On the inversion formula for two polynomials in two variables
type: Item
instance of: scholarly article
title: On the inversion formula for two polynomials in two variables
zbMATH Open document ID: 0664.13001
DOI: 10.1016/0022-4049(88)90138-7
author: James H. McKay
author: Stuart Sui-Sheng Wang
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: 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.
reviewed by: David E. Rush
Mathematics Subject Classification ID: 13B10
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 12D99
zbMATH DE Number: 4085779
zbMATH Keywords: automorphism of complex polynomial ring
zbMATH Keywords: face polynomials
zbMATH Keywords: Newton polygons
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90138-7
OpenAlex ID: W4210556641
cites work: Q4097032
cites work: An inversion formula for two polynomials in two variables
cites work: An elementay proof of the automorphism theorem for the polynomial ring in two variables
cites work: On face polynomials
description: scientific article; zbMATH DE number 4085779