Multiplicities of Maxwell sets of Pham singularities
type: Item
instance of: scholarly article
title: Multiplicities of Maxwell sets of Pham singularities
zbMATH Open document ID: 1368.32017
publication date: 2017-02-08T00:00:00Z
review text: Let \(f:(\mathbb{C}^{n},0)\rightarrow (\mathbb{C},0)\) be an isolated singularity and \(F:(\mathbb{C}^{n}\times \mathbb{C}^{k},0)\rightarrow ( \mathbb{C},0)\) a versal deformation of \(f.\) The \textit{caustic} (resp. \textit{Maxwell set}) of \(F\) is the set (resp. closure of the set) of all parameters \(\lambda \in \mathbb{C}^{k}\) such that the corresponding function \( f_{\lambda }=F(\cdot ,\lambda )\) has a non-Morse critical point (resp. has equal critical values at different critical points) close to the origin in \( \mathbb{C}^{n}.\) These sets are analytic. Denote by \(C(f)\) and \(M(f)\) the multiplicities of these sets at \(0\in \mathbb{C}^{k}\). The results are: 1. if \(f=z_{1}^{a_{1}+1}+\cdots +z_{n}^{a_{n}+1}\) is a Pham singularity then \[ \begin{aligned} C(f)& =(a_{1}\cdots a_{n})\left( \frac{a_{1}-1}{a_{1}}+\cdots +\frac{a_{n}-1 }{a_{n}}\right) , \\ 3C(f)+2M(f)& \leq \sum_{i=1}^{n}(a_{1}\cdots a_{i-1})\left( a_{i}^{2}-1\right) (a_{i+1}\cdots a_{n})^{2}, \end{aligned} \] 2. if additionally \(a_{1}=\cdots =a_{n}=0\) then \[ M(f)=\frac{a^{n-1}\left( \left( a+1\right) \left( a^{n}-1\right) -3n\left( a-1\right) \right) }{2}, \] 3. if \(f\) is an arbitrary isolated singularity then \[ 3C(f)+2M(f)\geq \frac{\mu (f)\left( \mu (f)-1\right) d}{d-1}, \] where \(\mu (f)\) is the Milnor number of \(f\) and \(d\) the order of \(f\) at \(0.\)
reviewed by: Tadeusz Krasiński
Mathematics Subject Classification ID: 32S05
zbMATH DE Number: 6681944
zbMATH Keywords: Maxwell set
zbMATH Keywords: caustic
zbMATH Keywords: isolated singularity
zbMATH Keywords: versal deformation
zbMATH Keywords: Pham singularity
author: V. A. Vassiliev
MaRDI profile type: Publication
full work available at URL: s10688-016-0150-3
OpenAlex ID: W2521437769
cites work: Dynamical systems VIII. Singularity theory. II: Classification and applications. Transl. from the Russian by J. S. Joel
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cites work: A few problems on monodromy and discriminants
DOI: 10.1007/S10688-016-0150-3
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 6681944