Minimization of differential equations and algebraic values of $E$-functions

Alin Bostan (1, 2) , Tanguy Rivoal (3, 4) , Bruno Salvy (5, 6)

1 MATHEXP - Calcul formel, mathématiques expérimentales et interactions
2 SPECFUN - Symbolic Special Functions : Fast and Certified
3 IF - Institut Fourier
4 CNRS - Centre National de la Recherche Scientifique
5 ENS de Lyon - École normale supérieure de Lyon
6 ARIC - Arithmétiques des ordinateurs, méthodes formelles, génération de code

A power series being given as the solution of a linear differential equation with appropriate initial conditions, minimization consists in finding a non-trivial linear differential equation of minimal order having this power series as a solution. This problem exists in both homogeneous and inhomogeneous variants; it is distinct from, but related to, the classical problem of factorization of differential operators. Recently, minimization has found applications in Transcendental Number Theory, more specifically in the computation of non-zero algebraic points where Siegel's $E$-functions take algebraic values. We present algorithms for these questions and discuss implementation and experiments.

Mots clés: Linear differential operators; Minimization; Factorization; Desingularization; $E$-functions; Siegel-Shidlovskii theorem; Beukers’ algorithm; Adamczewski-Rivoal algorithm

Domaines: Calcul formel [cs.SC]; Théorie des nombres [math.NT]

Alin Bostan, Tanguy Rivoal, Bruno Salvy. Minimization of differential equations and algebraic values of $E$-functions. 2022. ⟨hal-03771150v1⟩