1 means that a is infinite, that is, |a| C, and a ≺ 1 means that a is infinitesimal, that is, a ∈ O. It is crucial that the asymptotic relations above can be differentiated, provided we restrict to nonzero a, b with a 6 1, b 6 1: a b ⇐⇒ a0 b0 , a b ⇐⇒ a0 b0 , a ∼ b ⇐⇒ a0 ∼ b0 . For a 6= 0 we let a† := a0 /a be its logarithmic derivative, so (ab)† = a† + b† for a, b 6= 0. Elements a, b 1 are said to be comparable if a† b† ; if K is a Hardy field containing R as subfield, or K = T, this is equivalent to the existence of an n ⩾ 1 such that |a| ⩽ |b|n and |b| ⩽ |a|n . Comparability is an equivalence relation on the set of infinite elements of K, and the comparability classes Cl(a) of K are totally ordered by Cl(a) ⩽ Cl(b) :⇐⇒ a† 4 b† . EXAMPLE. For K = T, set e0 = x and en+1 = exp(en). Then the sequence (Cl(en)) is strictly increasing and cofinal in the set of comparability classes. More important are the `n defined recursively by `0 = x, and `n+1 = log `n. Then the sequence Cl(`0)
- 39. 10 INTRODUCTION AND OVERVIEW Cl(`1) Cl(`2) · · · Cl(`n) · · · is coinitial in the set of comparability classes of T. For later use it is worth noting at this point that `† n = 1 `0 · · · `n , −`†† n = 1 `0 + 1 `0`1 + · · · + 1 `0`1 · · · `n . We call K grounded if K has a smallest comparability class. Thus T is not grounded. If Γ contains an element α such that for every γ ∈ Γ we have nγ ⩾ α for some n ⩾ 1, then K is grounded; this condition on Γ is in particular satisfied if Γ 6= {0} and Γ has finite archimedean rank. If K is grounded, then K has only one Liouville closure (up to isomorphism over K). The H-field K is said to have asymptotic integration if K satisfies ∀a∃b(a b0 ), equivalently, {vb0 : b ∈ K} = Γ∞. It is obvious that every Liouville closed H-field has asymptotic integration; in particular, T has asymptotic integration. In general, at most one γ ∈ Γ lies outside {vb0 : b ∈ K}; if K is grounded, then such a γ exists, by results in Section 9.2, and so K cannot have asymptotic integration. STRATEGY AND MAIN RESULTS Model completeness of T concerns finite systems of algebraic differential equations over T with asymptotic side conditions in several differential indeterminates. Robinson’s strategy for establishing model completeness applied to T requires us to move beyond T to consider H-fields and their extensions. If we are lucky—as we are in this case—it will suffice to consider extensions of H-fields by one element y at a time. This leads to equations P(y) = 0 with an asymptotic side condition y ≺ g. Here P ∈ K{Y } is a univariate differential polynomial with coefficients in an H- field K with g ∈ K× , and K{Y } = K[Y, Y 0 , Y 00 , . . . ] is the differential domain of d-polynomials in the differential indeterminate Y over K. The key issue: when is there a solution in some H-field extension of K? A detailed study of such equations in the special case K = Tg and where we only look for solutions in Tg itself was undertaken in [194], using an assortment of techniques (for instance, various fixpoint theorems) heavily based on the particular structure of Tg. Generalizing these results to suitable H-fields is an important guideline in our work. Differential Newton diagrams Let K be an H-field, and consider a d-algebraic equation with asymptotic side condi- tion, (4) P(y) = 0, y ≺ g, where P ∈ K{Y }, P 6= 0, and g ∈ K× ; we look for nonzero solutions in H-field extensions of K. For the sake of concreteness we take K = Tg and look for nonzero solutions in Tg, focusing on the example below: (5) e− ex y2 y00 + y2 − 2xyy0 − 7 e−x y0 − 4 + 1 log x = 0, y ≺ x.
- 40. INTRODUCTION AND OVERVIEW 11 We sketch briefly how [194] goes about solving (5). First of all, we need to find the possible dominant terms of solutions y. This is done by considering possible can- cellations. For example, y2 and −4 might be the terms of least valuation in the left side of (5), with all other terms having greater valuation, so negligible compared to y2 and −4. This yields a cancellation y2 ∼ 4, so y ∼ 2 or y ∼ −2, giving 2 and −2 as potential dominant terms of a solution y. Another case: e− ex y2 y00 and y2 are the terms of least valuation. Then we get a cancellation e− ex y2 y00 ∼ −y2 , that is, y00 ∼ − eex , which leads to y ∼ − eex / e2x . But this possibility is discarded, since (5) also requires y ≺ x. (On the other hand, if the asymptotic condition in (5) had been y ≺ eex , we would have kept − eex / e2x as a potential dominant term of a solution y.) What makes things work in these two cases is that the cancellations arise from terms of different degrees in y, y0 , y00 , . . . . Such cancellations are reminiscent of the more familiar setting of algebraic equations where the dominant monomials of solutions can be read off from a Newton diagram and the corresponding dominant coefficients are zeros of the corresponding Newton polynomials; see Section 3.7. This method still works in our d-algebraic setting, for cancellations among terms of different degrees, but requires the construction of so-called equalizers. A different situation arises for cancellations between terms of the same degree. Con- sider for example the case that y2 and −2xyy0 have least valuation among the terms in the left side of (5), with all other terms of higher valuation. Then y2 ∼ 2xyy0 , so y† ∼ 1 2x . Now y† = 1 2x gives y = cx1/2 with c ∈ R× , but the weaker condition y† ∼ 1 2x only gives y = ux1/2 with u 6= 0, u† ≺ x−1 , that is, |v(u)| |v(x)|/n for all n ⩾ 1. Substituting ux1/2 for y in (5) and considering u as the new unknown, the condition on v(u) forces u 1, so after all we do get y ∼ cx1/2 with c ∈ R× , giving cx1/2 as a potential dominant term of a solution y. It is important to note that here an integration constant c gets introduced. Manipulations as we just did are similar to rewriting an equation H(y) = 0 with ho- mogeneous nonzero H ∈ K{Y } of positive degree as a (Riccati) equation R(y† ) = 0 with R of lower order than H. This technique can be shown to work in general for cancellations among terms of the same degree, provided we are also allowed to transform the equation to an equiv- alent one by applying a suitable iteration of the upward shift f(x) 7→ f(ex ). (For reasonable H-fields K one can apply instead compositional conjugation by positive active elements; see below for compositional conjugation and active.) Having determined a possible dominant term f = cm of a solution of (4), where c ∈ R× and m is a transmonomial, we next perform a so-called refinement (6) P(f + y) = 0, y ≺ f of (4). For instance, taking f = 2, the equation (5) transforms into e− ex y2 y00 + y2 − 2xyy0 + 4 e− ex yy00 + 4y − (4x + 7 e−x )y0 + 4 e− ex y00 + 1 log x = 0, y ≺ 2. Now apply the same procedure to this refinement, to find the “next” term.
- 41. 12 INTRODUCTION AND OVERVIEW Roughly speaking, this yields an infinite process to obtain all possible asymptotic ex- pansions of solutions to any asymptotic equation. How do we make this into a finite process? For this, it is useful to introduce the Newton degree of (4). This notion is similar to the Weierstrass degree of a multivariate power series and corresponds to the degree of the asymptotically significant part of the equation. If the Newton degree is 0, then (4) has no solution. The Newton degree of (5) turns out to be 2: this has to do with the fact that e− ex y2 y00 ≺ y2 whenever y ≺ x. We shall return soon to the precise definition of Newton degree for differential polynomials over rather general H-fields. As to the resolution of asymptotic equations over K = Tg, the following key facts were established in [194]: • The Newton degree stays the same or decreases under refinement. • If the Newton degree of the refinement (6) equals that of (4), we employ so-called unravelings; these resemble the Tschirnhaus transformations that overcome sim- ilar obstacles in the algebraic setting. Combining unravelings with refinements as described above, we arrive after finitely many steps at an asymptotic equation of Newton degree 0 or 1. • The H-field Tg is newtonian, that is, any asymptotic equation over Tg of Newton degree 1 has a solution in Tg. All in all, we have for any given asymptotic equation over Tg a more or less finite procedure for gaining an overview of the entire space of solutions in Tg. To define the Newton degree of an asymptotic equation (4) over rather general H-fields, we first need to introduce the dominant part of P and then, based on a process called compositional conjugation, the Newton polynomial of P. The dominant part Let K be an H-field. We extend the valuation v of K to the integral domain K{Y } by setting vP = min{va : a is a coefficient of P}, and we extend the binary relations and ∼ on K to K{Y } accordingly. It is also convenient to fix a monomial set M in K, that is, a subset M of K that is mapped bijectively by v onto the value group Γ of K. This allows us to define the dominant part DP (Y ) of a nonzero d-polynomial P(Y ) over K to be the unique element of C{Y } ⊆ K{Y } with P ∼ dP DP , where dP ∈ M is the dominant monomial of P determined by P dP . (Another choice of monomial set would just multiply DP by some positive constant.) For K = T we always take the set of transmonomials as our monomial set. EXAMPLE 1. Let K = T. For P = x5 + (2 + ex )Y + (3 ex + log x)(Y 0 )2 , we have dP = ex and DP = Y + 3(Y 0 )2 . For Q = Y 2 − 2xY Y 0 we have DQ = −2Y Y 0 .
