Abstract:
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 the Gauss–Manin connection and parameterized differential Galois groups.
S. Gorchinskiy was supported by the grants RFBR 11-01-00145, 12-01-31506, 12-01-3302, 13-01-12420, AG Laboratory NRU HSE, RF government grant, ag.11.G34.31.0023, and by Dmitry Zimin’s Dynasty Foundation.
A. Ovchinnikov was supported by the grants: NSF CCF-0952591 and PSC-CUNY No. 60001-40 41.
Received: 19.09.2012
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Article
Language: English
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This publication is cited in the following 11 articles:
Michael Wibmer, “Finiteness Properties of Affine Difference Algebraic Groups”, International Mathematics Research Notices, 2022:1 (2022), 506
Andrei Minchenko, Alexey Ovchinnikov, “Triviality of differential Galois cohomology of linear differential algebraic groups”, Communications in Algebra, 47:12 (2019), 5094
Lucia Di Vizio, “Action of an endomorphism on (the solutions of) a linear differential equation”, Publications mathématiques de Besançon. Algèbre et théorie des nombres, 2019, no. 1, 21
Andrei Minchenko, Alexey Ovchinnikov, “Calculating Galois groups of third-order linear differential equations with parameters”, Commun. Contemp. Math., 20:04 (2018), 1750038
Carlos E. Arreche, Michael F. Singer, “Galois groups for integrable and projectively integrable linear difference equations”, Journal of Algebra, 480 (2017), 423
Lucia Di Vizio, Charlotte Hardouin, Michael Wibmer, “DIFFERENCE ALGEBRAIC RELATIONS AMONG SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS”, J. Inst. Math. Jussieu, 16:1 (2017), 59
Carlos E. Arreche, “On the computation of the parameterized differential Galois group for a second-order linear differential equation with differential parameters”, Journal of Symbolic Computation, 75 (2016), 25
Andrey Minchenko, Alexey Ovchinnikov, Michael F. Singer, “Reductive Linear Differential Algebraic Groups and the Galois Groups of Parameterized Linear Differential Equations”, International Mathematics Research Notices, 2015:7 (2015), 1733
S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Russian Math. Surveys, 70:5 (2015), 901–951
Thomas Dreyfus, “A density theorem in parametrized differential Galois theory”, Pacific J. Math., 271:1 (2014), 87
Carlos E. Arreche, “Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation”, Advances in Applied Mathematics, 57 (2014), 44
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