An arithmetic study of the formal Laplace transform in several variables

Abstract

Let $K$ be a number field, and let $K(x_1,...,x_d)$ be the field of rational fractions in the variables $x_1,...,x_d$. In this paper, we introduce two kinds of Laplace transform adapted to solutions of the differential $K(x_1,...,x_d)$-modules with regular singularities, and give some of their basic differential and arithmetic properties. The purpose of this article is to provide some tools which might be useful, in particular, for the arithmetic study of the differential $K(x_1,...,x_d)$-modules associated to $E$-functions in several variables.