Fuchs' theorem on linear differential equations in arbitrary characteristic

TL;DR

This paper extends Fuchs' theorem to linear differential equations over fields of any characteristic, describing solutions using a differential ring with iterated logarithms and connecting to the $p$-curvature conjecture.

Contribution

It introduces a new differential ring framework for arbitrary characteristic and proves that solutions can be constructed using iterated logarithms, generalizing classical results.

Findings

Solutions involve all variables in positive characteristic

Explicit power series expansion of the exponential function in positive characteristic

Connections established with the Grothendieck-Katz $p$-curvature conjecture

Abstract

The paper generalizes Lazarus Fuchs' theorem on the solutions of complex ordinary linear differential equations with regular singularities to the case of ground fields of arbitrary characteristic, giving a precise description of the shape of each solution. This completes partial investigations started by Taira Honda and Bernard Dwork. The main features are the introduction of a differential ring $\mathcal{R}$ in infinitely many variables mimicking the role of the (complex) iterated logarithms, and the proof that adding these "logarithms" already provides sufficiently many primitives so as to solve any differential equation with regular singularity in $\mathcal{R}$. A key step in the proof is the reduction of the involved differential operator to an Euler operator, its normal form, to solve Euler equations in $\mathcal{R}$ and to lift their (monomial) solutions to solutions of the original equation. The first (and already very striking) example of this outset is the exponential function $\exp_p$ in positive characteristic, solution of $y' = y$. We prove that it necessarily involves all variables and we construct its explicit (and quite mysterious) power series expansion. Additionally, relations of our results to the Grothendieck-Katz $p$-curvature conjecture and related conjectures will be discussed.