An elementary proof of the automorphism theorem for the polynomial ring in two variables

AbstractIf k is a field, then the automorphism theorem for k[x, y] states that every k-algebra automorphism of k[x, y] is a finite compositional product of automorphisms of the type: (i) x↦x, y↦y+h(x) with h(x) ∈ k[x]; or (ii) x↦a11x+a12y+a13, y↦a21x+a22y+a23 with a11a22≠a21a12 and aij∈k. The proof presented in this paper, in the case of k being the complex numbers, uses the resultant formula for the inverse of an automorphism (McKay and Wang) and a differential equation associated with the fact that the Jacobian of the automorphism is a nonzero constant. Then the required divisibility condition on the degrees of the images of x and y follows from the fact that this differential equation must have a rational function solution