- 42. INTRODUCTION AND OVERVIEW 13 For K with small derivation we can use DP to get near the zeros a 1 of P: if P(a) = 0, a 1, then DP (c) = 0 where c is the unique constant with a ∼ c. We need to understand, however, the behavior of P(a) not only for a 1, that is, va = 0, but also for “sufficiently flat” elements a ∈ K, that is, for va approaching 0 ∈ Γ. For instance, in T, the iterated logarithms `0 = x, `1 = log x, `2 = log log x, . . . satisfy v(`n) → 0 in ΓT and likewise v(1/`n) → 0. The dominant term dP DP of P often provides a good approximation for P when evaluating at sufficiently flat elements, but not always: for K = T and Q as in Example 1 we note that for y = `2 we have: y2 = `2 2 2xyy0 = 2`2/`1, so Q(y) ∼ y2 6 (dQDQ)(y). In order to approximate P(y) by (dP DP )(y) for sufficiently flat y, we need one more ingredient: compositional conjugation. For K = T and Q as in Example 1, this amounts to a change of variables x = eee x , so that Q(y) = y2 − 2y(dy/de x) e−e x for y ∈ T. With respect to this new variable e x, the dominant term Y 2 of the adjusted d- polynomial Y 2 −2Y Y 0 e−e x is then an adequate approximation of Q when evaluating at sufficiently flat elements of T. Such changes of variable do not make sense for general H-fields, but as it turns out, compositional conjugation is a good substitute. Compositional conjugation We define this for an arbitrary differential field K. For φ ∈ K× we let Kφ be the differ- ential field obtained from K by replacing its derivation ∂ by the multiple φ−1 ∂. Then a differential polynomial P(Y ) ∈ K{Y } defines the same function on the common underlying set of K and Kφ as a certain differential polynomial Pφ (Y ) ∈ Kφ {Y }: for P = Y 0 , we have Pφ (Y ) = φY 0 (since over Kφ we evaluate Y 0 according to the derivation φ−1 ∂), for P = Y 00 we have Pφ (Y ) = φ0 Y 0 + φ2 Y 00 (with φ0 = ∂φ), and so on. This yields a ring isomorphism P 7→ Pφ : K{Y } → Kφ {Y } that is the identity on the common subring K[Y ]. It is also an automorphism of the common underlying K-algebra of K{Y } and Kφ {Y }, and studied as such in Chap- ter 12. We call Kφ the compositional conjugate of K by φ, and Pφ the compositional conjugate of P by φ. Note that K and Kφ have the same constant field C. If K is an H-field and φ ∈ K , then so is Kφ . It pays to note how things change under compositional conjugation, and what remains invariant. The Newton polynomial Suppose now that K is an H-field with asymptotic integration. For φ ∈ K we say that φ is active (in K) if φ a† for some nonzero a 6 1 in K; equivalently, the derivation φ−1 ∂ of Kφ is small. Let φ ∈ K range over the active elements of K in what follows, fix a monomial set M ⊆ K of K, and let P ∈ K{Y }, P 6= 0. The dominant part DP φ of Pφ lies in C{Y }, and we show in Section 13.1 that it
- 43. 14 INTRODUCTION AND OVERVIEW eventually stabilizes as φ varies: there is a differential polynomial NP ∈ C{Y } and an active φ0 ∈ K such that for all φ 4 φ0, DP φ = cφNP , cφ ∈ C . We call NP the Newton polynomial of P. It is of course only determined up to a factor from C , but this ambiguity is harmless. The (total) degree of NP is called the Newton degree of P. EXAMPLE 2. Let K = T. Then f ∈ K is active iff f `† n = 1 `0`1···`n for some n. If P is as in Example 1, then for each φ, Pφ = x5 + (2 + ex )Y + φ2 (3 ex + log x)(Y 0 )2 , so DP φ = Y if φ ≺ 1. This yields NP = Y , so P has Newton degree 1. It is an easy exercise to show that for Q = Y 2 − 2xY Y 0 we have NQ = Y 2 . A crucial result in [194] (Theorem 8.6) says that if K = Tg, then NP ∈ R[Y ](Y 0 )N . A major step in our work was to isolate a robust class of H-fields K with asymptotic integration for which likewise NP ∈ C[Y ](Y 0 )N for all nonzero P ∈ K{Y }. This required several completely new tools to be discussed below. The special cuts γ, λ and ω Recall that `n denotes the nth iterated logarithm of x in T, so `0 = x and `n+1 = log `n. We introduce the elements γn = `† n = 1 `0 · · · `n λn = −γ† n = 1 `0 + 1 `0`1 + · · · + 1 `0`1 · · · `n ωn = −2λ0 n − λ2 n = 1 `2 0 + 1 `2 0`2 1 + · · · + 1 `2 0`2 1 · · · `2 n of T. As n → ∞ these elements approach their formal limits γT = 1 `0`1`2 · · · λT = 1 `0 + 1 `0`1 + 1 `0`1`2 + · · · ωT = 1 `2 0 + 1 `2 0`2 1 + 1 `2 0`2 1`2 2 + · · · , which for now are just suggestive expressions. Indeed, our field T of transseries of finite logarithmic and exponential depth does not contain any pseudolimit of the pseu- docauchy sequence (λn), nor of the pseudocauchy sequence (ωn). There are, however, immediate H-field extensions of T where such pseudolimits exist, and if we let λT be
- 44. INTRODUCTION AND OVERVIEW 15 such a pseudolimit of (λn), then in some further H-field extension we have an element suggestively denoted by exp( R −λT) that can play the role of γT. Even though γT, λT and ωT are not in T, we can take them as elements of some H-field extension of T, as indicated above, and so we obtain sets Γ(T) = {a ∈ T : a γT} Λ(T) = {a ∈ T : a λT} Ω(T) = {a ∈ T : a ωT} that can be shown to be definable in T. For instance, Γ(T) = a ∈ T : ∀b ∈ T (b 1 ⇒ a 6= b† ) = {−a0 : a ∈ T, a ⩾ 0}. In other words, γT, λT and ωT realize definable cuts in T. For any ungrounded H-field K 6= C we can build a sequence (`ρ) of elements `ρ 1, indexed by the ordinals ρ less than some infinite limit ordinal, such that σ ρ ⇒ `† σ ≺ `† ρ, v(`ρ) → 0 in Γ. These `ρ play in K the role that the iterated logarithms `n play in T. In analogy with T they yield the elements γρ := `† ρ, λρ := −γ† ρ, ωρ := −2λ0 n − λ2 n, of K, and (λρ) and (ωρ) are pseudocauchy sequences. As with T this gives rise to definable sets Γ(K), Λ(K) and Ω(K) in K. The fact mentioned earlier that T does not contain γT, λT or ωT turns out to be very significant: in general, we have γK ∈ K ⇒ λK ∈ K ⇒ ωK ∈ K and each of the four mutually exclusive cases γK ∈ K, γK / ∈ K λK ∈ K, λK / ∈ K ωK ∈ K, ωK / ∈ K can occur; see Section 13.9. Here we temporarily abuse notations, since we should explain what we mean by γK ∈ K and the like; see the next subsections. On gaps and forks in the road Let K be an H-field. We say that an element γ ∈ K is a gap in K if for all a ∈ K with a 1 we have a† γ (1/a)0 . The existence of such a gap is the formal counterpart to the informal statement that γK ∈ K. If K has a gap γ, then γ has no primitive in K, so K is not closed under integration. If K has trivial derivation (that is, K = C), then K has a gap γ = 1.
- 45. 16 INTRODUCTION AND OVERVIEW There are also K with K 6= C (even Hardy fields) that have a gap. Not having a gap is equivalent to being grounded or having asymptotic integration. We already mentioned the result from [19] that K may have two Liouville closures that are not isomorphic over K (but fortunately not more than two). Indeed, if K has a gap γ, then in one Liouville closure all primitives of γ are infinitely large, whereas in the other γ has an infinitesimal primitive. Even if K has no gap, the above fork in the road can arise more indirectly: Assume that K has asymptotic integration and λ ∈ K is such that for all a ∈ K× with a ≺ 1, a0† −λ a†† . Then K has no element γ 6= 0 with λ = −γ† , but K has an H-field extension Khγi generated by an element γ with λ = −γ† , and any such γ is a gap in Khγi. It follows again that K has two Liouville closures that are not K-isomorphic. For real closed K with asymptotic integration, the existence of such an element λ corresponds to the informal statement that γK / ∈ K λK ∈ K. We define K to be λ-free if K has asymptotic integration and satisfies the sentence ∀a∃b b 1 a − b†† b† . It can be shown that for real closed K with asymptotic integration, λ-freeness is equiv- alent to the nonexistence of an element λ as above. More generally, K is λ-free iff K has asymptotic integration and (λρ) has no pseudolimit in K. The property of ω-freeness Even λ-freeness might not prevent a fork in the road for some d-algebraic extension. Let K be an H-field, and define ω = ωK : K → K, ω(z) := −2z0 − z2 . Assume that K is λ-free and ω ∈ K is such that for all b 1 in K, ω − ω(b†† ) ≺ (b† )2 . Then the first-order differential equation ω(z) = ω admits no solution in K, but K has an H-field extension Khλi generated by a solution z = λ to ω(z) = ω such that Khλi is no longer λ-free (and with a fork in its road towards Liouville closure). For λ-free K the existence of an element ω as above corresponds to the informal statement that λK / ∈ K ωK ∈ K. We say that K is ω-free if no such ω exists, more precisely, K has asymptotic integration and satisfies the sentence ∀a∃b b 1 a − ω(b†† ) (b† )2 . (It is easy to show that if K is ω-free, then it is λ-free.) For K with asymptotic in- tegration, ω-freeness is equivalent to the pseudocauchy sequence (ωρ) not having a pseudolimit in K. Thus T is ω-free. More generally, if K has asymptotic integration and is a union of grounded H-subfields, then K is ω-free by Corollary 11.7.15.
