Polynomial maps with maximal multiplicity and the special closure
type: Item
instance of: scholarly article
title: Polynomial maps with maximal multiplicity and the special closure
zbMATH Open document ID: 1426.32015
author: Carles Bivià-Ausina
author: Jorge A. C. Huarcaya
published in: Monatshefte für Mathematik
publication date: 2019-02-28T00:00:00Z
full work available at URL: https://repositorio.usil.edu.pe/handle/usil/3665
review text: For a polynomial \(F:\mathbb C^n\to\mathbb C^n\) one defines \(\mu(F)\) to be the dimension of the algebra obtained by quotienting out polynomials on \(\mathbb C^n\) by the components of \(F\). This number can also be interpreted as the expected number of solutions of the system \(F=0\). It is known that \(\mu(F)\) is bounded above by \(n! \mathrm{Vol}(\widetilde{\Gamma}_+(F))\), where \(\widetilde{\Gamma}_+(F)\) is a convex polyhedron obtained when studying which monomials appear with non-0 coefficients in \(F\). The main result of this paper is that in this estimate we have equality if an only if \(F\) is Newton non-degenerate at infinity -- an important condition on polynomials, involving the convex geometry of the faces of \(\widetilde{\Gamma}_+(F)\). In the last section the authors introduce and study the notion of \(F\) being non-degenerate with respect to a ``global Newton polyhedron''. In Theorem 4.9 these maps are proved to be the ones for which a certain \(\mu\)-vs-Volume estimate is sharp.
reviewed by: Richárd Rimányi
Mathematics Subject Classification ID: 32S99
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 13P99
zbMATH DE Number: 7031251
zbMATH Keywords: complex polynomial maps
zbMATH Keywords: Milnor number
zbMATH Keywords: multiplicity
zbMATH Keywords: Newton polyhedron
MaRDI profile type: Publication
OpenAlex ID: W2807880542
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DOI: 10.1007/S00605-018-1204-9
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description: scientific article; zbMATH DE number 7031251