Algebraic geometry zero set of polynomials
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REWRITE ZERO SET OF POLYNOMIALS!
![𝑘𝑒𝑟 𝑓 𝑥1, … , 𝑥 𝑛 ↦ 𝑓(𝑎1, … , 𝑎 𝑛)
= 𝑓 ∈ 𝑘 𝑥1, … , 𝑥 𝑛 | 𝑓 𝑎1, … , 𝑎 𝑛 = 0
= 𝑥1 − 𝑎1, … , 𝑥 𝑛 − 𝑎 𝑛 ∈ 𝑆𝑝𝑚(𝑘[𝑥1, … , 𝑥 𝑛])](https://image.slidesharecdn.com/algebraicgeometryzerosetofpolynomials-151214134141/85/Algebraic-geometry-zero-set-of-polynomials-3-320.jpg)
![𝑘𝑒𝑟 𝑓 𝑥1, … , 𝑥 𝑛 ↦ 𝑓(𝑎1, … , 𝑎 𝑛)
= 𝑓 ∈ 𝑘 𝑥1, … , 𝑥 𝑛 | 𝑓 𝑎1, … , 𝑎 𝑛 = 0
= 𝑥1 − 𝑎1, … , 𝑥 𝑛 − 𝑎 𝑛 ∈ 𝑆𝑝𝑚(𝑘[𝑥1, … , 𝑥 𝑛])](https://image.slidesharecdn.com/algebraicgeometryzerosetofpolynomials-151214134141/85/Algebraic-geometry-zero-set-of-polynomials-4-320.jpg)
![𝑘𝑒𝑟 𝑓 𝑥1, … , 𝑥 𝑛 ↦ 𝑓(𝑎1, … , 𝑎 𝑛)
= 𝑓 ∈ 𝑘 𝑥1, … , 𝑥 𝑛 | 𝑓 𝑎1, … , 𝑎 𝑛 = 0
= 𝑥1 − 𝑎1, … , 𝑥 𝑛 − 𝑎 𝑛 ∈ 𝑆𝑝𝑚(𝑘[𝑥1, … , 𝑥 𝑛])
∵
𝑓 𝑥1, … , 𝑥 𝑛 ↦ 𝑓 𝑎1, … , 𝑎 𝑛
𝑘𝑒𝑟 𝑓 𝑥1, … , 𝑥 𝑛 ↦ 𝑓 𝑎1, … , 𝑎 𝑛
≅ 𝑘, (𝑘 𝑖𝑠 𝒇𝒊𝒆𝒍𝒅)](https://image.slidesharecdn.com/algebraicgeometryzerosetofpolynomials-151214134141/85/Algebraic-geometry-zero-set-of-polynomials-5-320.jpg)
![REWRITE ZERO SET OF POLYNOMIALS!
𝑉 𝑓 = 𝑎1, . . , 𝑎 𝑛 ∈ 𝑘 𝑛 𝑓 𝑎1, . . , 𝑎 𝑛 = 0}
= 𝑀 ∈ 𝑆𝑝𝑚(𝐴) 𝑓 ⊂ 𝑀}
𝑊ℎ𝑒𝑟𝑒 𝑀 = 𝑥1 − 𝑎1, … , 𝑥 𝑛 − 𝑎 𝑛 , 𝐴 = 𝑘[𝑥1, … , 𝑥 𝑛]](https://image.slidesharecdn.com/algebraicgeometryzerosetofpolynomials-151214134141/85/Algebraic-geometry-zero-set-of-polynomials-7-320.jpg)
Algebraic geometry zero set of polynomials
- 1. ALGEBRAIC GEOMETRY ZERO SET OF POLYNOMIALS HANPEN ROBOT 2015/12/14 (MONDAY) IN HANOI
- 2. 𝑓 𝑥1, … , 𝑥 𝑛 ↦ 𝑓(𝑎1, … , 𝑎 𝑛) 𝑎1, … , 𝑎 𝑛 ∈ 𝑘, 𝑤ℎ𝑒𝑟𝑒 𝑘 𝑖𝑠 𝑓𝑖𝑒𝑙𝑑. CONSIDER THE FOLLOWING HOMOMORPHISM MAP
- 3. 𝑘𝑒𝑟 𝑓 𝑥1, … , 𝑥 𝑛 ↦ 𝑓(𝑎1, … , 𝑎 𝑛) = 𝑓 ∈ 𝑘 𝑥1, … , 𝑥 𝑛 | 𝑓 𝑎1, … , 𝑎 𝑛 = 0 = 𝑥1 − 𝑎1, … , 𝑥 𝑛 − 𝑎 𝑛 ∈ 𝑆𝑝𝑚(𝑘[𝑥1, … , 𝑥 𝑛])
- 4. 𝑘𝑒𝑟 𝑓 𝑥1, … , 𝑥 𝑛 ↦ 𝑓(𝑎1, … , 𝑎 𝑛) = 𝑓 ∈ 𝑘 𝑥1, … , 𝑥 𝑛 | 𝑓 𝑎1, … , 𝑎 𝑛 = 0 = 𝑥1 − 𝑎1, … , 𝑥 𝑛 − 𝑎 𝑛 ∈ 𝑆𝑝𝑚(𝑘[𝑥1, … , 𝑥 𝑛])
- 5. 𝑘𝑒𝑟 𝑓 𝑥1, … , 𝑥 𝑛 ↦ 𝑓(𝑎1, … , 𝑎 𝑛) = 𝑓 ∈ 𝑘 𝑥1, … , 𝑥 𝑛 | 𝑓 𝑎1, … , 𝑎 𝑛 = 0 = 𝑥1 − 𝑎1, … , 𝑥 𝑛 − 𝑎 𝑛 ∈ 𝑆𝑝𝑚(𝑘[𝑥1, … , 𝑥 𝑛]) ∵ 𝑓 𝑥1, … , 𝑥 𝑛 ↦ 𝑓 𝑎1, … , 𝑎 𝑛 𝑘𝑒𝑟 𝑓 𝑥1, … , 𝑥 𝑛 ↦ 𝑓 𝑎1, … , 𝑎 𝑛 ≅ 𝑘, (𝑘 𝑖𝑠 𝒇𝒊𝒆𝒍𝒅)
- 6. ∴ 𝑓 𝑎1, … , 𝑎 𝑛 = 0 ⇔ 𝑓 ∈ 𝑥1 − 𝑎1, … , 𝑥 𝑛 − 𝑎 𝑛
- 7. REWRITE ZERO SET OF POLYNOMIALS! 𝑉 𝑓 = 𝑎1, . . , 𝑎 𝑛 ∈ 𝑘 𝑛 𝑓 𝑎1, . . , 𝑎 𝑛 = 0} = 𝑀 ∈ 𝑆𝑝𝑚(𝐴) 𝑓 ⊂ 𝑀} 𝑊ℎ𝑒𝑟𝑒 𝑀 = 𝑥1 − 𝑎1, … , 𝑥 𝑛 − 𝑎 𝑛 , 𝐴 = 𝑘[𝑥1, … , 𝑥 𝑛]
- 8. REWRITE ZERO SET OF POLYNOMIALS! 𝑉 𝑓1, … , 𝑓𝑚 = 𝑀 ∈ 𝑆𝑝𝑚(𝐴) 𝑓1, … , 𝑓𝑚 ⊂ 𝑀} 𝐼𝑛 𝑔𝑒𝑛𝑒𝑟𝑎𝑙,