- 46. INTRODUCTION AND OVERVIEW 17 Much deeper and very useful is that if K is an ω-free H-field and L is a d-algebraic H-field extension of K, then L is also ω-free and has no comparability class smaller than all those of K; this is part of Theorem 13.6.1. Thus the property of ω-freeness is very robust: if K is ω-free, then forks in the road towards Liouville closure no longer occur, even for d-algebraic H-field extensions of K (Corollary 13.6.2). There are, however, Liouville closed H-fields that are not ω-free; see [22]. Another important consequence of ω-freeness is that Newton polynomials of dif- ferential polynomials then take the same simple shape as those over Tg: THEOREM 1. If K is ω-free and P ∈ K{Y }, P 6= 0, then NP ∈ C[Y ](Y 0 )N . The proof in Chapter 13 depends heavily on Chapter 12, where we determine the in- variants of certain automorphism groups of polynomial algebras in infinitely many vari- ables Y0, Y1, Y2, . . . over a field of characteristic zero. The function ω and the notion of ω-freeness are closely related to second order lin- ear differential equations over K. More precisely (Riccati), for y ∈ K× , 4y00 + fy = 0 is equivalent to ω(z) = f with z := 2y† ; so the second-order linear differential equa- tion 4y00 +fy = 0 reduces in a way to a first-order (but non-linear) differential equation ω(z) = f. (The factor 4 is just for convenience, to get simpler expressions below.) EXAMPLE. The differential equation y00 = −y has no solution y ∈ T× , whereas the Airy equation y00 = xy has two R-linearly independent solutions in T [308, Chapter 11, (1.07)]. Indeed, in Sections 11.7 and 11.8 we show that for f ∈ T, the differential equation 4y00 +fy = 0 has a solution y ∈ T× if and only if f ωT, that is, f ωn = 1 `2 0 + 1 `2 0`2 1 + · · · + 1 `2 0`2 1···`2 n for some n. This fact reflects classical results [167, 184] on the question: for which logarithmico-exponential functions f (in Hardy’s sense) does the equation 4y00 + fy = 0 have a non-oscillating real-valued solution (more precisely, a nonzero solution in a Hardy field)? Newtonianity This is the most consequential elementary property of T. An ω-free H-field K is said to be newtonian if every d-polynomial P(Y ) over K of Newton degree 1 has a zero in O. This turns out to be the correct analogue for valued differential fields like T of the property of being henselian for a valued field. We chose the adjective newtonian since it is this property that allows us to develop in Chapter 13 a Newton diagram method for differential polynomials. It is good to keep in mind that the role of newtonianity in the results of Chapters 14, 15, and 16 is more or less analogous to that of henselianity in the theory of valued fields and as the key condition in the Ax-Kochen-Eršov results. We already mentioned the result from [194] that Tg is newtonian. That T is newto- nian is a consequence of the following analogue in Chapter 15 of the familiar valuation- theoretic fact that spherically complete valued fields are henselian: THEOREM 2. If K is an H-field, ∂K = K, and K is a directed union of spherically complete grounded H-subfields, then K is (ω-free and) newtonian. EXAMPLE. Let K = T and consider for α ∈ R the differential polynomial P(Y ) = Y 00 − 2Y 3 − xY − α ∈ T{Y }.
- 47. 18 INTRODUCTION AND OVERVIEW For φ ∈ T× we have (Y 00 )φ = φ2 Y 00 + φ0 Y 0 for φ ∈ T× , so Pφ = φ2 Y 00 + φ0 Y 0 − 2Y 3 − xY − α. Now φ2 , φ0 ≺ 1 ≺ x for active φ ≺ 1 in T . Hence NP ∈ R× Y , so P has Newton degree 1. Thus the Painlevé II equation y00 = 2y3 + xy + α has a solution y ∈ OT. (It is known that P has a zero y 4 1 in the differential subfield R(x) of T iff α ∈ Z; see for example [156, Theorem 20.2].) The main results of Chapter 14 amount for H-fields to the following: THEOREM 3. If K is a newtonian ω-free H-field with divisible value group, then K has no proper immediate d-algebraic H-field extension. COROLLARY 1. Let K be a real closed newtonian ω-free H-field, and let Ka = K[i] (where i2 = −1) be its algebraic closure. Then: (i) each d-polynomial in Ka {Y } of positive degree has a zero in Ka ; (ii) each linear differential operator in Ka [∂] of positive order is a composition of such operators of order 1; (iii) each d-polynomial in K{Y } of odd degree has a zero in K; and (iv) each linear differential operator in K[∂] of positive order is a composition of such operators of order 1 and order 2. THEOREM 4. If K is an ω-free H-field with divisible value group, then K has an immediate d-algebraic newtonian H-field extension, and any such extension embeds over K into every ω-free newtonian H-field extension of K. An extension of K as in Theorem 4 is minimal over K and thus unique up to isomor- phism over K. We call such an extension a newtonization of K. THEOREM 5. If K is an ω-free H-field, then K has a d-algebraic newtonian Liouville closed H-field extension that embeds over K into every ω-free newtonian Liouville closed H-field extension of K. An extension of K as in Theorem 5 is minimal over K and thus unique up to isomor- phism over K. We call such an extension a Newton-Liouville closure of K. The main theorems We now come to the results in Chapter 16, which in our view justify this volume. First, the various elementary conditions we have discussed axiomatize a model complete theory. To be precise, construe H-fields in the natural way as L-structures where L := {0, 1, +, −, · , ∂, ⩽, 4}, and let Tnl be the L-theory whose models are the newtonian ω-free Liouville closed H-fields. THEOREM 6. Tnl is model complete.
- 48. INTRODUCTION AND OVERVIEW 19 The theory Tnl is not complete and has exactly two completions, namely Tnl small (small derivation) and Tnl large (large derivation). Thus newtonian ω-free Liouville closed H- fields with small derivation have the same elementary properties as T. Every H-field with small derivation can be embedded into a model of Tnl small; thus Theorem 6 yields the strong version of the model completeness conjecture from [19] stated earlier in this introduction. As Tnl small is complete and effectively axiomatized, it is decidable. In particular, there is an algorithm which, for any given d-polynomials P1, . . . , Pm in indeterminates Y1, . . . , Yn with coefficients from Z[x], decides whether there is a tuple y ∈ Tn such that P1(y) = · · · = Pm(y) = 0. Such an algorithm with T replaced by its differential subring R[[x−1 ]] is due to Denef and Lipshitz [101], but no such algorithm can exist with T replaced by R((x−1 )) or by any of various other natural H-subfields of T [20, 155]. Theorem 6 is the main step towards an elimination of quantifiers, in a slightly extended language: Let Lι Λ,Ω be L augmented by the unary function symbol ι and the unary predicates Λ, Ω, and extend Tnl to the Lι Λ,Ω-theory Tnl,ι Λ,Ω by adding as defining axioms for these new symbols the universal closures of a 6= 0 −→ a · ι(a) = 1 a = 0 −→ ι(a) = 0 , Λ(a) ←→ ∃y y 1 a = −y†† , Ω(a) ←→ ∃y y 6= 0 4y00 + ay = 0 . For a model K of Tnl this makes the sets Λ(K) and Ω(K) downward closed with respect to the ordering of K. For example, for f ∈ T, f ∈ Λ(T) ⇐⇒ f λn = 1 `0 + 1 `0`1 + · · · + 1 `0`1 · · · `n for some n, f ∈ Ω(T) ⇐⇒ f ωn = 1 `2 0 + 1 `2 0`2 1 + · · · + 1 `2 0`2 1 · · · `2 n for some n, that is, Λ(T) and Ω(T) are the cuts in T determined by λT, ωT introduced earlier. We can now state what we view as the main result of this volume: THEOREM 7. The theory Tnl,ι Λ,Ω admits elimination of quantifiers. We cannot omit here either Λ or Ω. In Chapter 16 we do include for convenience one more unary predicate I in Lι Λ,Ω: for a model K of Tnl and a ∈ K, I(a) ←→ ∃y a 4 y0 y 4 1 ←→ a = 0 ∨ a 6= 0 ¬Λ(−a† ) , where the first equivalence is the defining axiom for I, and the second shows that I is superfluous in Theorem 7. We note here that this predicate I governs the solvability of first-order linear differential equations with asymptotic side condition. More precisely, for K as above and f ∈ K, g, h ∈ K× , the following are equivalent: (a) there exists y ∈ K with y0 = fy + g and y ≺ h; (b) (f −h† ) ∈ I(K) and (g/h) ∈ I(K) or (f −h† ) / ∈ I(K) and (g/h) ≺ f −h† .
