Action of an endomorphism on (the solutions of) a linear differential equation
Lucia Di Vizio
Publications Mathématiques de Besançon no. 1  (2019), p. 21-39

The purpose of this survey is to provide the reader with a user friendly introduction to the two articles [8] and [9], which give a Galoisian description of the action of an endomorphism of a differential field (K,) on the solutions of a linear differential equation defined over (K,). After having introduced the theory, we give some concrete examples.

Le but de ce survol est de présenter d’une façon accessible le contenu des articles [8] et [9], qui donnent une description galoisienne de l’action d’un endomorphisme d’un corps différentiel (K,) sur les solutions d’une équation différentielle linéaire à coefficients dans (K,). Après une présentation de la théorie nous donnons quelques exemples d’applications.

Received : 2018-01-05
Published online : 2019-10-15
Classification:  12H10,  12H20,  34M15
Keywords: Differential Galois theory, discrete parameter, difference algebra
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     author = {Lucia Di Vizio},
     title = {Action of an endomorphism on (the solutions of) a linear differential equation},
     journal = {Publications Math\'ematiques de Besan\c con},
     publisher = {Presses universitaires de Franche-Comt\'e},
     number = {1},
     year = {2019},
     pages = {21-39},
     language = {en},
     url = {https://pmb.centre-mersenne.org/item/PMB_2019___1_21_0}
}
Di Vizio, Lucia. Action of an endomorphism on (the solutions of) a linear differential equation. Publications Mathématiques de Besançon, no. 1 (2019), pp. 21-39. pmb.centre-mersenne.org/item/PMB_2019___1_21_0/

[1] Frits Beukers Differential Galois theory, From number theory to physics (Les Houches, 1989), Springer (1992), pp. 413-439 | Zbl 0813.12001

[2] Phyllis J. Cassidy; Michael F. Singer Galois Theory of Parameterized Differential Equations and Linear Differential Algebraic Groups, Differential Equations and Quantum Groups, European Mathematical Society (IRMA Lectures in Mathematics and Theoretical Physics) Tome 9 (2007), pp. 113-157 | Zbl 1230.12003

[3] Zoé Chatzidakis; Ehud Hrushovski; Yaʼacov Peterzil Model theory of difference fields. II. Periodic ideals and the trichotomy in all characteristics, Proc. Lond. Math. Soc., Tome 85 (2002) no. 2, pp. 257-311 | Article | MR 1912052

[4] Richard M. Cohn Difference algebra, Interscience Publishers (1965), xiv+355 pages | MR MR0205987 | Zbl 0127.26402

[5] Teresa Crespo; Zbigniew Hajto Algebraic groups and differential Galois theory, American Mathematical Society, Graduate Studies in Mathematics, Tome 122 (2011), xiv+225 pages

[6] Lucia Di Vizio Approche galoisienne de la transcendance différentielle, Transendance et irrationalité, Société Mathématique de France (SMF Journée Annuelle) (2012), pp. 1-20 | Zbl 1342.12007

[7] Lucia Di Vizio; Charlotte Hardouin Descent for differential Galois theory of difference equations: confluence and q-dependence, Pac. J. Math., Tome 256 (2012) no. 1, pp. 79-104

[8] Lucia Di Vizio; Charlotte Hardouin; Michael Wibmer Difference Galois theory of linear differential equations, Adv. Math., Tome 260 (2014), pp. 1-58 | Zbl 1328.12013

[9] Lucia Di Vizio; Charlotte Hardouin; Michael Wibmer Difference algebraic relations among solutions of linear differential equations, J. Inst. Math. Jussieu, Tome 16 (2017) no. 1, pp. 59-119 | Article | MR 3591962

[10] Bernard Dwork; Giovanni Gerotto; Francis J. Sullivan An introduction to G-functions, Princeton University Press, Annals of Mathematics Studies, Tome 133 (1994)

[11] Charlotte Hardouin Galoisian approach to differential transcendence, Galois theories of linear difference equations: an introduction, American Mathematical Society (Mathematical Surveys and Monographs) Tome 211 (2016), pp. 43-102 | Zbl 1347.39001

[12] Charlotte Hardouin; Michael F. Singer Differential Galois theory of linear difference equations, Math. Ann., Tome 342 (2008) no. 2, pp. 333-377

[13] Ellis R. Kolchin Differential algebra and algebraic groups, Academic Press Inc., Pure and Applied Mathematics, Tome 54 (1973), xviii+446 pages | MR MR0568864 | Zbl 0264.12102

[14] Piotr Kowalski; Anand Pillay On algebraic σ-groups, Trans. Am. Math. Soc., Tome 359 (2007) no. 3, pp. 1325-1337

[15] Peter Landesman Generalized differential Galois theory, Trans. Am. Math. Soc., Tome 360 (2008) no. 8, pp. 4441-4495

[16] Alexander Levin Difference algebra, Springer, Algebra and Applications, Tome 8 (2008)

