REWRITE 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,

9. THANKYOU FOR READING!