- 49. 20 INTRODUCTION AND OVERVIEW This equivalence is part of Corollary 11.8.12 and exemplifies Theorem 7 (but is not derived from that theorem, nor used in its proof). In the proof of Theorem 7, and throughout the construction of suitable H-field extensions, the predicates I, Λ and Ω act as switchmen. Whenever a fork in the road occurs due to the presence of a gap γ, then I(γ) tells us to take the branch where R γ 4 1, while ¬I(γ) forces R γ 1. Likewise, the predicates Λ and Ω control what happens when adjoining elements γ and λ with γ† = −λ and ω(λ) = ω. From the above defining axioms for Λ and Ω it is clear that these predicates are (uniformly) existentially definable in models of Tnl . By model completeness of Tnl they are also uniformly universally definable in these models; Section 16.5 deals with such algebraic-linguistic issues. Next we list some more intrinsic consequences of our elimination theory. COROLLARY 2. Let K be a newtonian ω-free Liouville closed H-field, and suppose the set X ⊆ Kn is definable. Then X has empty interior in Kn (with respect to the order topology on K and the product topology on Kn ) if and only if for some nonzero P ∈ K{Y1, . . . , Yn} we have X ⊆ y ∈ Kn : P(y) = 0 . In (i) below the intervals are in the sense of the ordered field K. COROLLARY 3. Let K be a newtonian ω-free Liouville closed H-field. Then: (i) K is o-minimal at infinity: if X ⊆ K is definable in K, then for some a ∈ K, either (a, +∞) ⊆ X, or (a, +∞) ∩ X = ∅; (ii) if X ⊆ Kn is definable in K, then X ∩ Cn is semialgebraic in the sense of the real closed constant field C of K; (iii) K has NIP. (See Appendix B for this very robust property.) It is hard to imagine obtaining these results for K = T without Theorem 7. Item (i) relates to classical bounds on solutions of algebraic differential equations over Hardy fields; see [20, Section 3]. To illustrate item (ii) of Corollary 3, we note that the set of real parameters (λ0, . . . , λn) ∈ Rn+1 for which the system λ0y + λ1y0 + · · · + λny(n) = 0, 0 6= y ≺ 1 has a solution in T is a semialgebraic subset of Rn+1 ; in fact, it agrees with the set of all (λ0, . . . , λn) ∈ Rn+1 such that the polynomial λ0 + λ1Y + · · · + λnY n ∈ R[Y ] has a negative zero in R; see Corollary 11.8.26. To illustrate item (iii), let Y = (Y1, . . . , Yn) be a tuple of distinct differential indeterminates; for an m-tuple σ = (σ1, . . . , σm) of elements of {≺, , } we say that P1, . . . , Pm ∈ T{Y } realize σ if there exists a ∈ Tn such that Pi(a) σi 1 holds for i = 1, . . . , m. Then a special case of (iii) says that for fixed d, n, r ∈ N, the number of tuples σ ∈ {≺, , }m realized by some P1, . . . , Pm ∈ T{Y } of degree at most d and order at most r grows only polynomially with m, even though the total number of tuples is 3m . These manifestations of (ii) and (iii), though instructive, are perhaps a bit misleading, since they can be obtained without appealing to (ii) and (iii). In the course of proving Theorem 6 we also get:
- 50. INTRODUCTION AND OVERVIEW 21 THEOREM 8. If K is a newtonian ω-free Liouville closed H-field, then K has no proper d-algebraic H-field extension with the same constant field. For K = Tg this yields: every f ∈ T Tg is d-transcendental over Tg. We can also enlarge T. For example, the series P∞ n=0 e−1 n , with en the nth iterated exponential of x, does not lie in T but does lie in a certain completion Tc of T. This completion Tc is naturally an ordered valued differential field extension of T, and by Corollary 14.1.6 we have T 4 Tc . ORGANIZATION Here we discuss the somewhat elaborate organization of this volume into chapters, some technical ingredients not mentioned so far, and some material that goes beyond the setting of H-fields. Indeed, the supporting algebraic theory deserves to be devel- oped in a broad way, and there are more notions to keep track of than one might expect. Background chapters To make our work more accessible and self-contained, we provide in the first five chap- ters background on commutative algebra, valued abelian groups, valued fields, differ- ential fields, and linear differential operators. This material has many sources, and we thought it would be convenient to have it available all in one place. In addition we have an appendix with the construction of T, and an appendix exposing the (small) part of model theory that we need. The basic facts on Hahn products, pseudocauchy sequences and spherical com- pleteness in these early chapters are used throughout the volume. Some readers might prefer to skip in a first reading cauchy sequences, completeness (for valued abelian groups and valued fields) and step-completeness, which are not needed for the main results in this volume (but see Corollary 14.1.6). Some parts, like Sections 2.3 and 5.4, fit naturally where we put them, but are mainly intended for use in the next volume. On the other hand, Section 5.7 on compositional conjugation is elementary and frequently referred to in subsequent chapters, but this material seems virtually absent from the literature. Valued differential fields We also profited from examining arbitrary valued differential fields K with small de- rivation, that is, ∂O ⊆ O for the maximal ideal O of the valuation ring O of K. This yields the continuity of the derivation ∂ with respect to the valuation topology and gives ∂O ⊆ O, and so induces a derivation on the residue field. To our surprise, we could establish in Chapters 6 and 7 some useful facts in this very general setting when the induced derivation on the residue field is nontrivial, for example the Equalizer Theorem 6.0.1. We need this result in deriving an “eventual” version of it for ω-free H-fields in Chapter 13, which in turn is crucial in obtaining our main results, via its role in constructing an appropriate Newton diagram for d-polynomials.
- 51. 22 INTRODUCTION AND OVERVIEW Asymptotic couples A useful gadget is the asymptotic couple of an H-field K. This is the value group Γ of K equipped with the map γ 7→ γ† : Γ6= → Γ defined by: if γ = vf, f ∈ K× , then γ† = v(f† ). This map is a valuation on Γ, and we extend it to a map Γ → Γ∞ by setting 0† := ∞. Two key facts are that α† β+β† for all α, β 0 in Γ, and α† ⩾ β† whenever 0 α ⩽ β in Γ. The condition on an H-field of having small derivation can be expressed in terms of its asymptotic couple; the same holds for having a gap, for being grounded, and for having asymptotic integration, but not for being ω-free. Asymptotic couples were introduced by Rosenlicht [364] for d-valued fields. In Chapter 6 we assign to any valued differential field with small derivation an asymptotic couple, with good effect. Asymptotic couples play also an important role in Chapters 9, 10, 11, 13, and 16. Differential-henselian fields Valued differential fields with small derivation include the so-called monotone dif- ferential fields defined by the condition a0 4 a. In analogy with the notion of a henselian valued field, Scanlon [382] introduced differential-henselian monotone dif- ferential fields. Using the Equalizer Theorem we extend this notion and basic facts about it to arbitrary valued differential fields with small derivation in Chapter 7. (We abbreviate differential-henselian to d-henselian.) This material plays a role in Chap- ter 14, using the following relation between d-henselian and newtonian: an ω-free H-field K is newtonian iff for every active φ ∈ K the compositional conjugate Kφ is d-henselian, with the valuation v on Kφ replaced by the coarser valuation π ◦ v where π: v(K× ) = Γ → Γ/∆ is the canonical map to the quotient of Γ by its convex subgroup ∆ := {γ ∈ Γ : γ† vφ}. We pay particular attention to two special cases: v(C× ) = {0} (few constants), and v(C× ) = Γ (many constants). The first case is relevant for newtonianity, the second case is considered in a short Chapter 8, where we present Scanlon’s extension of the Ax-Kochen-Eršov theorems to d-henselian valued fields with many constants, and add some things on definability. While d-henselianity is defined in terms of solving differential equations in one unknown, it implies the solvability of suitably non-singular systems of n differential equations in n unknowns: this is shown at the end of Chapter 7, and has a nice conse- quence for newtonianity: Proposition 14.5.7. Asymptotic differential fields To keep things simple we confined most of the exposition above to H-fields, but this setting is a bit too narrow for various technical reasons. For example, a differential subfield of an H-field with the induced ordering is not always an H-field, and passing to an algebraic closure like T[i] destroys the ordering, though T[i] is still a d-valued field. On occasion we also wish to change the valuation of an H-field or d-valued field
- 52. INTRODUCTION AND OVERVIEW 23 by coarsening. For all these reasons we introduce in Chapter 9 the class of asymptotic differential fields, which is larger and more flexible than Rosenlicht’s class of d-valued fields. Many basic facts about H-fields and d-valued fields do have good analogues for asymptotic differential fields. This is shown in Chapter 9, which also contains a lot of basic material on asymptotic couples. Chapter 10 deals more specifically with H-fields. Immediate extensions Indispensable for attaining our main results is the fact that every H-field with divisible value group and with asymptotic integration has a spherically complete immediate H- field extension. This is part of Theorem 11.4.1, and proving it about five years ago removed a bottleneck. It provides the only way known to us of extending every H-field to an ω-free H-field. Possibly more important than Theorem 11.4.1 itself are the tools involved in its proof. In view of the theorem’s content, it is ironic that models of Tnl are never spherically complete, in contrast to all prior positive results on elementary theories of valued fields with or without extra structure, cf. [28, 29, 41, 131, 382]. The differential Newton diagram method Chapters 13 and 14 present the differential Newton diagram method in the general con- text of asymptotic fields that satisfy suitable technical conditions, such as ω-freeness. Before tackling these chapters, the reader may profit from first studying our exposition of the Newton diagram method for ordinary one-variable polynomials over henselian valued fields of equicharacteristic zero in Section 3.7. Some of the issues encountered there (for example, the unraveling technique) appear again, albeit in more intricate form, in the differential context of these chapters. In the proofs of a few crucial facts about the special cuts λ and ω in Chapter 13 we use some results from the preceding Chapter 12 on triangular automorphisms. Chapter 12 is a bit special in being essentially independent of the earlier chapters. Proving newtonianity Chapter 15 contains the proof of Theorem 2, and thus establishes that T is a model of our theory Tnl small. This theorem is also useful in other contexts: In [43], Berarducci and Mantova construct a derivation on Conway’s field No of surreal numbers [92, 150] turning it into a Liouville closed H-field with constant field R. From Theorem 2 and the completeness of Tnl it follows that No with this derivation and T are elementarily equivalent, as we show in [24]. Quantifier elimination In Chapter 16 we first prove Theorem 6 on model completeness, next we consider H-fields equipped with a ΛΩ-structure, and then deduce Theorem 7 about quantifier elimination with various interesting consequences, such as Corollaries 2 and 3. The
- 53. 24 INTRODUCTION AND OVERVIEW introduction to this chapter gives an overview of the proof and the role of various em- bedding and extension results in it. THE NEXT VOLUME The present volume focuses on achieving quantifier elimination (Theorem 7), and so we left out various things we did since 1995 that were not needed for that. In a second volume we intend to cover these things as required for developing our work further. Let us briefly survey some highlights of what is to come. Linear differential equations We plan to consider linear differential equations in much greater detail, comprising the corresponding differential Galois theory, in connection with constructing the linear surjective closure of a differential field, factoring linear differential operators over suit- able algebraically closed d-valued fields, and explicitly constructing the Picard-Vessiot extension of such an operator. Concerning the latter, the complexification T[i] of T is no longer closed under exponential integration, since oscillatory “transmonomials” such as eix are not in T [i]. Adjoining these oscillatory transmonomials to T[i] yields a d-valued field that contains a Picard-Vessiot extension of T for each operator in T[∂]. Hardy fields We also wish to pay more attention to Hardy fields, and this will bring up analytic issues. For example, every Hardy field containing R can be shown to extend to an ω-free Hardy field. Using methods from [195], we also hope to prove that it always extends to a newtonian ω-free Hardy field. Indeed, that paper proves among other things the following pertinent result (formulated here with our present terminology): Let Tda g consist of the grid-based transseries that are d-algebraic over R. Then Tda g is a newtonian ω-free Liouville closed H-subfield of Tg and is isomorphic over R to a Hardy field containing R. Embedding into fields of transseries Another natural question we expect to deal with is whether every H-field can be given some kind of transserial structure. This can be made more precise in terms of the axiomatic definition of a field of transseries in terms of a transmonomial group M in Schmeling’s thesis [388]. For instance, one axiom there is that for all m ∈ M we have supp log m ⊆ M . We hope that any H-field can be embedded into such a field of transseries. This would be a natural counterpart of Kaplansky’s theorem [209] embedding certain valued fields into Hahn fields, and would make it possible to think of H-field elements as generalized transseries.
