Let u,v in C[x,y] and assume that (u,v) is a maximal ideal of C[x,y]. Question: Is it true that there exists an automorphism g of C[x,y] s.t. u=g(x), v=g(y)?
Recall that if I is a maximal ideal of C[x,y], then by Hilbert's Nullstellensatz I=(x-a,y-b), for some a,b in C. Therefore here (u,v)=(x-a,y-b) for some a,b in C. Then u=A(x-a)+B(y-b) and v=D(x-a)+E(y-b), for some A,B,D,E in C[x,y] and these A,B,D,E satisfy AE-BD is a scalar in C, for convenience assume it is 1. Now I guess we can somehow construct an automorphism g, in terms of A,B,D,E, such that u=g(x) and v=g(y). It seems that AE-BD should be the Jacobian of g. We know that x-a=Eu-Bv and y-b=-Du+Av.
Thank you very much!
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