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

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 $\mathrm{\Delta }={\mathrm{\Delta }}_1\oplus {\mathrm{\Delta }}_2$ 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.