Simplifying matrix differential equations with general coefficients

Abstract

We show that the n × n matrix differential equation δ(Y) = AY with n² 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.