Lê numbers and Newton diagram

Abstract

We give an algorithm to compute the Lê numbers of (the germ of) a Newton non-degenerate complex analytic function $f:({\mathbb{C}}^n,0)\to (\mathbb{C},0)$ 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 ${\alpha }_1,\dots ,{\alpha }_d$ 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 Lê numbers.

17 pages, 1 figure