# Finitness theorem for Multi-K-Bi-Lipschtiz equivalence of map-germs

## TL;DR

This paper proves that, for polynomial map germs of bounded degree, the set of equivalence classes under multi-K-bi-Lipschitz equivalence is finite, extending understanding of classification in singularity theory.

## Contribution

It establishes a finiteness theorem for the classification of polynomial map germs under multi-K-bi-Lipschitz equivalence, a new result in singularity theory.

## Key findings

- Finiteness of equivalence classes for polynomial map germs of bounded degree.
- Extension of classification results to multi-K-bi-Lipschitz equivalence.
- Provides a foundational result for further studies in Lipschitz geometry of singularities.

## Abstract

Let $P^{k}(n,p) $ be the set of all real polynomial map germs $f = ( f_1 , ..., f_p ) : (\mathbb{R}^{n},0) \rightarrow (\mathbb{R}^{p},0)$ with degree of $f_1 , ...,f_p$ less than or equal to $ k \in \mathbb{N}$. The main result of this paper shows that the set of equivalence classes of $ P^{k}(n,p)$, with respect to multi-$\mathcal{K}$-bi-Lipschitz equivalence, is finite.