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Real and complex integral closure, Lipschitz equisingularity and applications on square matrices

Thiago F. da Silva, Nivaldo G. Grulha, Miriam S. Pereira

Abstract

Recently the authors investigated the Lipschitz triviality of simple germs of matrices. In this work, we improve some previous results and we present an extension of an integral closure result for the real setting. These tools are applied to investigate classes of square matrices singularities classified by Bruce and Tari.

Real and complex integral closure, Lipschitz equisingularity and applications on square matrices

Abstract

Recently the authors investigated the Lipschitz triviality of simple germs of matrices. In this work, we improve some previous results and we present an extension of an integral closure result for the real setting. These tools are applied to investigate classes of square matrices singularities classified by Bruce and Tari.

Paper Structure

This paper contains 3 sections, 13 theorems, 21 equations.

Table of Contents

  1. Notation and Background
  2. Real integral closure and Lipschitz Equisingularity
  3. Applications in some classes of square matrices

Key Result

Theorem \oldthetheorem

Let $I$ be an ideal of ${\mathcal{A}}_{X,x}$ and $h\in {\mathcal{A}}_{X,x}$. Then: $h\in\overline{I}$ if and only if for each choice of generators $\{f_i\}$ there exist a positive constant $C$ and a neighborhood $U$ of $x$ such that for all $z\in U$.

Theorems & Definitions (27)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem: G3
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 17 more