Invariance of the global monodromies in families of polynomials of two complex variables
type: Item
instance of: scholarly article
title: Invariance of the global monodromies in families of polynomials of two complex variables
zbMATH Open document ID: 0902.32018
author: Hà Huy Vui
author: Tien Son Pham
published in: Acta Mathematica Vietnamica
publication date: 1998-12-03T00:00:00Z
review text: Let \(f_\alpha(x,y)\) be a family of primitive polynomials in two complex variables with coefficients which are smooth complex-valued functions of \(\alpha \in [0,1].\) For such a polynomial \(f\) the author considers the following four numerical invariants of \(f :\) the total Milnor number \(\mu,\) the degree \(d,\) the number of critical values at infinity \(\sigma,\) and certain invariant \(\lambda\) that characterizes the intersection of the degree \(d\) homogeneous part \(f_d\) of the polynomial \(f\) and the line at infinity. In particular, for any \(M\)-tame polynomial \(f\) we have \(\lambda(f) = \sigma(f) = 0\) [see \textit{Ha Huy Vui} and \textit{A. Zaharia}, Math. Ann. 304, No. 3, 481-488 (1996; Zbl 0844.32021)]. It is proved that the constancy \(\mu(f_\alpha), d(f_\alpha), \sigma(f_\alpha)\) and \(\lambda(f_\alpha)\) on the interval \([0,1]\) implies the conjugacy of the geometric global monodromies of the polynomials \(f_0\) and \(f_1.\) In authors' opinion the above result can be considered as a global analog of the Lê-Ramanujam theorem [\textit{Lê Dung Trang} and \textit{C. P. Ramanujam}, Am. J. Math. 98, 67-79 (1976; Zbl 0351.32009)].
Mathematics Subject Classification ID: 32S55
Mathematics Subject Classification ID: 32S15
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 57R70
Mathematics Subject Classification ID: 14E15
zbMATH DE Number: 1145964
zbMATH Keywords: geometric monodromy
zbMATH Keywords: global monodromy
zbMATH Keywords: bifurcation set
zbMATH Keywords: critical value at infinity
zbMATH Keywords: Lojasiewicz number
zbMATH Keywords: Milnor number
zbMATH Keywords: Milnor fibration
zbMATH Keywords: global monodromy fibration
zbMATH Keywords: \(M\)-tame polynomial
zbMATH Keywords: primitive polynomial
reviewed by: Aleksandr G. Aleksandrov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1145964