On the symmetry of Stokes regions of a simple singularity
type: Item
instance of: scholarly article
title: On the symmetry of Stokes regions of a simple singularity
zbMATH Open document ID: 0788.58007
DOI: 10.1007/BF01445116
author: Jianming Yu
published in: Mathematische Annalen
publication date: 1994-05-31T00:00:00Z
full work available at URL: https://eudml.org/doc/165096
review text: Let \(f(x)\) be a quasihomogeneous polynomial with a simple singularity at \(x=0 \in \mathbb{C}^ n\) of the Milnor number \(k\). Let \(\varphi (x,t)\) with \(t \in T=\mathbb{C}^{k-1}\) be a universal unfolding of \(f(x)\) which we can also assume to be quasihomogeneous. Then for each parameter \(t \in T\) \(\varphi_ t(x)=\varphi (x,t)\) has \(k\) critical values (accounting multiplicities) which we denote by \(w_ 1(t),\dots,w_ k(t)\). We define the following two sets: \[ \begin{aligned} P & =\{t \in T | \exists i \neq j,\text{ s.t. Re} w_ i(t)=\text{Re} w_ j(t)\} \\ Q & =\{t \in T | \exists i \neq j,\text{ s.t. } w_ i(t)=w_ j(t), \text{ or } \exists i,j,l, \text{ s.t. Re} w_ i(t)=\text{Re} w_ j(t)=\text{Re} w_ l(t)\}. \end{aligned} \] The components of \(T \backslash P\) are called Stokes regions and the components of \(P \backslash Q\) are called walls of the Stokes regions which form the smooth part of the boundary of a Stokes region. Two Stokes regions are adjacent if they have a wall in common. We can construct a graph \(\Gamma\) which describes the adjacency of the Stokes regions as follows: take for each Stokes region one vertex and combine two vertices with the same number of edges as the walls that the corresponding Stokes regions have in common. The aim of this paper is to study the symmetry of this graph and to prove the following result: Theorem: If \(f(x)\) has a simple singularity of the type \(A_ k\), then the symmetry group of the graph \(\Gamma\) is the dihedral group \(D_ q\), where \(q=3\) if \(k=2\) and \(q=2(k+1)\) if \(k \geq 3\).
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
zbMATH DE Number: 537263
zbMATH Keywords: quasihomogeneous polynomial
zbMATH Keywords: Stokes regions
zbMATH Keywords: symmetry
zbMATH Keywords: simple singularity
MaRDI profile type: Publication
cites work: Q3203214
cites work: Q3478608
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description: scientific article; zbMATH DE number 537263