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Tame deformations of highly singular function germs

Cezar Joiţa, Matteo Stockinger, Mihai Tibăr

TL;DR

The paper addresses deformations of real analytic germs with non-isolated singularities and seeks fibre constancy via a tameness framework that yields a Milnor-Hamm tube fibration.It introduces an analytic condition eq:cond and a Jacobian-criterion based on inclusions of Jacobian ideals, linking them to tameness and fibre constancy through a partial Thom regularity notion.Tameness implies the existence of a tube fibration and fibre constancy; conversely, the analytic and algebraic criteria provide practical routes to certify tameness, with nuanced behavior between real and complex settings demonstrated via examples.Overall, the work integrates geometric regularity concepts with algebraic criteria to advance equisingularity results for non-isolated singularities and to illuminate when deformations preserve fibre structure.

Abstract

We give analytic and algebraic conditions under which a deformation of real analytic functions with non-isolated singular locus is a deformation with fibre constancy.

Tame deformations of highly singular function germs

TL;DR

The paper addresses deformations of real analytic germs with non-isolated singularities and seeks fibre constancy via a tameness framework that yields a Milnor-Hamm tube fibration.It introduces an analytic condition eq:cond and a Jacobian-criterion based on inclusions of Jacobian ideals, linking them to tameness and fibre constancy through a partial Thom regularity notion.Tameness implies the existence of a tube fibration and fibre constancy; conversely, the analytic and algebraic criteria provide practical routes to certify tameness, with nuanced behavior between real and complex settings demonstrated via examples.Overall, the work integrates geometric regularity concepts with algebraic criteria to advance equisingularity results for non-isolated singularities and to illuminate when deformations preserve fibre structure.

Abstract

We give analytic and algebraic conditions under which a deformation of real analytic functions with non-isolated singular locus is a deformation with fibre constancy.
Paper Structure (11 sections, 17 theorems, 38 equations)

This paper contains 11 sections, 17 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Image and discriminant of a deformation
  3. The image problem
  4. The discriminant of a deformation
  5. Tame deformations
  6. Local tube fibration
  7. A condition for the tameness
  8. The partial Thom stratification
  9. Theorem and its proof
  10. The Jacobian criterium
  11. Examples

Key Result

Theorem 1.3

Let $F(x,t) = F_{t}(x)$ be a ${\mathbb K}$-analytic deformation of $F_{0}$ which satisfies condition eq:cond of Definition d:flat. Then the deformation $F$ of $F_{0}$ is a deformation with fibre constancy in the sense of Definition d:topconstantaway. Moreover, $\widetilde{F}$ has a Milnor-Hamm tube

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof : Proof of Proposition \ref{['p:image']}
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 3.1
  • ...and 26 more