Milnor number at infinity, topology and Newton boundary of a polynomial function
type: Item
instance of: scholarly article
title: Milnor number at infinity, topology and Newton boundary of a polynomial function
zbMATH Open document ID: 0956.32025
author: A. Melle-Hernández
author: Enrique Artal Bartolo
author: Ignacio Luengo
published in: Mathematische Zeitschrift
publication date: 2000-10-16T00:00:00Z
full work available at URL: 2000Milnor-number-infinty.pdf
review text: For a complex polynomial \(f:\mathbb{C}^n\to \mathbb{C}\) the set of values where \(f\) fails to be a \(C^{\infty}\)-locally trivial fibration (the bifurcation set) includes not only the critical values of \(f\) but may also contain values corresponding to `critical points at infinity' [for a reference see, e.g., \textit{A. H. Durfee}, Prog. Math. 162, 345-360 (1996; Zbl 0919.32021)]. At such an atypical value at infinity the fibre is different from the generic one. The paper studies a topological invariant of fibres, namely Euler characteristic. The authors prove the formulas (similar to those of Ha, Le, Siersma, Tibar): \[ \chi(\text{generic fibre})=1+(-1)^n(\mu(f)+\lambda(f)) \] \[ \chi(\text{fibre over }b)=\chi(\text{generic fibre})+(-1)^n (\mu^a_f(b)+\lambda_f(b)), \] where the \(\mu\)'s are sums of local Milnor numbers and the \(\lambda\)'s measure singularities at infinity. Their definition of \(\lambda\) relies on the generalized Milnor number of Parusiński (see also works of Parusiński-Pragacz), so is well computable -- which is an advantage to earlier works, and is well demonstrated in many examples in the paper. In case \(f\) has non-degenerate Newton principal part at infinity they relate the Euler characteristic of the generic fibre to \(\nu(f)\), the Newton number of \(f\). This, together with the formulas above, gives the main result of the paper \(\mu(f)+\lambda(f)=\nu(f)\), which generalizes Kushnirenko's theorem and also some results of Cassou-Nogues.
reviewed by: Richárd Rimányi
Mathematics Subject Classification ID: 32S15
Mathematics Subject Classification ID: 32S60
Mathematics Subject Classification ID: 14F45
Mathematics Subject Classification ID: 14L24
zbMATH DE Number: 1463421
MaRDI profile type: Publication
OpenAlex ID: W1975192203
DOI: 10.1007/S002090050493
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description: scientific article; zbMATH DE number 1463421