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.
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Acknowledgements
This work contains results from the author’s thesis. The author thanks Anand Pillay and R´emi Jaoui for spotting a problem in an early version of Proposition 3.5, David Harbater for his generous input and helpful comments especially towards Proposition 3.5, the Kolchin Seminar in Differential Algebra for the opportunity to present an early version of this work, Michael Wibmer for helpful conversations, Mark van Hoeij and the referee for helpful comments, and Julia Hartmann for her patient advising and ample support.
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Tsui, M.C. Simplifying matrix differential equations with general coefficients. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2599-0
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DOI: https://doi.org/10.1007/s11856-023-2599-0