On the generalized critical values of a polynomial mapping
type: Item
instance of: scholarly article
title: On the generalized critical values of a polynomial mapping
zbMATH Open document ID: 1033.14036
author: Z. Jelonek
published in: Manuscripta Mathematica
publication date: 2003-06-02T00:00:00Z
review text: Let \(f: k^n \to k^m\) be a polynomial dominant mapping, where \(k= {\mathbb R}\) or \(k= {\mathbb C}\). It is known that \(f\) is a locally trivial fibration outside the bifurcation set \(B(f)\), which has measure 0. The set \(K(f)\) of generalized critical values of \(f\) is defined as \(K(f) = K_0(f) \cup K_\infty(f)\), where \(K_0(f)\) is the set of classical critical values of \(f\) and \(K_\infty(f)\) is the set of asymptotic critical values at infinity. This set \(K(f)\) is known to be a semialgebraic set (or algebraic in the complex case) [ see the paper by \textit{K. Kurdyka, P. Orro} and \textit{S. Simmon}, J. Differ. Geom. 56, 67--92 (2000; Zbl 1067.58031)]. One of the results of the paper under review is that \(B(f) \subset K(f)\), when \(f\) is a \(k\)-smooth mapping. This is a particular case of a more general result of \textit{P. J. R. Rabier} [ Ann. Math. 146, 647--691 (1997; Zbl 0919.58003)]. The interest in the proof given here is that it is direct and short. The main result of the paper states that if \(f=(f_1, \ldots,f_m): {\mathbb C}^n \to {\mathbb C}^m\) is a polynomial dominant mapping then the set \(K(f)\) is a proper algebraic subset of \({\mathbb C}^m\) and it is contained in a hypersurface of degree at most \(D = (d+s(m-1)(d-1))^n\), where \(s = {n\choose m} \), and \(\text{ deg} \, f_i \leq d\). An algorithm to compute the set \(K(f)\) effectively is given in the last section by using Gröbner basis.
Mathematics Subject Classification ID: 14R10
Mathematics Subject Classification ID: 14Q20
Mathematics Subject Classification ID: 51N10
Mathematics Subject Classification ID: 51N20
Mathematics Subject Classification ID: 15A04
zbMATH DE Number: 1922375
zbMATH Keywords: generalized critical values
zbMATH Keywords: asymptotic critical values
zbMATH Keywords: polynomial dominant mappings
zbMATH Keywords: proper algebraic subset
zbMATH Keywords: algorithm
zbMATH Keywords: Gröbner basis
reviewed by: Antonio Diaz-Cano
MaRDI profile type: Publication
full work available at URL: s00229-002-0320-x
OpenAlex ID: W2046777039
DOI: 10.1007/S00229-002-0320-X
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description: scientific article; zbMATH DE number 1922375