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Newton Polyhedra and Whitney Equisingularity for Isolated Determinantal Singularities

Thaís M. Dalbelo, Luiz Hartmann, Maicom Varella

TL;DR

The paper develops a criterion for Whitney equisingularity of families of isolated determinantal singularities (IDS) by leveraging Newton polyhedra and Newton non-degeneracy. It frames IDS deformations and their polar multiplicities, highlights the Lê–Greuel relation tying the top polar multiplicity to vanishing Euler characteristics, and then imposes conditions of Newton non-degeneracy and $t$-independent convenient Newton polyhedra to guarantee equisingularity. The main result shows that, under these hypotheses, the IDS family is Whitney equisingular, with corollaries asserting the constancy of the vanishing Euler characteristic and top polar multiplicities, as well as a uniform Milnor ball radius. These findings connect polyhedral data to topological invariants and provide a practical criterion for equisingularity in determinantal settings.

Abstract

Using Newton polyhedra and non-degeneracy of matrices we present conditions which guarantee the Whitney equisingularity of families of isolated determinantal singularities.

Newton Polyhedra and Whitney Equisingularity for Isolated Determinantal Singularities

TL;DR

The paper develops a criterion for Whitney equisingularity of families of isolated determinantal singularities (IDS) by leveraging Newton polyhedra and Newton non-degeneracy. It frames IDS deformations and their polar multiplicities, highlights the Lê–Greuel relation tying the top polar multiplicity to vanishing Euler characteristics, and then imposes conditions of Newton non-degeneracy and -independent convenient Newton polyhedra to guarantee equisingularity. The main result shows that, under these hypotheses, the IDS family is Whitney equisingular, with corollaries asserting the constancy of the vanishing Euler characteristic and top polar multiplicities, as well as a uniform Milnor ball radius. These findings connect polyhedral data to topological invariants and provide a practical criterion for equisingularity in determinantal settings.

Abstract

Using Newton polyhedra and non-degeneracy of matrices we present conditions which guarantee the Whitney equisingularity of families of isolated determinantal singularities.

Paper Structure

This paper contains 5 sections, 4 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Determinantal deformation and Newton polyhedra
  3. Determinantal deformation
  4. Newton polyhedra
  5. Whitney equisingularity

Key Result

Theorem 1.1

Let $\left\{(X_{A_t}^s, 0) \right\}_{ t\in D }$, be a d-dimensional family of determinantal singularities, defined by the germ of matrices $A_t = ((a_{ij})_t):(\mathbb{C}^m,0)\rightarrow (M_{n,k},0)$ with holomorphic entries. Suppose that $X_{A_0}^s$ has an isolated singularity at $0$ and, for all $ Then the family $\left\{(X_{A_t}^s, 0) \right\}_{ t\in D}$ is Whitney equisingular.

Theorems & Definitions (14)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Corollary 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • ...and 4 more