Growth at infinity and index of polynomial maps
type: Item
instance of: scholarly article
title: Growth at infinity and index of polynomial maps
zbMATH Open document ID: 1310.58027
author: Carles Bivià-Ausina
author: Jorge A. C. Huarcaya
published in: Journal of Mathematical Analysis and Applications
publication date: 2014-10-31T00:00:00Z
review text: Let \(F:\mathbb{K}^{n}\mathbb{\rightarrow K}^{n}\) be a polynomial mapping such that \(F^{-1}(0)\) is compact, where \(\mathbb{K=R}\) or \(\mathbb{C}\). The authors give an estimation of the Łojasiewicz exponent at infinity of \(F\) (defined as the supremum of those real numbers \(\alpha \) for which there exist constants \(C,M>0\) such that \(\left| \left| x\right| \right| ^{\alpha }\leq C\left| \left| F(x)\right| \right| \) for \(\left| \left| x\right| \right| \geq M),\) under some non-degeneracy conditions for some Newton polyhedra. This estimation allows a uniform estimation of the Łojasiewicz exponent at infinity for the mappings in a homotopy \(F+tG,\) \(t\in [ 0,1]\), where \(G \) is another polynomial mapping satisfying appropriate conditions. This, in turn, implies the invariance of the global index of the mappings in the homotopy \(F+tG.\) In particular, \(F\) and \(F+G\) have the same global index.
reviewed by: Tadeusz Krasiński
Mathematics Subject Classification ID: 58K45
Mathematics Subject Classification ID: 32S65
Mathematics Subject Classification ID: 14R99
Mathematics Subject Classification ID: 14P99
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 6362884
zbMATH Keywords: polynomial mapping
zbMATH Keywords: topological index
zbMATH Keywords: Newton polyhedron
zbMATH Keywords: Łojasiewicz exponent at infinity
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.09.062
OpenAlex ID: W2013081356
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DOI: 10.1016/J.JMAA.2014.09.062
description: scientific article; zbMATH DE number 6362884