同构微分方程与微分范畴
Isomonodromic differential equations and differential categories
摘要：
利用微分范畴研究了参数化线性微分方程组的同构性和线性微分代数群的共轭性。我们证明了在参数的函数域上，在过滤线性闭的假设下，等单性等价于对每个参数的等单性。我们的结果意味着人们不再需要求解任何非线性微分方程来检验同构性。我们通过一个反例说明，削弱对参数的要求不能进一步加强这个结果。同时，我们证明了同构单值性等价于与参数化微分伽罗瓦群的常数共轭，推广了P. Cassidy和M.辛格，我们也明确证明。我们通过一系列的例子来说明我们的主要结果，特别是使用高斯-马宁连接和参数化微分伽罗瓦群之间的关系。
英文摘要：
We study isomonodromicity of systems of parameterized linear differential equations and related conjugacy properties of linear differential algebraic groups by means of differential categories. We prove that isomonodromicity is equivalent to isomonodromicity with respect to each parameter separately under a filtered-linearly closed assumption on the field of functions of parameters. Our result implies that one does not need to solve any non-linear differential equations to test isomonodromicity anymore. This result cannot be further strengthened by weakening the requirement on the parameters as we show by giving a counterexample. Also, we show that isomonodromicity is equivalent to conjugacy to constants of the associated parameterized differential Galois group, extending a result of P. Cassidy and M. Singer, which we also prove categorically. We illustrate our main results by a series of examples, using, in particular, a relation between Gauss-Manin connection and parameterized differential Galois groups.
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