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

TL;DR

This paper introduces a new arithmetic characterization for certain algebraic power series coefficients, extending recent results and providing a shorter proof for a Golyshev algebraicity prediction.

Contribution

It offers a novel arithmetic criterion for algebraic power series with specific differential equation properties, advancing understanding in algebraic function theory.

Findings

Extended Delaygue and Rivoal's result on algebraic power series

Provided a shorter proof of Golyshev's algebraicity prediction

Characterized coefficients of algebraic solutions to differential equations

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.