On the inversion formula for two polynomials in two variables (Q1115485)

Let \({\mathbb{C}}\) denote the complex numbers and let f and g be two polynomials in two variables over \({\mathbb{C}}\) with zero constant terms. Assume that the system \(z=f(x,y)\), \(w=g(x,y)\) has a unique solution (x,y) for each \(z,w\in {\mathbb{C}}\). In a previous paper the authors derived an inversion formula which expresses x and y as polynomials in z and w, and their formula only involves the face polynomials f(x,0), f(0,y), g(x,0) and g(0,y). In this paper it is shown that for \(n\geq 2\), a \({\mathbb{C}}\)-isomorphism \(\phi\) of the polynomial ring \({\mathbb{C}}[X_ 1,...,X_ n]={\mathbb{C}}[X]\) is completely determined by the face polynomials of the \(\phi (X_ i)\). The proof given does not use the inversion formula, and has as a consequence a simpler proof of this formula. Various conditions for a family of polynomials to be the set of face polynomials of an automorphism of \({\mathbb{C}}(X]\) are also investigated. For this last question the emphasis is on the two variable case where conditions are given in terms of the inversion and Newton polygons.