[17] Andy R. Magid Lectures on differential Galois theory, American Mathematical Society, University Lecture Series, Tome 7 (1994)

[18] Alexey Ovchinnikov; Michael Wibmer σ-Galois theory of linear difference equations, Int. Math. Res. Not. (2015) no. 12, pp. 3962-4018

[19] C. Praagman The formal classification of linear difference operators, Indag. Math., Tome 45 (1983) no. 2, pp. 249-261 | Zbl 0519.39003

[20] Marius van der Put; Michael F. Singer Galois theory of linear differential equations, Springer (2003), viii+180 pages

[21] Julien Roques La théorie de Galois différentielle, Gaz. Math., Soc. Math. Fr. (2017) no. 152, pp. 59-63 | Zbl 06938832

[22] Michael F. Singer Introduction to the Galois theory of linear differential equations, Algebraic theory of differential equations, Cambridge University Press (London Mathematical Society Lecture Note Series) Tome 357 (2009), pp. 1-82

[23] Michael Wibmer A Chevalley theorem for difference equations, Math. Ann., Tome 354 (2012) no. 4, pp. 1369-1396 | Article | MR 2992999

[24] Benjamin Antieau; Alexey Ovchinnikov; Dmitry Trushin Galois theory of difference equations with periodic parameters, Commun. Algebra, Tome 42 (2014) no. 9, pp. 3902-3943 | Article | MR 3200069

[25] Carlos E. Arreche A Galois-theoretic proof of the differential transcendence of the incomplete Gamma function, J. Algebra, Tome 389 (2013), pp. 119-127 | Article | MR 3065995

[26] Carlos E. Arreche Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation, Adv. Appl. Math., Tome 57 (2014), pp. 44-59 | Article | MR 3206521

[27] Carlos E. Arreche On the computation of the parameterized differential Galois group for a second-order linear differential equation with differential parameters, J. Symb. Comput., Tome 75 (2016), pp. 25-55 | Article | MR 3451330

[28] Carlos E. Arreche Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation, Commun. Contemp. Math., Tome 19 (2017) no. 6, 1650056, 42 pages | Article | MR 3691498 | Zbl 1384.65091

[29] Carlos E. Arreche; Michael F. Singer Galois groups for integrable and projectively integrable linear difference equations, J. Algebra, Tome 480 (2017), pp. 423-449 | Article | MR 3633315

[30] Annette Bachmayr; Michael Wibmer Torsors for Difference Algebraic Groups (2016) (https://arxiv.org/abs/1607.07035)

[31] Daniel Bertrand Théories de Galois différentielles et transcendance, Ann. Inst. Fourier, Tome 59 (2009) no. 7, pp. 2773-2803 | MR 2649338

[32] David Blázquez-Sanz; Weimar Muñoz Generalized linear cellular automata in groups and difference Galois theory, J. Difference Equ. Appl., Tome 21 (2015) no. 2, pp. 127-154 | Article | MR 3292134 | Zbl 1379.37028

[33] Guy Casale Irréductibilité de la première équation de Painlevé, C. R. Math. Acad. Sci. Paris, Tome 343 (2006) no. 2, pp. 95-98 | Article | MR 2242039

[34] Guy Casale Le groupoïde de Galois de P 1 et son irréductibilité, Comment. Math. Helv., Tome 83 (2008) no. 3, pp. 471-519 | Article | MR 2410777

[35] Guy Casale; Julien Roques Non-integrability by discrete quadratures, J. Reine Angew. Math., Tome 687 (2014), pp. 87-112 | Article | MR 3176608

[36] Shaoshi Chen; Frédéric Chyzak; Ruyong Feng; Guofeng Fu; Ziming Li On the existence of telescopers for mixed hypergeometric terms, J. Symb. Comput., Tome 68 (2015), pp. 1-26 | Article | MR 3283835

[37] Shaoshi Chen; Michael F. Singer Residues and telescopers for bivariate rational functions, Adv. Appl. Math., Tome 49 (2012) no. 2, pp. 111-133 | Article | MR 2946428

[38] Teresa Crespo; Zbigniew Hajto; Marius van der Put Real and p-adic Picard-Vessiot fields, Math. Ann., Tome 365 (2016) no. 1-2, pp. 93-103 | Article | MR 3498905

[39] Lucia Di Vizio; Charlotte Hardouin Courbures, groupes de Galois génériques et D-groupoïde de Galois d’un système aux q-différences, C. R. Math. Acad. Sci. Paris, Tome 348 (2010) no. 17-18, pp. 951-954 | Article | MR 2721777

[40] Thomas Dreyfus Computing the Galois group of some parameterized linear differential equation of order two, Proc. Am. Math. Soc., Tome 142 (2014) no. 4, pp. 1193-1207 | Article | MR 3162242

