Effective algebraic independence of values of E-functions

Abstract

E-functions are entire functions with algebraic Taylor coefficients satisfying certain arithmetic conditions, and which are also solutions of linear differential equations with polynomial coefficients. They were introduced by Siegel in 1929 to generalize the Diophantine properties of the exponential and Bessel's functions. The Siegel-Shidlovskii Theorem (1956) deals with the algebraic (in)dependence of values at algebraicpoints of E-functions solutions of a differential system. In this paper, we prove the existence of an algorithm to perfom the following three tasks. Given as inputs some E-functions $F_1(z), ..., F_p(z)$, (1) it computes a system of generators of the ideal of polynomial relations between $F_1(z), ..., F_p(z)$; (2) given any algebraic number $\alpha$, it computes a system of generators of the ideal of polynomial relations between the values $F_1(\alpha), ..., F_p(\alpha)$ with algebraic coefficients;(3) if $z,F_1(z), ..., F_p(z)$ are algebraically independent, it determines the finite set of all algebraic numbers $\alpha$ such that the values $F_1(\alpha), ..., F_p(\alpha)$ are algebraically dependent. The existence of this algorithm relies on a variant of the Hrushovski-Feng algorithm (to compute polynomial relations between solutions of differential systems) and on Beukers' lifting theorem (an optimal refinement of the Siegel-Shidlovskii theorem) in order to reduce the problem to an effective elimination procedure in multivariate polynomial rings. The latter is then performed using Gr{\"o}bner bases.