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- Title
On Abel's Problem and Gauss Congruences.
- Authors
Delaygue, É; Rivoal, T
- Abstract
A classical problem due to Abel is to determine if a differential equation |$y^{\prime}=\eta y$| 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 $| admits a Puiseux expansion with rational coefficients at some point in |$\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^{\prime}=\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 |$y^{\prime}=\eta y$| 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 two other applications, namely to diagonals of rational fractions and to directed two-dimensional walks.
- Subjects
ALGEBRAIC functions; PRIME numbers; HYPERGEOMETRIC series; DIFFERENTIAL equations; HYPERGEOMETRIC functions; ARITHMETIC; GEOMETRIC congruences
- Publication
IMRN: International Mathematics Research Notices, 2024, Vol 2024, Issue 5, p4301
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnad229