A numerical approach to some basic theorems in singularity theory

TL;DR

This paper provides explicit bounds for key theorems in Singularity theory, aiming to enhance their applicability in Numerical Analysis and computational fields.

Contribution

It introduces explicit bounds for fundamental theorems in Singularity theory, facilitating their practical use in numerical and computational contexts.

Findings

Derived explicit bounds for the Inverse, Implicit, and Rank Theorems.

Established bounds for the Splitting and Morse Lemmas.

Demonstrated the density and openness of Morse functions with these bounds.

Abstract

In this paper, we give the explicit bounds for the data of objects involved in some basic theorems of Singularity theory: the Inverse, Implicit and Rank Theorems for Lipschitz mappings, Splitting Lemma and Morse Lemma, the density and openness of Morse functions. We expect that the results will make Singularities more applicable and will be useful for Numerical Analysis and some fields of computing.