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Article Dans Une Revue International Mathematics Research Notices Année : 2023

On Abel's problem and Gauss congruences

Résumé

A classical problem due to Abel is to determine if a differential equation $y'=\eta y$ admits a non-trivial solution $y$ algebraic over $\mathbb C(x)$ when $\eta$ is a given algebraic function over $\mathbb C(x)$. Risch designed an algorithm that, given $\eta$, determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when $\eta$ admits a Puiseux expansion with {\em rational} coefficients at some point in $\mathbb C\cup \{\infty\}$, which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of $y'=\eta y$ if and only if the coefficients of the Puiseux expansion of $x\eta(x)$ at $0$ satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations $y'=\eta y$ with an algebraic solution when $x\eta(x)$ is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present three other applications, in particular to diagonals of rational fractions and to directed two-dimensional walks.
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Dates et versions

hal-03772016 , version 1 (07-09-2022)
hal-03772016 , version 2 (19-04-2023)

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Eric Delaygue, Tanguy Rivoal. On Abel's problem and Gauss congruences. International Mathematics Research Notices, 2023, 2024 (5), pp.4301-4327. ⟨10.1093/imrn/rnad229⟩. ⟨hal-03772016v2⟩
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