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Preprints, Working Papers, ... Year : 2023

Algebraic solutions of linear differential equations: an arithmetic approach

Abstract

Given a linear differential equation with coefficients in $\mathbb{Q}(x)$, an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck-Katz $p$-curvature conjecture.
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Dates and versions

hal-04065092 , version 1 (11-04-2023)

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  • HAL Id : hal-04065092 , version 1

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Alin Bostan, Xavier Caruso, Julien Roques. Algebraic solutions of linear differential equations: an arithmetic approach. 2023. ⟨hal-04065092⟩
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