An arithmetic characterization of some algebraic functions and a new proof of an algebraicity prediction by Golyshev

TL;DR

The paper characterizes when the coefficient sequence $a_n$ of an algebraic power series with factorial-ratio form leads to an algebraic $Q(t)=\exp(\int F(t)/t\,dt)$, sharpening Golyshev's algebraicity prediction. It proves a general equivalence: for $f(t)=\sum_{n>0} a_n t^{n-1}$ algebraic over $\mathbb{Q}(t)$, the algebraicity of all solutions to $y'=f y$ is equivalent to congruence properties $a_{np}-a_n \in p\mathbb{Z}_{(p)}$ (for almost all $p$) and to $a_{np}-a_n \in n p \mathbb{Z}_{(p)}$ (for almost all $p$). The main novelty is a shorter proof of Golyshev's prediction using the Chudnovsky–Chudnovsky criterion and Wilson’s theorem to establish $a_{np} \equiv a_n \pmod p$, avoiding heavier p-adic machinery. The results extend to general algebraic $f$ and open questions about finite-version criteria, algebraicity degrees, and potential refinements of the underlying congruence framework.

Abstract

We provide a new arithmetic characterization for the sequence of coefficients of algebraic power series $f(t)$ having the property that the differential equation $y'(t) = f(t) y(t)$ has algebraic solutions only. This extends a recent result by Delaygue and Rivoal, and also provides a new and shorter proof of an algebraicity result predicted by Golyshev.

Theorems & Definitions (10)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Corollary 2.3
• proof
• Corollary 2.4
• proof
• Theorem 3.1
• proof