# Invariants of the bi-Lipschitz contact equivalence of continuous definable function germs

## TL;DR

This paper introduces an invariant for classifying continuous definable function germs under bi-Lipschitz contact equivalence, using asymptotic expansion coefficients along tangency components.

## Contribution

It constructs a new invariant based on asymptotic expansions for definable function germs, aiding in their classification under bi-Lipschitz contact equivalence.

## Key findings

- Invariant effectively distinguishes bi-Lipschitz contact classes.
- Applicable to polynomially bounded o-minimal structures, including semialgebraic functions.
- Provides a method to compute invariants from asymptotic data.

## Abstract

We construct an invariant of the bi-Lipschitz contact equivalence of continuous function germs definable in a polynomially bounded o-minimal structure, such as semialgebraic functions. For a germ $f,$ the invariant is given in terms of the leading coefficients of the asymptotic expansions of $f$ along the connected components of the tangency variety of $f.$