L\^e numbers and Newton diagram

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.