An algorithm for computing differential equations for invariant curves

TL;DR

This paper presents an algorithm leveraging Picard-Vessiot theory to compute differential equations that parametrize algebraic invariant curves under finite linear algebraic groups.

Contribution

It introduces a novel algorithm that constructs differential equations for invariant curves using Picard-Vessiot theory, bridging algebraic group actions and differential equations.

Findings

Algorithm successfully constructs differential equations for invariant curves.

The method applies to curves invariant under finite linear algebraic groups.

Provides a systematic way to connect algebraic invariants with differential equations.

Abstract

In this paper we describe an algorithm based on the Picard-Vessiot theory that constructs, given any curve invariant under a finite linear algebraic group over the complex numbers, an ordinary linear differential equation whose Schwarz map parametrizes it.