On locally invertible systems of polynomials
type: Item
instance of: scholarly article
title: On locally invertible systems of polynomials
zbMATH Open document ID: 0819.13006
author: Kossivi Adjamagbo
published in: Journal of Algebra
publication date: 1994-09-27T00:00:00Z
review text: Let \(K\) be a commutative integral domain, \(K^*\) its group of units and \(F = (F_ 1, \dots, F_ n) \in K[X]^ n\), \(X = (X_ 1, \dots, X_ n)\). \(F\) is called locally invertible if for every point \(a \in K^ n\) the system of polynomials \(F_ a (X) : = F(X + a) - F(a)\) has a formal inverse, i.e. there exists \(G_ a \in K X^ n\) such that \(F_ a \circ G_ a = G_ a \circ F_ a = X\) (this is equivalent to the jacobian condition, i.e. \(\text{Jac} (F) (a) \in K^*\) for every \(a \in K^ n)\). The main theorem of the paper is the following upper bound for the cardinality of the fibers of \(F\): if \(K\) is of characteristic zero and \(F \in K [X]^ n\) is locally invertible, then the field \(K(X)\) is a finite algebraic extension of the field \(K(F)\) and for any point \(b \in K^ n\) holds: \(\# F^{-1} (b) \leq [K(X) : K(F(X))]\).
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 14H37
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 645068
zbMATH Keywords: polynomial map
zbMATH Keywords: formal series
zbMATH Keywords: integral domain
zbMATH Keywords: formal inverse
reviewed by: Tadeusz Krasiński
MaRDI profile type: Publication
full work available at URL: jabr.1994.1218
OpenAlex ID: W1977605542
DOI: 10.1006/JABR.1994.1218
description: scientific article; zbMATH DE number 645068