Testing sets for properness of polynomial mappings
type: Item
instance of: scholarly article
title: Testing sets for properness of polynomial mappings
zbMATH Open document ID: 0946.14039
author: Z. Jelonek
published in: Mathematische Annalen
publication date: 2000-10-22T00:00:00Z
review text: Fix \(m,n\in \mathbb{N}\). A set \(S\subset \mathbb{C}^{m}\) is a testing set for polynomial mappings \(f:\mathbb{C}^{n}\rightarrow \mathbb{C}^{m}\) if and only if for every polynomial mapping \(f:\mathbb{C}^{n}\rightarrow \mathbb{C} ^{m}\) if \(f|f^{-1}(S):f^{-1}(S)\rightarrow S\) is proper then \(f\) is proper. The first aim of the author is to study testing sets. He gives many characterizations and concrete classes of testing sets, even in the more general case of arbitrary affine varieties (instead of \(\mathbb{C}^{n}\) and \( \mathbb{C}^{m}\)). The second aim of the author is to study the sets \(S_{f}\subset Y\) where a regular mapping \(f:X\rightarrow Y\) is not proper (\(y\in S_{f}\) if and only if there is no neighbourhood \(U\) of \(y\) such that \(f^{-1}(\bar{U})\) is compact). The author proves that if \(X,Y\) are affine sets then \(S_{f}\) is empty or a hypersurface in \(Y\) and (when \(X\) is dominated by \(\mathbb{C}^{n}\)) is always \(\mathbb{C}\)-uniruled. As applications the author: 1. solves the complementary conjecture of McKay and Wang on characterization of \(\mathbb{C}\)-automorphisms of \(\mathbb{C}[z_{1},\dots,z_{n}],\) 2. gives a characterization of \(\mathbb{C}^{n}\) (it is a particular case of a Russel-Kraft conjecture), 3. shows that any irreducible affine variety \(X\) can be always extended to another irreducible affine variety \(Y\) of the same dimension such that \( X\varsubsetneq Y\).
Mathematics Subject Classification ID: 14R15
Mathematics Subject Classification ID: 14R10
zbMATH DE Number: 1347387
zbMATH Keywords: proper mapping
zbMATH Keywords: testing set for polynomial mappings
zbMATH Keywords: affine varieties
zbMATH Keywords: uniruled variety
zbMATH Keywords: regular mappings
zbMATH Keywords: Jacobian conjecture
reviewed by: Tadeusz Krasiński
MaRDI profile type: Publication
full work available at URL: s002080050316
OpenAlex ID: W2023715740
DOI: 10.1007/S002080050316
description: scientific article; zbMATH DE number 1347387