On the reduction of the degree of linear differential operators

TL;DR

This paper investigates how to find the minimal degree linear differential operator with algebraic coefficients that annihilates solutions of a given operator, with applications to Picard-Fuchs equations in dynamical systems.

Contribution

It provides a method to determine the minimal degree operator over the algebraic closure for solutions of linear differential equations, extending previous results.

Findings

Explicit minimal degree operators for specific Picard-Fuchs equations

Application to perturbations of Lotka-Volterra systems

Enhanced understanding of differential operator reduction

Abstract

Let L be a linear differential operator with coefficients in some differential field k of characteristic zero with algebraically closed field of constants. Let k^a be the algebraic closure of k. For a solution y, Ly=0, we determine the linear differential operator of minimal degree M and coefficients in k^a, such that My=0. This result is then applied to some Picard-Fuchs equations which appear in the study of perturbations of plane polynomial vector fields of Lotka-Volterra type.