Let f1,…,fr be complex polynomials in the variables x1,…,xn let V be the variety of their common zeros, and let I be the ideal of the polynomial ring

Yasmin

Yasmin

Answered question

2021-06-03

Let f1,…,fr be complex polynomials in the variables x1,…,xn let V be the variety of their common zeros, and let I be the ideal of the polynomial ring R=C[x1,,xn] that they generate. Define a homomorphism from the quotient ring R=R x2F; I to the ring RR of continuous, complex-valued functions on V.

Answer & Explanation

i1ziZ

i1ziZ

Skilled2021-06-04Added 92 answers

Every polynomial with m variables and complex coefficients can be considered a continuous function C">©. Wecan also restrict its domain to V! Denote C=VC for better visibility. So we first define a. mapping
ϕ:C[x1xn]C
given by ϕ(f(x1...xn))=f.It is clearly a homomorphism. Now we want to prove that IK, where K=kerϕ.
Let f(x1...xn) I. Then for every (y1,...,yn) V we have that f(y1,...,yn)=0 by the definition of V. But this means that
ϕ(f([x1xn]))=0C
(the null function)! Therefore, f([x1...xn])  K, so
IK
By the Theorem 11.4.2 (a) we conclude that there exists « (unique) homomorphism ϕ:C[x1xn]IC.

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