"Combinatorics of 4-dimensional resultant polytopes"

1. Combinatorics of 4-dimensional Resultant Polytopes Vissarion Fisikopoulos Joint work with Alicia Dickenstein (U. Buenos Aires) & Ioannis Z. Emiris (UoA) Dept. of Informatics & Telecommunications, University of Athens ISSAC 2013

2. Resultant polytopes Algebra: generalization of the resultant polynomial degree Geometry: Minkowski summands of secondary polytopes Applications: support computation → discriminant and resultant computation

3. Polytopes and Algebra Given n + 1 polynomials on n variables. Supports (set of exponents of monomials with non-zero coeﬃcient) A0, A1, . . . , An ⊂ Zn . The resultant R is the polynomial in the coeﬃcients of a system of polynomials which vanishes if there exists a common root in the torus of the given polynomials. The resultant polytope N(R), is the convex hull of the support of R, i.e. the Newton polytope of the resultant. f0(x) = ax2 + b f1(x) = cx2 + dx + e

4. Polytopes and Algebra Given n + 1 polynomials on n variables. Supports (set of exponents of monomials with non-zero coeﬃcient) A0, A1, . . . , An ⊂ Zn . The resultant R is the polynomial in the coeﬃcients of a system of polynomials which vanishes if there exists a common root in the torus of the given polynomials. The resultant polytope N(R), is the convex hull of the support of R, i.e. the Newton polytope of the resultant. A0 A1 f0(x) = ax2 + b f1(x) = cx2 + dx + e

5. Polytopes and Algebra Given n + 1 polynomials on n variables. Supports (set of exponents of monomials with non-zero coeﬃcient) A0, A1, . . . , An ⊂ Zn . The resultant R is the polynomial in the coeﬃcients of a system of polynomials which vanishes if there exists a common root in the torus of the given polynomials. The resultant polytope N(R), is the convex hull of the support of R, i.e. the Newton polytope of the resultant. A0 A1 R(a, b, c, d, e) = ad2 b + c2 b2 − 2caeb + a2 e2 f0(x) = ax2 + b f1(x) = cx2 + dx + e

6. Polytopes and Algebra Given n + 1 polynomials on n variables. Supports (set of exponents of monomials with non-zero coeﬃcient) A0, A1, . . . , An ⊂ Zn . The resultant R is the polynomial in the coeﬃcients of a system of polynomials which vanishes if there exists a common root in the torus of the given polynomials. The resultant polytope N(R), is the convex hull of the support of R, i.e. the Newton polytope of the resultant. A0 A1 N(R)R(a, b, c, d, e) = ad2 b + c2 b2 − 2caeb + a2 e2 f0(x) = ax2 + b f1(x) = cx2 + dx + e

7. Polytopes and Algebra The case of linear polynomials A0 A1 N(R) A2 4-dimensional Birkhoff polytope f0(x, y) = ax + by + c f1(x, y) = dx + ey + f f2(x, y) = gx + hy + i a b c d e f g h i R(a, b, c, d, e, f, g, h, i) =

8. Polytopes and Algebra A0 A1 Q: How N(R) looks like in the general case A2 f0(x, y) = axy2 + x4 y + c f1(x, y) = dx + ey f2(x, y) = gx2 + hy + i

9. Resultant polytopes: Motivation Algebra: useful to express the solvability of polynomial systems, generalizes the notion of the degree of the resultant Geometry: Minkowski summands of secondary polytopes, equival. classes of secondary vertices, generalize Birkhoﬀ polytopes Applications: support computation → discriminant and resultant computation, implicitization of parametric hypersurfaces

10. Existing work [GKZ’90] Univariate case / general dimensional N(R) [Sturmfels’94] Multivariate case / up to 3 dimensional N(R)

11. Existing work [GKZ’90] Univariate case / general dimensional N(R) [Sturmfels’94] Multivariate case / up to 3 dimensional N(R)

12. One step beyond... 4-dimensional N(R) Polytope P ⊆ R4 ; f-vector is the vector of its face cardinalities. Call vertices, edges, ridges, facets, the 0,1,2,3-d, resp., faces of P. f-vectors of 4-dimensional N(R) (5, 10, 10, 5) (6, 15, 18, 9) (8, 20, 21, 9) (9, 22, 21, 8) . . . (17, 49, 48, 16) (17, 49, 49, 17) (17, 50, 50, 17) (18, 51, 48, 15) (18, 51, 49, 16) (18, 52, 50, 16) (18, 52, 51, 17) (18, 53, 51, 16) (18, 53, 53, 18) (18, 54, 54, 18) (19, 54, 52, 17) (19, 55, 51, 15) (19, 55, 52, 16) (19, 55, 54, 18) (19, 56, 54, 17) (19, 56, 56, 19) (19, 57, 57, 19) (20, 58, 54, 16) (20, 59, 57, 18) (20, 60, 60, 20) (21, 62, 60, 19) (21, 63, 63, 21) (22, 66, 66, 22)

15. Computation of resultant polytopes respol software [Emiris-F-Konaxis-Pe˜naranda ’12] lower bounds C++, CGAL (Computational Geometry Algorithms Library) http://sourceforge.net/projects/respol Alternative algorithm that utilizes tropical geometry (GFan library) [Jensen-Yu ’11]

