On Abel's problem and Gauss congruences

TL;DR

This paper characterizes when certain differential equations have algebraic solutions using Gauss congruences on Puiseux series coefficients, with applications to hypergeometric series, motives, diagonals, and lattice walks.

Contribution

It introduces an arithmetic criterion based on Gauss congruences for algebraic solutions of differential equations with Puiseux expansions, extending Abel's problem.

Findings

Characterization of algebraic solutions via Gauss congruences.

Complete determination of hypergeometric equations with algebraic solutions.

Applications to motives, diagonals of rational functions, and lattice walks.

Abstract

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 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.