- 54. INTRODUCTION AND OVERVIEW 25 More on the model theory of T In the second volume we hope to deal with further issues around T of a model-theoretic nature: for example, identifying the induced structure on its value group (conjectured to be given by its H-couple, as specified in [18]); and determining the definable closure of a subset of a model of Tnl , in order to get a handle on what functions are definable in T. A by-product of the present volume is a full description of several important 1- types over a given model of Tnl , but the entire space of such 1-types remains to be surveyed. Theorem 8 suggests that the model-theoretic notions of non-orthogonality to C or C-internality may be significant for models of Tnl ; see also [25]. FUTURE CHALLENGES We now discuss a few more open-ended avenues of inquiry. Differentiation and exponentiation The restriction to OT of the exponential function on T is easily seen to be definable in T, but by part (ii) of Corollary 3, the restriction to R of this exponential function is not definable in T. This raises the question whether our results can be extended to the differential field T with exponentiation, or with some other extra o-minimal structure on it. Logarithmic transseries A transseries is logarithmic if all transmonomials in it are of the form `r0 0 · · · `rn n with r0, . . . , rn ∈ R. (See Appendix A.) The logarithmic transseries make up an ω-free newtonian H-subfield Tlog of T that is not Liouville closed. We conjecture that Tlog as a valued differential field is model complete. The asymptotic couple of Tlog has been successfully analyzed by Gehret [146], and turns out to be model-theoretically tame, in particular, has NIP [147]. (There is also the notion of a transseries being expo- nential. The exponential transseries form a real closed H-subfield Texp of T in which the set Z is existentially definable, see [20]. It follows that the differential field Texp does not have a reasonable model theory: it is as complicated as so-called second-order arithmetic.) Accelero-summable transseries The paper [195] on transserial Hardy fields yields on the one hand a method to as- sociate a genuine function to a suitable formal transseries, and in the other direction also provides means to associate concrete asymptotic expansions to elements of Hardy fields. We expect that more can be done in this direction. Écalle’s theory of analyzable functions has a more canonical procedure that asso- ciates a function to an accelero-summable transseries. These transseries make up an H-subfield Tas of T. This procedure has the advantage that it does not only preserve
- 55. Another Random Document on Scribd Without Any Related Topics
- 59. The Project Gutenberg eBook of Happy House
- 60. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Happy House Author: Freifrau von Betsey Riddle Hutten zum Stolzenberg Release date: May 22, 2013 [eBook #42771] Most recently updated: October 23, 2024 Language: English Credits: E-text prepared by Annie McGuire from page images generously made available by the Google Books Library Project (http://books.google.com) *** START OF THE PROJECT GUTENBERG EBOOK HAPPY HOUSE ***
- 61. The Project Gutenberg eBook, Happy House, by Betsey Riddle, Freifrau von Hutten zum Stolzenberg Note: Images of the original pages are available through the Google Books Library Project. See http://www.google.com/books?id=jE8gAAAAMAAJ HAPPY HOUSE
- 62. The BARONESS
- 63. VON HUTTEN
- 64. HAPPY HOUSE BY The BARONESS VON HUTTEN AUTHOR OF PAM, PAM DECIDES, SHARROW, KINGSMEAD, ETC. NEW YORK GEORGE H. DORAN COMPANY COPYRIGHT, 1920, BY GEORGE H. DORAN COMPANY TO MISS LILY BETTS my dear lily: We three, one of us in a chair, and two of us upside down on the grass-plot, have decided that this book must be dedicated to you, in memory of how we did not work on it at Sennen Cove, and how we did work on it here. So here it is, with our grateful love, from Your affectionate R ichard, and Hetty, and B. v. H. PENZANCE
- 65. CONTENTS CHAPTER I CHAPTER II CHAPTER III CHAPTER IV CHAPTER V CHAPTER VI CHAPTER VII CHAPTER VIII CHAPTER IX CHAPTER X CHAPTER XI CHAPTER XII CHAPTER XIII CHAPTER XIV CHAPTER XV CHAPTER XVI CHAPTER XVII CHAPTER XVIII CHAPTER XIX CHAPTER XX CHAPTER XXI CHAPTER XXII CHAPTER XXIII CHAPTER XXIV CHAPTER XXV CHAPTER XXVI CHAPTER XXVII
- 66. HAPPY HOUSE
- 67. CHAPTER I Mrs. Walbridge stood at the top of the steps, a pink satin slipper in her hand, looking absently out into the late afternoon. The July sunlight spread in thick layers across the narrow, flagged path to the gate, and the shadows under the may tree on the left were motionless, as if cut out of lead. The path was strewn with what looked like machine-made snowflakes, and a long piece of white satin ribbon had caught on the syringa bush on the right of the green gate, and hung like a streak of whiter light across the leaves. Someone inside the house was playing a fox-trot, and sounds of tired laughter were in the air, but the well-known author, Mrs. Walbridge, did not hear them. She was leaning against the side of the door, recklessly crushing her new grey frock, and her eyes were fixed on the gate in the unseeing stare of utter fatigue. Presently the music stopped and the sudden silence seemed to rouse her, for, with a deep sigh and a little shake of the head that was evidently characteristic, she turned and went slowly into the house. A few minutes later a brisk-looking young man in a new straw hat came down the street and paused at the gate, peering up at the fanlight to verify his whereabouts. Number eighty-eight did not seem to satisfy him, but suddenly his eyes fell on the gate. On its shabby green were painted the words, very faded, almost undecipherable, Happy House, and with a contented nod the young man opened the gate and went quickly up the steps. No one answered his ring, so he rang again. Again the silence was unbroken, but from somewhere far off he heard the sound of laughter and talking, and, peering forward into the little hall, he took a small notebook from his pocket and wrote a few words in it, whistling softly between his teeth. He was a freckled-faced young man with a tip-tilted nose, not in the least like the petals of a flower, and with a look of cheery cheekiness. After a moment he went into the passage and thrust his head into the open drawing- room door. The room was filled with flowers, and though the windows were wide open, it smelt close, as if it had already been full of people. The walls were covered with pink and white moiré paper, whose shiny surface was broken by various pictures. Watts's Hope in a gilt frame dominated the mantelpiece; a copy of The Fighting Téméraire faced it, and there were a good many photographs elaborately framed, grouped, like little families, in clusters. Between the windows hung an old, faded photogravure of The Soul's Awakening, and Alone at Last revealed its artless passion over a walnut chiffonier laden with small pieces of china. The young man in the straw hat, which was now pushed far back on his sweat-darkened fair hair, stood in the middle of the room and looked round, scratching his head with his pencil. His bright eyes missed nothing, and
- 68. although he was plainly a young man full of buoyant matter-of-factness, there was scorn, not unkindly, but decided, in his merry but almost porcine eyes as he made mental notes of his surroundings. Poor old girl, he muttered. Hang that 'bus accident. I wish I'd been here in time for the party—— Then his shrewd face softened as the deeper meaning of the room reached him. It was ugly; it was commonplace, but it was more of a home than many a room his journalistic activities had acquainted him with. By a low, shabby, comfortable-looking arm-chair that stood near the flower-filled grate was a dark-covered table on which stood five photographs, all in shiny silver or leather frames. Mr. Wick stood over the table tapping his teeth softly with his pencil, and moving his lips in a way that produced a hollow tune. So that's the little lot, he said to himself in a cheerful, confidential voice. Three feminines and two masculines, as the Italians say. And very nice too. Her own corner, I bet. Yes, there's her fountain pen. He took it up and made a note of its make and laid it carefully down. There was a little fire-screen in the shape of a banner of wool embroidery on the table. That's how she keeps the firelight out of her eyes when she's working in the winter. Poor old girl. What ghastly muck it is, too—— Good thing for her the public likes it. Now, then, what about that bell? Guess I'll go and have another tinkle at it. He started to the door, when it was pushed further open and the owner of the house came in. Mr. Wick knew at the first glance that it was the owner of the house. A fattish, middle-aged man in brand new shepherd's plaid trousers and a not quite so new braided morning-coat. Hallo! I—I beg your pardon—— the new-comer began, not at all in the voice of one who begs pardon. Mr. Wick waved his hand kindly. Oliver Wick's my name, he explained. I come from Round the Fire for an account of the wedding, but I got mixed up with a rather good 'bus smash in Oxford Street, and that's why I'm late. Oh, I see. Want a description of the wedding, do you? Clothes and so on? I'm afraid I'm not much good for that, but if you'll come into the garden I'll get one of my daughters to tell you. Some of the young people are still there, as a matter of fact. Mr. Walbridge had stopped just short of being a tall man. His figure had thickened and spread as he grew older and his hips were disproportionately broad, which gave him a heavy, clumsy look. In his reddish, rather swollen face were traces of what had been great beauty, and he had the unpleasant manner of a man who consciously uses his charm as a means to attain his own ends. Come into the dining-room first and have a glass of the widow, he suggested, as he led the way down the narrow passage towards an open door at the back of the house.
