Automorphisms from face polynomials via two Gröbner bases
type: Item
instance of: scholarly article
title: Automorphisms from face polynomials via two Gröbner bases
zbMATH Open document ID: 0805.13009
DOI: 10.1016/0022-4049(92)90010-D
published in: Journal of Pure and Applied Algebra
publication date: 1993-04-01T00:00:00Z
review text: It is now a well-known fact that a polynomial automorphism \(F = (F_ 1, \dots, F_ n) : k^ n \to k^ n\), where \(k\) is a field, \(n \geq 2\) is completely determined by its face polynomials \(F_ i |_{X_ j = 0}\), \(i,j=1,\dots,n\). Several proofs were given, by McKay and Wang and Li. Explicit formulae to reconstruct the automorphism for the case \(n=2\) were given by McKay and Wang and Adjamagbo and van den Essen. In J. Pure Appl. Algebra 80, No. 3, 327-336 (1992; Zbl 0763.14004), \textit{A. van den Essen} and the author show that these formulae do not always work in higher dimensions. Instead, an algorithm for reconstructing the inverse, based on a theorem from \textit{A. van den Essen} [Commun. Algebra 18, No. 10, 3183-3186 (1990; Zbl 0718.13008)], is given there. That algorithm involves \(n+1\) Gröbner basis computations. The main theorem presented here expresses the inverse of \(F\) in terms of the reduced Gröbner basis of an ideal constructed from face polynomials. Thus, it leads to an algorithm for the computation of \(F\) which involves only two Gröbner basis computations.
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 14H37
zbMATH DE Number: 149934
zbMATH Keywords: inverse of polynomial automorphism
zbMATH Keywords: Gröbner basis
zbMATH Keywords: face polynomials
author: Michal Kwienciński
MaRDI profile type: Publication
cites work: A resultant criterion and formula for the inversion of a polynomial map in two variables
cites work: Q3714165
cites work: A Gröbner basis criterion for isomorphisms of algebraic varieties
cites work: On a problem about face polynomials
cites work: An inversion formula for two polynomials in two variables
cites work: On the inversion formula for two polynomials in two variables
cites work: On face polynomials
cites work: The theory of Gröbner bases
cites work: A criterion to decide if a polynomial map is invertible and to compute the inverse
cites work: Q3975240
cites work: On the reconstruction of polynomial automorphisms from their face polynomials
full work available at URL: 0022-4049(92)90010-d
OpenAlex ID: W2002345180
description: scientific article; zbMATH DE number 149934