简化具有一般系数的矩阵微分方程

Pub Date : 2023-12-18 DOI:10.1007/s11856-023-2599-0

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引用次数: 0

摘要

摘要我们证明了具有 n2 个一般系数的 n × n 矩阵微分方程 δ(Y) = AY 无法通过使用其系数为 A 的矩阵项中的有理函数及其导数的规整变换简化为小于 n 个参数的方程。我们的证明使用了微分伽罗瓦理论和本质维度的微分类似方法。我们还限定了描述某些一般皮卡-维西奥扩展所需的最小参数数。

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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.

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