Exceptional values of E-functions at algebraic points

TL;DR

This paper proves the existence of an algorithm that determines all algebraic points where a transcendental E-function takes algebraic values, thereby solving a longstanding problem in the transcendence theory of E-functions.

Contribution

It introduces an algorithm that identifies all algebraic points where a given transcendental E-function assumes algebraic values, building on Beukers' refinement of the Siegel-Shidlovskii Theorem.

Findings

Algorithm outputs finite list of algebraic points with algebraic function values

Solves the problem of deciding algebraic or transcendental nature of E-function values

Builds on Beukers' refinement of the Siegel-Shidlovskii Theorem

Abstract

E-functions are entire functions with algebraic Taylor coefficients satisfying certain arithmetic conditions, and which are also solutions of linear differential equations with rational functions coefficients. They were introduced by Siegel in 1929 to generalize Diophantine properties of the exponential function, and studied further by Shidlovskii in 1956. The celebrated Siegel-Shidlovskii Theorem deals with the algebraic (in)dependence of values at algebraic points of E-functions solutions of a differential system. However, somewhat paradoxically, this deep result may fail to decide whether a given E-fuction assumes an algebraic or a transcendental value at some given algebraic point. Building upon Andr\'e's theory of E-operators, Beukers refined in 2006 the Siegel-Shidlovskii Theorem in an optimal way. In this paper, we use Beukers' work to prove the following result: there exists an algorithm which, given a transcendental E-function $f(z)$ as input, outputs the finite list of all exceptional algebraic points $\alpha$ such that $f(\alpha)$ is also algebraic, together with the corresponding list of values $f(\alpha)$. This result solves the problem of deciding whether values of E-functions at algebraic points are transcendental.