On Abel's Problem about Logarithmic Integrals in Positive Characteristic

TL;DR

This paper investigates solutions to linear differential equations over fields of positive characteristic, demonstrating algebraic properties and infinite product representations, thus extending Abel's problem into characteristic p settings.

Contribution

It introduces a characteristic p analogue of Abel's problem, showing algebraicity of solutions' projections and establishing infinite product representations, with the development of $p^i$-curvatures as a key tool.

Findings

Existence of algebraic solutions' projections in positive characteristic

Infinite product representations of solutions are established

Introduction of $p^i$-curvatures as a generalization of $p$-curvature

Abstract

Linear differential equations with polynomial coefficients over a field $K$ of positive characteristic $p$ with local exponents in the prime field have a basis of solutions in the differential extension $\mathcal{R}_p=K(z_1, z_2, \ldots)(\!( x)\!)$ of $K(x)$, where $x'=1, z_1'=1/x$ and $z_i'=z_{i-1}'/z_{i-1}$. For differential equations of order $1$ it is shown that there exists a solution $y$ whose projections $y\vert_{z_{i+1}=z_{i+2}=\cdots=0}$ are algebraic over the field of rational functions $K(x, z_1, \ldots, z_{i})$ for all $i$. This can be seen as a characteristic $p$ analogue of Abel's problem about the algebraicity of logarithmic integrals. Further, the existence of infinite product representations of these solutions is shown. As a main tool $p^i$-curvatures are introduced, generalizing the notion of the $p$-curvature.