Polygones de Newton de certaines sommes de caract\`eres et s\'eries de Poincar\'e

TL;DR

This paper investigates the asymptotic behavior of Newton polygons associated with L-functions from character sums of multivariable Laurent polynomials, using convex polytope operations to understand their limits.

Contribution

It introduces a method using free sums of convex polytopes to determine the limits of Newton polygons for sums of Laurent polynomials, a novel approach in multivariable cases.

Findings

Established the asymptotic behavior of Newton polygons for certain multivariable polynomials.

Linked the limits of Newton polygons to the free sum operation on convex polytopes.

Provided the first results on asymptotic Newton polygon behavior in multivariable polynomial contexts.

Abstract

In this paper, we shall precise the asymptotic behaviour of Newton polygons of $L$ functions associated to character sums, coming from some $n$ variable Laurent polynomials. In order to do this, we use the free sum on convex polytopes. This operation allows the determination of the limit of generic Newton polygons for the sum $\Delta=\Delta_1\oplus \Delta_2$ when we know the limit of generic Newton polygons for each factor. To our knowledge, these are the first results concerning the asymptotic behaviour of Newton polygons for multivariable polynomials.