16. Computation of resultant polytopes respol software [Emiris-F-Konaxis-Pe˜naranda ’12] lower bounds C++, CGAL (Computational Geometry Algorithms Library) http://sourceforge.net/projects/respol Alternative algorithm that utilizes tropical geometry (GFan library) [Jensen-Yu ’11]

17. Main result Theorem Given A0, A1, . . . , An ⊂ Zn with N(R) of dimension 4. Then N(R) are degenerations of the polytopes in following cases. (i) All |Ai| = 2, except for one with cardinality 5, is a 4-simplex with f-vector (5, 10, 10, 5). (ii) All |Ai| = 2, except for two with cardinalities 3 and 4, has f-vector (10, 26, 25, 9). (iii) All |Ai| = 2, except for three with cardinality 3, maximal number of ridges is ˜f2 = 66 and of facets ˜f3 = 22. Moreover, 22 ≤ ˜f0 ≤ 28, and 66 ≤ ˜f1 ≤ 72. The lower bounds are tight. Degenarations can only decrease the number of faces. Focus on new case (iii), which reduces to n = 2 and each |Ai| = 3. Previous upper bound for vertices yields 6608 [Sturmfels’94].

22. Tool (1): N(R) faces and subdivisions A subdivision S of A0 + A1 + · · · + An is mixed when its cells have expressions as Minkowski sums of convex hulls of point subsets in Ai’s. Example mixed subdivision S of A0 + A1 + A2 A0 A1 A2 Proposition (Sturmfels’94) A regular mixed subdivision S of A0 + A1 + · · · + An corresponds to a face of N(R) which is the Minkowski sum of the resultant polytopes of the cells (subsystems) of S.

23. Tool (1): N(R) faces and subdivisions A subdivision S of A0 + A1 + · · · + An is mixed when its cells have expressions as Minkowski sums of convex hulls of point subsets in Ai’s. Example mixed subdivision S of A0 + A1 + A2 A0 A1 A2 Proposition (Sturmfels’94) A regular mixed subdivision S of A0 + A1 + · · · + An corresponds to a face of N(R) which is the Minkowski sum of the resultant polytopes of the cells (subsystems) of S.

24. Tool (1): N(R) faces and subdivisions Example white, blue, red cells → N(R) vertex purple cell → N(R) segment turquoise cell → N(R) triangle Mink. sum of N(R) triangle and N(R) segmentsubd. S of A0 + A1 + A2

27. Tool (2): Input genericity Proposition Input genericity maximizes the number of resultant polytope faces. Proof idea N(R∗ ) f-vector: (18, 52, 50, 16) N(R) f-vector: (14, 38, 36, 12) p p∗ A0 A1 A2 A0 A1 A2

28. Tool (2): Input genericity Proposition Input genericity maximizes the number of resultant polytope faces. Proof idea N(R∗ ) f-vector: (18, 52, 50, 16) N(R) f-vector: (14, 38, 36, 12) p p∗ A0 A1 A2 A0 A1 A2 → For upper bounds on the number of N(R) faces consider generic inputs, i.e. no parallel edges.

29. Facets of 4-d resultant polytopes Lemma All the possible types of N(R) facets are resultant facet: 3-d N(R) prism facet: 2-d N(R) (triangle) + 1-d N(R) cube facet: 1-d N(R) + 1-d N(R) + 1-d N(R) 3D

30. Facets of 4-d resultant polytopes Lemma All the possible types of N(R) facets are resultant facet: 3-d N(R) prism facet: 2-d N(R) (triangle) + 1-d N(R) cube facet: 1-d N(R) + 1-d N(R) + 1-d N(R) 3D 2D

31. Counting facets Lemma There can be at most 9, 9, 4 resultant, prism, cube facets, resp., and this is tight. Proof idea Unique subdivision that corresponds to 4 cube facets

32. Faces of 4-d resultant polytopes Lemma The maximal number of ridges of N(R) is ˜f2 = 66. Moreover, ˜f1 = ˜f0 + 44, 22 ≤ ˜f0 ≤ 28, and 66 ≤ ˜f1 ≤ 72. The lower bounds are tight. Elements of proof [Kalai87] f1 + i≥4 (i − 3)fi 2 ≥ df0 − d + 1 2 , where fi 2 is the number of 2-faces which are i-gons.

33. Open problems & a conjecture Open The maximum f-vector of a 4d-resultant polytope is (22, 66, 66, 22). Open Explain symmetry of f-vectors of 4d-resultant polytopes. Conjecture f0(d) ≤ 3 · S =d−1 i∈S ˜f0(i) where S is any multiset with elements in {1, . . . , d − 1}, S := i∈S i, and ˜f0(i) is the maximum number of vertices of a i-dimensional N(R). The only bound in terms of d is (3d − 3)2d2 [Sturmfels’94], yielding ˜f0(5) ≤ 1250 whereas our conjecture yields ˜f0(5) ≤ 231.

38. Thank you!

"Combinatorics of 4-dimensional resultant polytopes"

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"Combinatorics of 4-dimensional resultant polytopes"