Simplifying matrix differential equations with general coefficients

TL;DR

This paper proves that certain matrix differential equations with general coefficients cannot be simplified to fewer parameters using rational gauge transformations, using differential Galois theory and essential dimension concepts.

Contribution

It establishes a lower bound on the number of parameters needed to simplify matrix differential equations with general coefficients, introducing new bounds via differential Galois theory.

Findings

Matrix differential equations with general coefficients require at least n parameters to simplify.

The proof employs differential Galois theory and essential dimension concepts.

Bounds are provided for the parameters needed in generic Picard-Vessiot extensions.

Abstract

We show that the $n\times n$ matrix differential equation $\delta(Y)=AY$ with $n^2$ general coefficients cannot be simplified to an equation in less than $n$ parameters by using gauge transformations whose coefficients are rational functions in the matrix entries of $A$ and their derivatives. Our proof uses differential Galois theory and a differential analogue of essential dimension. We also bound the minimum number of parameters needed to describe some generic Picard-Vessiot extensions.