# L\^e numbers and Newton diagram

## TL;DR

This paper presents an algorithm to compute Lê numbers for Newton non-degenerate complex analytic functions using invariants from their Newton diagrams, extending Kouchnirenko's theorem to non-isolated singularities.

## Contribution

The authors introduce a novel algorithm linking Lê numbers to Newton diagrams for non-isolated singularities, generalizing Kouchnirenko's theorem.

## Key findings

- Algorithm computes Lê numbers from Newton diagram invariants.
- Functions with identical Newton diagrams share the same Lê numbers.
- Extension of Kouchnirenko's theorem to non-isolated singularities.

## Abstract

We give an algorithm to compute the L\^e numbers of (the germ of) a Newton non-degenerate complex analytic function $f\colon(\mathbb{C}^n,0) \rightarrow (\mathbb{C},0)$ in terms of certain invariants attached to the Newton diagram of the function $f+z_1^{\alpha_1}+\cdots +z_d^{\alpha_d}$, where $d$ is the dimension of the critical locus of $f$ and $\alpha_1,\ldots, \alpha_d$ are sufficiently large integers. This is a version for non-isolated singularities of a famous theorem of A. G. Kouchnirenko. As a corollary, we obtain that Newton non-degenerate functions with the same Newton diagram have the same L\^e numbers.