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A note on 1-parameter stable unfoldings

Ignacio Breva Ribes, Raúl Oset Sinha

Abstract

We give two characterisations of when a map-germ admits a 1-parameter stable unfolding, one related to the $\mathscr K_e$-codimension and another related to the normal form of a versal unfolding. We then prove that there are infinitely many finitely determined map-germs of multiplicity 4 from $\mathbb K^3$ to $\mathbb K^3$ which do not admit a 1-parameter stable unfolding.

A note on 1-parameter stable unfoldings

Abstract

We give two characterisations of when a map-germ admits a 1-parameter stable unfolding, one related to the -codimension and another related to the normal form of a versal unfolding. We then prove that there are infinitely many finitely determined map-germs of multiplicity 4 from to which do not admit a 1-parameter stable unfolding.

Paper Structure

This paper contains 5 sections, 4 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minimal stable unfoldings
  4. Corank 1 multiplicity 4 germs from $\mathbb K^3$ to $\mathbb K^3$.
  5. Declarations

Key Result

Lemma 3.3

For any unfolding $F(x,u) = (f_u(x),u)$ of $f$, there is an isomorphism $\beta\colon N\mathscr{K}_e F\to N\mathscr{K}_e f$ that takes the class of $\frac{\partial }{\partial X_i}$ to the class in $N\mathscr{K}_e f$ of $\frac{\partial }{\partial X_i}$ for each $i=1,\ldots,p$, and the class of $\frac{

Theorems & Definitions (14)

  • Definition 1.1
  • Definition 2.1
  • Example 3.1
  • Definition 3.2
  • Lemma 3.3
  • Theorem 3.4
  • proof
  • Example 3.5
  • Proposition 3.6
  • proof
  • ...and 4 more