On open real polynomial maps
type: Item
instance of: scholarly article
title: On open real polynomial maps
zbMATH Open document ID: 0874.14051
DOI: 10.1016/0022-4049(95)00106-9
author: Felice Ronga
author: Jose Manuel Gamboa
published in: Journal of Pure and Applied Algebra
publication date: 1997-11-10T00:00:00Z
review text: It is well known that a polynomial map \(f:\mathbb{C}^n\to \mathbb{C}^p\) is open if and only if its fibers have pure dimension \(n-p\). In this paper we prove the following result: Theorem. Let \(f:\mathbb{R}^n\to \mathbb{R}^n\) be a polynomial map. Denote by \(J(f):\mathbb{R}^n\to \mathbb{R}\) the jacobian of \(f\). The following statements are equivalent: (1) \(f\) is an open map. (2) The fibers of \(f\) are finite and the sign of \(J(f)\) does not change. (3) The fibers of \(f\) are finite and the sign of \(J(f)(x)\) does not change on the fibers of \(f\). Remark. The theorem and its proof remain valid if \(f\) is assumed to be a Nash map. The proof of this theorem will occupy section 1. -- In section 2 we take a first step of open polynomial maps between spaces of different dimension: We prove that an open map \(f:\mathbb{R}^n\to \mathbb{R}^2\), \(n\geq 3\) cannot have zero-dimensional fibers.
Mathematics Subject Classification ID: 14P99
Mathematics Subject Classification ID: 14E99
Mathematics Subject Classification ID: 14P20
zbMATH DE Number: 896576
zbMATH Keywords: open real polynomial maps
zbMATH Keywords: Nash map
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00106-9
OpenAlex ID: W1984684067
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description: scientific article; zbMATH DE number 896576