- 69. Mr. Wick, who had no inhuman prejudice against conviviality, followed him into the dining-room and partook, as his quick eyes made notes of everything on which they rested, of a glass of warmish, rather doubtful wine. I suppose Mrs. Walbridge will give me five minutes? the young man asked, setting down his glass and taking a cigarette from the very shiny silver case offered him by his host. Mr. Walbridge laughed, showing the remains of a fine set of teeth artfully reinforced by a skilled dentist. Oh, yes. My wife will quite enjoy being interviewed. Women always like that kind of thing, and, between you and me and the gate-post, he poured some champagne into a tumbler and drank it before he went on, interviewers don't come round quite as they used in her younger days. Mr. Wick despised the novels of the poor lady he had come to interview, but he was a youth not without chivalry, and something in his host's manner irritated him. She has a very good book public, anyhow, has Violet Walbridge. You mustn't mind me calling her that. I shouldn't call Browning Mr. Browning, you know, or Victoria Cross Miss Cross. Walbridge nodded. Oh, yes, they're pretty stories, pretty stories, though I like stronger stuff myself. Just re-reading 'L'Assommoir' again. Met Zola once when I was living in Paris. Always wondered how he smashed his nose. Well, if you're ready, let's come down into the garden where the ladies are. The garden of Happy House was a long narrow strip almost entirely covered by a grass tennis court, and bounded by a narrow, crowded, neglected herbaceous border. As he stood at the top of the steep flight of steps leading down to where the group of young people were sprawled about in dilapidated old deck-chairs or on the grass, Mr. Wick's quick eyes saw the herbaceous border, and, what is more, they understood it. It was a meagre, squeezed, depressed looking attempt, and the young man from Brondesbury knew instinctively that, whereas the tennis court was loved by the young people of the family, the wild and pathetic flowers belonged to the old lady he had come to interview. Somehow he seemed to know, as he told his mother later, quite a lot about Violet Walbridge, just through looking at her border. The sun was setting now, and a little wind had come up, stirring the leaves on the old elm under whose shade, erratic and scant, the little group were seated. Three or four young men were there, splendid, if rather warm, in their wedding garments, and several young women and girls, the pretty pale colours of their fine feathers harmonising charmingly with the evening. At the far end of the garden a lady was walking, with a blue silk sunshade over her shoulder. As the two men came down the steps Mr. Walbridge pointed to her.
- 70. There's my wife, he said. Shall I come and introduce you? No, thank you. No, no, I'll go by myself, the young man answered hastily, and as he went down across the lawn he heard a girl's voice saying laughingly: Reporter to interview Mrs. Jellaby. The others laughed, not unkindly, but their laughter lent to Mr. Wick's approach to Mrs. Walbridge a deference it might otherwise not have had. She had not heard him coming, and was standing with her back to him, her head and shoulders hidden by the delphinium-blue sunshade, and when she turned, starting nervously at the sound of his voice, he realised with painful acuteness that delphinium blue is not the colour to be worn by daylight by old ladies. Her thin, worn face, in which the bones showed more than in any face he had ever seen, was flooded with the blue colour that seemed to fill all the hollows and lines with indigo, and her large sunken eyes, on which the upper eyelids fitted too closely, must have been, the young man noticed, beautiful eyes long ago. They were of that most rare eye-colour, a really dark violet, and the eyebrows on the very edge of the clearly defined frontal bone were slightly arched and well marked over the temples. When he had told her who he was and his errand, she flushed with pleasure and held out her hand to him, and he, whose profession is probably second only to that of dentistry in its unpopularity, was touched by her simple pleasure. My Chief thought the public would be interested in the wedding. He tells me this daughter—the bride, I mean—was the original of—of—one of your chief heroines. Violet Walbridge led the way to an old, faded green garden seat, on which they sat down. Yes, she's the original of 'Rose Parmenter,' she helped him out gently, without offence at his having forgotten the name. I wish you had seen her. But you can say that she was looking beautiful, because she was—— Mr. Wick whipped out his notebook and his beautifully sharpened pencil, contrived a little table of his knees, and looked up at her. 'Rose Parmenter'—oh, yes. That's one of your best-known books, isn't it? Yes, that and 'Starlight and Moonlight.' They sold best, though 'One Maid's Word' has done very well. That, she added slowly, has been done into Swedish, as well as French and German. 'Queenie's Promise' has been done into six languages. Her voice was very low, and peculiarly toneless, but he noticed a little flush of pleasure in her thin cheeks—a flush that induced him, quite unexpectedly to himself, to burst out with the information that a friend of his sister—Jenny her name was—just revelled in his companion's works. Give me a box o' chocs, Kitty
- 71. will say, and one of Violet Walbridge's books, and I wouldn't change places with Queen Mary. Without being urged, Mrs. Walbridge gave the young man details he wanted— that her daughter's name was Hermione Rosalind; that she was the second daughter and the third child, and that she had married a man named Gaskell- Walker—William Gaskell-Walker. He belongs to a Lancashire family, and they've gone to the Lakes for their honeymoon. The author waved her thin hand towards the group of young people at the other end of the lawn. There's the rest of my flock, she said, her voice warming a little. The tall man who's looking at his watch is my other son-in-law, Dr. Twiss of Queen Anne Street, Cavendish Square. He married my eldest daughter, Maud, four years ago. Their little boy was page to-day. He's upstairs asleep now. As she spoke one of the girls in the group left the others and came towards her and Wick. This is your daughter, too? the young man asked, a little throb of pleasure in his voice. Yes, this, Mrs. Walbridge answered, taking the girl's hand, is my baby, Griselda. Grisel, dear, this is Mr.—Mr.—— Wick, said the young man. Oliver Wick. You've come to interview Mum? Miss Walbridge asked, a little good-natured raillery in her voice. The young man bowed. Yes. I represent Round the Fire, and my Chief thought that the public would be interested in an account of the wedding—— His eyes were glued to the young girl's face. She was very small, and, he thought to himself, the blackest white girl he had ever seen; so dark that if he had not known who she was he might have wondered whether she were not the whitest black girl—her hair was coal-black and her long eyes like inkwells, and her skin, smooth as vellum, without a touch of colour, was a rich golden brown. She was charmingly dressed in canary-coloured chiffon, and round her neck she wore a little necklet of twisted strands of seed pearls, from which hung a large, beautifully cut pearl-shaped topaz. I came to tell you, Mum, she went on, glancing over her shoulder at one of the upper windows, that Hilary's awake and bawling his head off, and Maud wants you to go up to him. Mrs. Walbridge rose and Wick noticed, although he could not have explained it, how very different were her grey silk draperies from the yellow ones of her
- 72. daughter. She had, moreover, sat down carelessly, and the back of her frock was crushed and twisted. It's my little grandson, she explained. He's always frightened when he wakes up. I'll go to him. Perhaps you'd like my daughter to show you the wedding presents, Mr. Wick. Oliver Wick was very young, and he was an ugly youth as well, but something about him held the girl's attention, in spite of his being only a reporter. This something, though she did not know it, was power, so it was perfectly natural that the little, spoilt beauty should lead him into the house to the room upstairs where the presents were set forth. His flowery article in the next number of Round the Fire expressed great appreciation of the gifts, but there was no detailed account of them, and that was because, although he looked at them and seemed to see what he was looking at, he really saw nothing but Miss Walbridge's enchanting little face. Do you ever read any of Mum's novels? the girl asked him at last, as they stood by the window, looking down over the little garden into the quiet, tree-bordered road. The young man hesitated, and she burst out laughing, pointing a finger of scorn at him. You've not? she cried. Own up. You needn't mind. I'm sure I don't blame you; they're awful rubbish—poor old Mum! I often wonder who it is does read them. As she finished speaking, the door into the back room opened, and Mrs. Walbridge came out, carrying the little boy who had been crying. His long, fat legs, ending in shiny patent leather slippers, hung limply down, and his towsled fair head leant on her shoulder. He was dressed in cavalier costume of velvet and satin, and his fat, stupid face was blotted and blurred with tears. He looked so very large and heavy, and Mrs. Walbridge looked so small and old and tired that the young man went towards with his arms held out. Let me carry him down for you, he said. He's too heavy—— Griselda laughed. My mother won't let you, she said gaily. She always carries him about. She's much stronger than she looks. Mrs. Walbridge didn't speak, but, with a little smile, went out of the room and slowly downstairs. Her daughter shrugged her shoulders. Mum's not only superannuated as to novels, she announced, smoothing her hair in front of a glass; she's the old-fashioned mother and grandmother. She won't let us do a thing.
- 73. Her bright beauty had already cast a small spell on the young man, but nevertheless he answered her in a flash: Do you ever try? She stared for a moment. In spite of his journalistic manner and what is really best described as his cheek, Oliver Wick was a gentleman, and the girl had instinctively accepted him as such. But at the abrupt, frank censure in his voice she drew herself up and assumed a new manner. Now that you've seen the presents, she said, in what he knew she thought to be a haughty tone, I think I must get back to my friends. He grinned. Righto! Sorry to have detained you. But I haven't quite finished my talk with Mrs. Walbridge. I'm sure she won't mind giving me a few tips about her next book. Our people love that kind of thing—eat it. He cast his eye about the pleasant sunny room, and then, as he reached the door, stopped. I suppose this is your room? he asked, with bland disregard of her manner. What do you mean? Well—different kinds of pictures, you know; brown wallpaper, and that's a good Kakemono. Hanabosa Iccho, isn't it? Miss Walbridge's face expressed surprise too acute to be altogether courteous. I—I don't know, she said. I know it's a very good one. Mother bought it for Paul—that's my brother—he's very fond of such things—for his birthday and at Christmas—his room is being painted, so some of his things are in here. The young man looked admiringly at the grey and white study of monkeys and leaves. I've got an uncle who collects them, he said, and that's a jolly good one. I suppose that Mrs. Walbridge goes in for Japanese art too? Poor mother! The girl laughed. She doesn't know a Kakemono from a broomstick. Paul found that one at some sale and asked her to give it to him. They went slowly down the stairs, the girl's pretty white hand sliding lightly along the polished rail in a way that put all thought of Japanese art out of the young man's active mind. He was going to be a great success, for he had the conquering power of concentrating not only his thoughts but his feelings on one thing at a time; and for the moment the only thing in the world was Griselda Walbridge's left hand.
