Linear differential equations and Hurwitz series

TL;DR

This paper explores solutions to linear differential equations using Hurwitz series, providing explicit formulas for solutions and their automorphism groups without restrictions on the field's algebraic closure or characteristic.

Contribution

It introduces explicit recursive solutions and automorphism group formulas for linear differential equations over general fields, expanding previous theoretical frameworks.

Findings

Derived recursive expressions for solutions

Explicit formulas for automorphism groups

Applicable to fields of arbitrary characteristic

Abstract

In this article, we study the set of all solutions of linear differential equations using Hurwitz series. We first obtain explicit recursive expressions for solutions of such equations and study the group of differential automorphisms of the set of all solutions. Moreover, we give explicit formulas that compute the group of differential automorphisms. We require neither that the underlying field be algebraically closed nor that the characteristic of the field be zero.