An arithmetic study of the formal Laplace transform in several variables

TL;DR

This paper introduces two types of Laplace transforms tailored for solutions of differential modules over multivariable rational function fields, exploring their properties to aid in the arithmetic analysis of related $E$-functions.

Contribution

It develops new Laplace transform tools for multivariable differential modules, enhancing the arithmetic study of $E$-functions in several variables.

Findings

Defined two types of Laplace transforms for multivariable differential modules.

Established basic differential and arithmetic properties of these transforms.

Provided tools for future arithmetic analysis of $E$-functions.

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.