- 75. CHAPTER II Happy House was a big old house with rooms on both sides of the door, and a good many bedrooms, but it was old-fashioned in the wrong way, like a man's straw hat, say, of the early seventies. It was inconvenient without being picturesque. There was only one bathroom, and the passages were narrow. Most of the children had been born there, indeed all of them except Paul, for the prudent Mrs. Walbridge had bought it out of the proceeds of her first book, Queenie's Promise—a book that is even now dear to thousands of romantic hearts in obscure homes. Paul had been born in the little house at Tooting Bec, for there it was that the Great Success had been written. In those days might have been seen walking under the fine trees of the common, a little dowdy figure with a bustle and flowing unhygienic draperies, that was the newly married Mrs. Ferdinand Walbridge, in the throes of literary invention. But just before the birth of Maud Evelyn the removal had been made; the hastily gathered, inexpensive household gods had been carried by the faithful Carter Paterson to Walpole Road and set up in their over-large, rather dwarfing shrine. Those were the days of limitless ambition and mad, rosy dreams, when Ferdinand was still regarded by his young wife much in the way that Antony Trollope's heroines worshipped their husbands a short time before. The romantic light of the runaway match still hung round him and his extraordinary good looks filled her with unweakened pride. They hung up Mr. Watts's Hope, the beautiful and touching Soul's Awakening (which, indeed, bore a certain resemblance to Walbridge at that time), she arranged her little odds and ends of china, and her few books that her father had sent her after the half-hearted reconciliation following Paul's birth, and one of the first things they bought was a gilt clock, representing two little cupids on a see- saw. Mrs. Walbridge's taste was bad, but it was no worse than the taste of the greater part of her contemporaries of her own class, for she belonged body and soul to the Philistines. She hadn't even an artistic uncle clinging to the uttermost skirts of the pre-Raphaelites to lighten her darkness, and, behold, when she had made it, her little kingdom looked good to her. She settled down light-heartedly and without misgivings, to her quadruple rôle of wife, mother, housekeeper and writer. She had no doubt, the delicate little creature of twenty, but that she could manage and she had been managing ever since. She managed to write those flowery sentimental books of hers in a room full of crawling, experimental, loud- voiced babies; she managed to break in a series of savage handmaidens, who married as soon as she had taught them how to do their work; she managed to make flowers grow in the shabby, weed-grown garden; she managed to mend
- 76. stair-carpets, to stick up fresh wallpapers, to teach her children their prayers and how to read and write; she managed to cook the dinner during the many servantless periods. The fate of her high-born hero and heroine tearing at her tender heart, while that fabulous being, the printer's devil, waited, in a metaphorical sense, on her doorstep. But most of all, she managed to put up with Ferdinand. She had loved him strongly and truly, but she was a clear-sighted little woman, and she could not be fooled twice in the same way, which, from some points of view, is a misfortune in a wife. So gradually she found him out, and with every bit of him that crumbled away, something of herself crumbled too. Nobody knew very much about those years, for she was one of those rare women who have no confidante, and she was too busy for much active mourning. Ferdinand was an expensive luxury. She worked every day and all day, believed in her stories with a pathetic persistence, cherishing all her press notices—she pasted them in a large book, and each one was carefully dated. She had a large public, and made a fairly large, fairly regular income, but there never was enough money, because Walbridge not only speculated and gambled in every possible way, but also required a great deal for his own personal comforts and luxuries. For years it was the joy of the little woman's heart to dress him at one of the classic tailors in Savile Row; his shirts and ties came from a Jermyn Street shop, his boots from St. James's Street, and his gloves (he had very beautiful hands) were made specially for him in the Rue de Rivoli. For many years Ferdinand Walbridge (or Ferdie, as he was called by a large but always changing circle of admiring friends) was one of the most carefully dressed men in town. He had an office somewhere in the city, but his various attempts at business always failed sooner or later, and then after each failure he would settle down gently and not ungratefully to a long period of what he called rest. When the three elder children were eight, six and three, a very bad time had come to Happy House. Little had been known about it except for the main fact that Mr. Walbridge was made a bankrupt. But Caroline Breeze, the only woman who was anything like an intimate friend of the household, knew that there was, over and above this dreadful business, a worse trouble. Caroline Breeze was one of those women who are not unaffectionately called a perfect fool by their friends, but she was a close-mouthed, loyal soul, and had never talked about it to anyone. But years afterwards, when the time had come for her to speak, she spoke, out of her silent observation, to great purpose. For a long time after his bankruptcy Ferdie Walbridge walked about like a moulting bird; his jauntiness seemed to have left him, and without it he wilted and became as nothing. During this three years Mrs. Walbridge for the first time did her writing in the small room in the attic—the small room with the sloping roof and the little view of the tree-tops and sky of which she grew so fond, and which, empty and desolate though it was, had gradually grown to be called the study; and that was
- 77. the time when Caroline Breeze was of such great use to her. For Caroline used to come every day and take the children, as she expressed it, off their mother's hands. In '94 Mrs. Walbridge produced Touchstones, in '95 Under the Elms and in '96 Starlight and Moonlight. It was in '98 that there appeared in the papers a small notice to the effect that Mr. Ferdinand Walbridge was discharged from his bankruptcy, having paid his creditors twenty shillings in the pound. Naturally, after this rehabilitation, Mr. Walbridge became once more his charming and fascinating self, and was the object of many congratulations from the entirely new group of friends that he had gathered round him since his misfortune. Most chaps would have been satisfied to pay fifteen shillings in the pound, more than one of these gentlemen declared to him, and Ferdie Walbridge, as he waved his hand and expressed his failure to comprehend such an attitude, really almost forgot that it was his wife and not himself who had provided the money that had washed his honour clean. Caroline Breeze, faithful and best of friends, lived up three pairs of stairs in the Harrow Road, and one of her few pleasures was the keeping of an accurate and minute record of her daily doings. Perhaps some selections from the diary will help to bring us up to date in the story of Happy House. October, 1894—Tuesday.—Have been with poor Violet. Mr. Walbridge has been most unfortunate, and someone has made him a bankrupt. It is a dreadful blow to Violet, and poor little Hermy only six weeks old. Brought Maud home for the night with me. She's cutting a big tooth. Gave her black currant jam for tea. Do hope the seeds won't disagree with her.... Wednesday.—Not much sleep with poor little Maud. Took her round and got Hermy in the pram, and did the shopping. Saw Mr. Walbridge for a moment. He looks dreadfully ill, poor man. Told me he nearly shot himself last night. I told him he must bear up for Violet's sake.... A week later.—Went to Happy House and took care of the children while Violet was at the solicitors. She looks frightfully ill and changed, somehow. I don't quite understand what it is all about. Several people I know have gone bankrupt, and none of their wives seem as upset as Violet.... November 5th.—Spent the day at Happy House looking after the children. Violet had to go to the Law Courts with Mr. Walbridge. He looked so desperate this morning that I crept in and hid his razors. He dined at the King's Arms with some of his friends, and Violet and I had high tea together. She looks dreadfully ill, and the doctor says she must wean poor little Hermy. She said very little, but I'm afraid she blames poor Mr. Walbridge. I begged her to be gentle with him, and
- 78. she promised she would, but she looked so oddly at me that I wished I hadn't said it. November 20th.—Violet has moved into the top room next the nursery to be nearer the children. I must say I think this is wrong of her. She ought to consider her husband. He looks a little better, but my heart aches for him. February, 1895.—Violet's new book doing very well. Third edition out yesterday. She's getting on well with the one for the autumn. Such a pretty title—Under the Elms. It's about a foundling, which I think is always so sweet. She's very busy making over the children's clothes. Ferdie (he says it is ridiculous that such an intimate friend as I am should go on calling him Mr. Walbridge) has gone to Torquay for a few weeks as he's very run down. Mem.—I lent him ten pounds, as dear Violet really doesn't seem quite to understand that a gentleman needs a little extra money when he's away. He was sweet about her. Told me how very good she was, and said that her not understanding about the pocket money is not her fault, as, of course, she is not quite so well born as he. He is very well connected indeed, though he doesn't care to have much to do with his relations. He's to pay me back when his two new pastels are sold. They are at Jackson's in Oxford Street, and look lovely in the window.... November, 1895.—Violet's new book out to-day—Under the Elms—a sweet story. She gave me a copy with my name in it, and I sat up till nearly two, with cocoa, reading it. Very touching, and made me cry, but has a happy ending. I wish I had such a gift. January 13th, 1896.—Just had a long talk with poor Ferdie. He is really very unlucky. Had his pocket picked on his way home from the city yesterday with £86 15s. 4d. in his purse. Does not wish to tell poor Violet. It would distress her so. He had bought some shares in some kind of mineral—I forget the name—and they had gone up, and he had been planning to buy her a new coat and skirt, and a hat, and lovely presents for all the children. He's such a kind man. He was even going to buy six pairs of gloves for me. The disappointment is almost more than he can bear. Sometimes I think Violet is rather hard on him. I couldn't bear to see him so disappointed, so I am lending him £50 out of the Post Office Savings Bank. He's going to pay me six per cent. It's better than I can get in any other safe investment. He's to pay me at midsummer. N.B.—That makes £60. February 12th, 1896.—Paul's birthday. Went to tea to Happy House. Violet made a beautiful cake with white icing, and had squeezed little pink squiggles all over it in a nice pattern. She gave him a fine new pair of boots and a bath sponge. His daddy gave him a drum—a real one—and a large box of chocolates. February 13th, 1896.—Ferdie came round at seven this morning to ask me to help nurse Paul. He was ill all night with nettle-rash in his throat, and nearly choked, poor little boy. I've been there all day. Susan told me Ferdie's grief in the night
- 79. was something awful. It's a good thing Violet does not take things so to heart. Odd about the chocolate. It seems it's always given him nettle-rash. September 4th, 1896.—Darling Hermy's second birthday. Her mother made her a really lovely coat out of her Indian shawl. I knitted her a petticoat. Dear Ferdie gave her a huge doll with real hair, that talks, and a box of chocolates, which we took away from her, as Paul cried for some. Ferdie had quite forgotten that chocolates poison Paul. He was very wonderful this evening after the children had gone to bed. He had made some money (only a little) by doing some work in the city, and he had bought Violet a lovely pair of seed-pearl earrings. I suppose she was very tired, because she was really quite ungracious about them, and hurt his feelings dreadfully. There was also some trouble about the gas man, which I didn't quite understand. But afterwards, when I had gone upstairs to take a last look at the children, they had a talk, and as I came downstairs I saw him kneeling in front of her with his head in her lap. He has such pretty curly hair, and when I came in he came to me and took my hand and said he didn't mind my seeing his tears, as I was the same as a sister, and asked me to help influence her to forgive him, and to begin over again. It was very touching, and I couldn't help crying a little. I was so sorry for him. Violet is really rather hard. I suggested to her that after all many nice people go bankrupt, and that other women have far worse things to bear, and she looked at me very oddly for a moment, almost as if she despised me, though it can't have been that.... September 30th, 1896.—Have been helping Violet move her things back into the downstairs room. Ferdie was so pleased. He brought home a great bunch of white lilac—in September!—and put it in a vase by the bed. I thought it was a lovely little attention. July 4th, 1897.—A beautiful little boy came home this morning to Happy House. They are going to call him Guy, which is Ferdie's favourite name. He was dreadfully disappointed it wasn't a little girl, so that she could be named Violet Peace. He's so romantic. What a pity there is no masculine name meaning Peace....
