# A note on the local Lipschitz triviality of values of complex polynomial functions

## TL;DR

This paper investigates when complex polynomial functions are locally bi-Lipschitz trivial at certain values, concluding that only univariate polynomials have this property.

## Contribution

It establishes a precise characterization of complex polynomials with locally bi-Lipschitz trivial values, showing they must be univariate.

## Key findings

- Only univariate complex polynomials have locally bi-Lipschitz trivial values.
- Multivariate polynomials do not admit such triviality at any value.
- The result clarifies the geometric structure of polynomial mappings.

## Abstract

We address the question of the bi-Lipschitz local triviality of a complex polynomial function over a complex value. Our main result state that a non constant complex polynomial admits a locally bi-Lipschitz trivial value if and only if it is a polynomial in a single complex variable.