Integrability conditions for parameterized linear difference equations

We study integrability conditions for systems of parameterized linear difference equations and related properties of linear differential algebraic groups. We show that isomonodromicity of such a system is equivalent to isomonodromicity with respect to each parameter separately under a linearly differentially closed assumption on the field of differential parameters. Due to our result, it is no longer necessary to solve non-linear differential equations to verify isomonodromicity, which will improve efficiency of computation with these systems. Moreover, it is not possible to further strengthen this result by removing the requirement on the parameters, as we show by giving a counterexample. We also discuss the relation between isomonodromicity and the properties of the associated parameterized difference Galois group.

[1]  L’INSTITUT Fourier,et al.  On a general difference Galois theory I , 2010 .

[2]  A. Granier.  A Galois $D$-groupoid for $q$-difference equations , 2011 .

[3]  A. Ovchinnikov,et al.  Zariski closures of reductive linear differential algebraic groups , 2010, 1005.0042.

[4]  Thomas Dreyfus,et al.  Computing the Galois group of some parameterized linear differential equation of order two , 2011, 1110.1053.

[5]  Michael Wibmer,et al.  Existence of $\partial$-parameterized Picard-Vessiot extensions over fields with algebraically closed constants , 2011, 1104.3514.

[6]  Lucia Di Vizio.  Approche galoisienne de la transcendance diff\'erentielle , 2014, 1404.3611.

[7]  Alexey Ovchinnikov,et al.  Parameterized Picard-Vessiot extensions and Atiyah extensions , 2013 .

[8]  C. Hardouin,et al.  On the Grothendieck conjecture on p-curvatures for q-difference equations , 2012, 1205.1692.

[9]  渋谷 泰隆.  Linear differential equations in the complex domain : problems of analytic continuation , 1990 .

[10]  A. Ovchinnikov,et al.  Extensions of differential representations of SL2 and tori , 2011, Journal of the Institute of Mathematics of Jussieu.

[11]  Michael F Singer,et al.  Linear algebraic groups as parameterized Picard-Vessiot Galois groups , 2011, 1108.0406.

[12]  Michael F. Singer,et al.  Unipotent differential algebraic groups as parameterized differential Galois groups , 2013, Journal of the Institute of Mathematics of Jussieu.

[13]  Michio Jimbo,et al.  Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and τ-function , 1981 .

[14]  Charlotte Hardouin,et al.  Descent for differential Galois theory of difference equations: confluence and q-dependence , 2011, 1103.5067.

[15]  Michio Jimbo,et al.  Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III , 1981 .

[16]  M. Kamensky.  Tannakian formalism over fields with operators , 2011, 1111.7285.

[17]  Lucia Di Vizio,et al.  Parameterized generic Galois groups for q-difference equations, followed by the appendix "The Galois D-groupoid of a q-difference system" by Anne Granier , 2012 .

[18]  Marius van der Put,et al.  Galois Theory of Linear Differential Equations , 2012 .

[20]  Moulay A. Barkatou,et al.  Computing closed form solutions of integrable connections , 2012, ISSAC.

[21]  Tannakian Approach to Linear Differential Algebraic Groups , 2007, math/0702846.

[22]  Phyllis J. Cassidy,et al.  Differential algebraic groups , 1972 .

[23]  Carlos E. Arreche.  Computing the differential Galois group of a one-parameter family of second order linear differential equations , 2012, 1208.2226.

[24]  Yasutaka Sibuya,et al.  Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation , 1990 .

[25]  O. Hölder,et al.  Ueber die Eigenschaft der Gammafunction keiner algebraischen Differentialgleichung zu genügen , 1886 .

[26]  Charlotte Hardouin,et al.  Hypertranscendance des systèmes aux différences diagonaux , 2008, Compositio Mathematica.

[27]  M. Singer,et al.  A Jordan–Hölder Theorem for differential algebraic groups , 2010, 1003.3274.

[28]  Nalini Joshi,et al.  Exact solutions of a q-discrete second Painlevé equation from its iso-monodromy deformation problem. II. Hypergeometric solutions , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[29]  Claude Sabbah,et al.  The work of Andrey Bolibrukh on isomonodromic deformations , 2006 .

[30]  Michio Jimbo,et al.  Deformation of linear ordinary differential equations, II , 1980 .

[31]  Charlotte Hardouin,et al.  Courbures, groupes de Galois gnriques et D-groupode de Galois d'un systme aux q-diffrences , 2010 .

[32]  P. Cassidy.  The differential rational representation algebra on a linear differential algebraic group , 1975 .

[33]  Michael F. Singer,et al.  Differential Galois theory of linear difference equations , 2008, 0801.1493.

[34]  E. Kolchin.  Differential Algebra and Algebraic Groups , 2012 .

[35]  Michael F. Singer,et al.  Reductive Linear Differential Algebraic Groups and the Galois Groups of Parameterized Linear Differential Equations , 2013, 1304.2693.

[36]  Lucia Di Vizio,et al.  Galois theories of q-difference equations: comparison theorems , 2012, Confluentes Mathematici.

[37]  Nalini Joshi,et al.  Exact solutions of a q-discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[38]  Joris van der Hoeven,et al.  Characteristic set method for differential-difference polynomial systems , 2009, J. Symb. Comput..

[39]  Alexey Ovchinnikov,et al.  Isomonodromic differential equations and differential categories , 2012, 1202.0927.