- 80. CHAPTER III Mr. Oliver Wick's ideas of courtship were primitive and unshakable. On one or two clever, ingenious pretexts he visited Happy House twice within the month after his first visit, in order, as he expressed it to himself, to look over Miss Walbridge in the light of a possible wife. That he was in love with her he recognised, to continue using his own language, from the drop of the hat, from the first gun. But although he belonged to the most romantic race under the sun, Mr. Wick was no fool, and whereas anything like a help-meet would have displeased him almost to the point of disgust, he had certain standards to which any one with claims to be the future Mrs. Oliver Wick must more or less conform. He didn't care a bit about money—he felt that money was his job, not the girl's—but she'd got to be straight, she'd got to be a good looker, and she'd got to be good-tempered. No shrew-taming for him—at least not in his own domestic circle. One evening, shortly after his third visit to Happy House, the young man was standing at the tallboys in his mother's room in Spencer Crescent, Brondesbury, tying a new tie over an immaculate dress shirt. I'm going to do the trick to-night, he declared, filled with pleasant confidence, or bust. Mrs. Wick, who looked more like her son's grandmother than his mother, sat in a low basket chair by the window, stretching, with an old, thin pair of olive-wood glove stretchers, the new white gloves that were to put the final touch of splendour to the wooer's appearance. She was a pleasant-faced old woman, with a strong chin and keen, clear eyes, and when she smiled she showed traces of past beauty. Well, of course, she said, snapping the glove-stretchers at him thoughtfully, you know everything—you always did—and far be it from me to make any suggestions to you. He turned round, grinning, his ugly face full of subtle likeness to her handsome one. Oh, go on, he jeered, you wonderful old thing! Some day your pictures will be in the penny papers as the mother of Baron Wick of Brondesbury. Of course I know everything! Look at this tie, for instance. A Piccadilly tie, built for dukes, tied in Brondesbury by Fleet Street. What's his name—D'Orsay—couldn't do it better. But what were you going to say?
- 81. She laughed and held out the gloves. Here you are, son. Only this. I bet you sixpence she won't look at you. She'll turn you down; refuse you; give you the cold hand; icy mit—what d'you call it? And then, you'll come back and weep on my shoulder. Mr. Wick, who had taken the gloves, stood still for a minute, his face full of sudden thought. She may, he said, she may. I don't care if she does. I tell you she's lovely, mother. She'd look like a fairy queen if the idiots who paint 'em realised that fairies ought to be dark, and not tow-coloured. Of course she'll refuse me a few times, but her father'll be on my side. Why? Because he's a rather clever old scoundrel, and he'll know that I'm a succeeder— a getter. The old woman looked thoughtful. I haven't liked anything you told me about him, Olly. But, after all, he has paid up, and lots of good men have been unfortunate in business. The young fellow took up his dress-coat, which was new and richly lined, and drew it on with care. Oh, I'm not marrying into this family because I admire my future father-in-law, he answered. I haven't any little illusions about him, old lady. It's his wife who's done the paying, or I'm very much mistaken. She's an honest woman—poor thing. There was such deep sympathy in his voice that his mother, who had risen, and was patting and smoothing the new coat into place on his broad shoulders, pulled him round till he faced her, and looked down at him, for she was taller than he. Why are you so sorry for her? He hesitated for a moment, and his hesitation meant much to her. I don't know. She never says anything, of course. She seems happy enough, but I believe—I believe she's found him out—— God help her, Mrs. Wick answered. The young man remembered this episode as he sat opposite his hostess at dinner an hour and a half later. The dining-room had been re-papered since he had drunk that glass of luke-warm wine in it the day of Hermione's wedding, and his sharp eyes noticed the absence of several ugly things that had been there then. Stags no longer hooted to each other across mountain chasms over the sideboard, and one or two good line drawings hung in their place.
- 82. How do you like it? Griselda asked him. Paul and I have been cheering things up a bit. Splendid, he replied promptly. I say, how beautiful your sister is! Griselda's rather hard little face softened charmingly as she looked across the table, where the bride was sitting. Hermione Gaskell-Walker was a very handsome young woman in an almost classical way, and her short-sighted, clever-looking husband, who sat nearly opposite her, evidently thought so too, for he peered over the flowers at her in adoration that was plain and pleasing to see. They've such a jolly house in Campden Hill. His father was Adrian Gaskell-Walker, the landscape painter, and collected things. Mr. Wick nodded, but did not answer, for he was busy making a series of those mental photographs, whose keenness and durability so largely contributed to his success in life. He had an amazing power of storing up records of incidents that somehow or other might come in useful to him, and this little dinner party, which he had decided to be a milestone on his road, interested him acutely in its detail. By candlelight, in perfect evening dress, Ferdinand Walbridge's slightly dilapidated charms were very manifest. On his right sat an elderly lady about whom Mr. Wick's apparatus recorded only one word—pearls. Next to her came Paul Walbridge, looking older than his twenty-nine years—thin, delicate, rather high shouldered, with remarkably glossy dark hair and immense soft, dove-coloured eyes. He looked far better bred, the young man decided, than he had any right to look; his hands, in particular, might have been modelled by Velasquez. Supercilious—— Wick thought, and then paused, not adding the ass that had come into his mind, for he knew that Paul Walbridge was not an ass, although he would have liked to call him one. Next Paul came the beautiful Hermione, with magnificent shoulders white as flour, and between her and her mother sat a man named Walter Crichell, a portrait painter, one of the best in the secondary school—a man with over-red lips and short white hands with unpleasant, pointed fingers. That fellow's a stinker, Wick decided, never to change his mind. Next came the hostess, thin, worn, rather silent, in the natural isolation of an old woman sitting between two young men, each of whom had youth and beauty on his far side. Then, of course, came Oliver himself and Grisel. Next to Grisel, Gaskell-Walker, the lower part of whose face was clever, but who would probably find himself handicapped by the qualities belonging to too high, too straight a forehead; and
- 83. next him, consequently on the host's left, sat Crichell's wife. Young Wick could not look at her very comfortably without leaning forward, but he caught one or two glimpses of her face as Walbridge bent over her, and promised himself a good look in the drawing-room. She was worth it, he knew. A soft, velvety brown creature, a little on the fat side, but rather beautiful. It was plain, too, that the old man admired her. Mr. Wick studied his host's face for a moment as he thus completed his circle of observation, and so strong were his feelings as he looked at Mr. Walbridge that quite unintentionally he said Ugh! aloud. What did you say? It was Mrs. Walbridge who spoke—her first remark for quite a quarter of an hour—and in her large eyes was the anxious, guilty look of one who has allowed herself to wool-gather in public. Wick started, blushed scarlet, and then burst out laughing at his dilemma. I didn't say anything, he answered. I was only thinking. I beg your pardon, Mrs. Walbridge. Her worn face softened into a kind smile, and he noticed that her teeth were even and very white. It is awful, isn't it, she said, to—to get thinking about things when one ought to be talking? I'm afraid I'm very dull for a young man to sit next. Oh, come, Mrs. Walbridge, he protested, when you know how they all lapped up that article I wrote about you. She bridled gently. It was a very nice article. After a minute she added anxiously, her thin fingers pressing an old blue enamel brooch that fastened the rather crumpled lace at her throat: Tell me, Mr. Wick, do you—do you really think that—that people like my books as much as they used to? You must have a very big public, he answered, wishing she had not put the question. Yes, I know I have, but—you see, of course I'm not young any more, and the children—they know a great many people, and bring some of them here and—I've noticed that while they are all very kind, they don't seem to have—to have really read my books. Don't they? said Wick, full of sympathy. Dear me! She shook her head. No, they really don't, and I've been wondering if—if it is that they're beginning to find me—a little old-fashioned. What he wanted to say in return for this was: But, bless your heart, you are old- fashioned, the old-fashionest old dear that ever lived! What he did say was: Well, I suppose lots of people think Thackeray and Dickens old-fashioned——
- 84. Welcome to our website – the perfect destination for book lovers and knowledge seekers. We believe that every book holds a new world, offering opportunities for learning, discovery, and personal growth. That’s why we are dedicated to bringing you a diverse collection of books, ranging from classic literature and specialized publications to self-development guides and children's books. More than just a book-buying platform, we strive to be a bridge connecting you with timeless cultural and intellectual values. With an elegant, user-friendly interface and a smart search system, you can quickly find the books that best suit your interests. Additionally, our special promotions and home delivery services help you save time and fully enjoy the joy of reading. Join us on a journey of knowledge exploration, passion nurturing, and personal growth every day! ebookbell.com