[41] Thomas Dreyfus A density theorem in parametrized differential Galois theory, Pac. J. Math., Tome 271 (2014) no. 1, pp. 87-141 | Article | MR 3259762

[42] Thomas Dreyfus; Charlotte Hardouin; Julien Roques Hypertranscendence of solutions of Mahler equations (2015) (https://arxiv.org/abs/1507.03361)

[43] Thomas Dreyfus; Charlotte Hardouin; Julien Roques Functional relations of solutions of q-difference equations (2016) (https://arxiv.org/abs/1603.06771)

[44] Thomas Dreyfus; Charlotte Hardouin; Julien Roques; Michael F. Singer Walks in the quarter plane, genus zero case (2017) (https://arxiv.org/abs/1710.02848)

[45] Thomas Dreyfus; Kilian Raschel Differential transcendence & algebraicity criteria for the series counting weighted quadrant walks, Publ. Math. Besançon, Algèbre Théorie Nombres, Tome 2019 (2019) no. 1, pp. 41-80

[46] Thomas Dreyfus; Julien Roques Galois groups of difference equations of order two on elliptic curves, SIGMA, Symmetry Integrability Geom. Methods Appl., Tome 11 (2015), 003, 23 pages | Article | MR 3313679

[47] Ruyong Feng; Michael F. Singer; Min Wu An algorithm to compute Liouvillian solutions of prime order linear difference-differential equations, J. Symb. Comput., Tome 45 (2010) no. 3, pp. 306-323 | Article | MR 2578341

[48] Ruyong Feng; Michael F. Singer; Min Wu Liouvillian solutions of linear difference-differential equations, J. Symb. Comput., Tome 45 (2010) no. 3, pp. 287-305 | Article | MR 2578340

[49] James Freitag Completeness in partial differential algebraic geometry, J. Algebra, Tome 420 (2014), pp. 350-372 | Article | MR 3261465

[50] Henri Gillet; Sergey Gorchinskiy; Alexey Ovchinnikov Parameterized Picard-Vessiot extensions and Atiyah extensions, Adv. Math., Tome 238 (2013), pp. 322-411 | Article | MR 3033637

[51] Oleg Golubitsky; Marina Kondratieva; Alexey Ovchinnikov; Agnes Szanto A bound for orders in differential Nullstellensatz, J. Algebra, Tome 322 (2009) no. 11, pp. 3852-3877 | Article | MR 2556127

[52] Sergey Gorchinskiy; Alexey Ovchinnikov Isomonodromic differential equations and differential categories, J. Math. Pures Appl., Tome 102 (2014) no. 1, pp. 48-78 | Article | MR 3212248

[53] Charlotte Hardouin; Andrey Minchenko; Alexey Ovchinnikov Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence, Math. Ann., Tome 368 (2017) no. 1-2, pp. 587-632 | Article | MR 3651584

[54] Moshe Kamensky Tannakian formalism over fields with operators, Int. Math. Res. Not. (2013) no. 24, pp. 5571-5622 | MR 3144174

[55] Omar León Sánchez Relative D-groups and differential Galois theory in several derivations, Trans. Am. Math. Soc., Tome 367 (2015) no. 11, pp. 7613-7638 | Article | MR 3391895

[56] Omar León Sánchez On the model companion of partial differential fields with an automorphism, Isr. J. Math., Tome 212 (2016) no. 1, pp. 419-442 | Article | MR 3504332

[57] Omar León Sánchez; Joel Nagloo On parameterized differential Galois extensions, J. Pure Appl. Algebra, Tome 220 (2016) no. 7, pp. 2549-2563 | Article | MR 3457983

[58] Omar León Sánchez; Anand Pillay Some definable Galois theory and examples, Bull. Symb. Log., Tome 23 (2017) no. 2, pp. 145-159 | Article | MR 3664720 | Zbl 06786768

[59] Ziming Li; Min Wu Transforming linear functional systems into fuzzy integrable systems, J. Symb. Comput., Tome 47 (2012) no. 6, pp. 711-732 | Article | MR 2908590

[60] Annette Maier On the parameterized differential inverse Galois problem over k((t))(x), J. Algebra, Tome 428 (2015), pp. 43-53 | Article | MR 3314284

[61] Stéphane Malek On complex singularity analysis for linear partial q-difference-differential equations using nonlinear differential equations, J. Dyn. Control Syst., Tome 19 (2013) no. 1, pp. 69-93 | Article | MR 3010267

[62] Andreas Maurischat A categorical approach to Picard–Vessiot theory, Theory Appl. Categ., Tome 32 (2017), pp. 488-525 | MR 3633711 | Zbl 1388.13022

[63] Andrey Minchenko; Alexey Ovchinnikov Zariski closures of reductive linear differential algebraic groups, Adv. Math., Tome 227 (2011) no. 3, pp. 1195-1224 | Article | MR 2799605

[64] Andrey Minchenko; Alexey Ovchinnikov Extensions of differential representations of SL 2 and tori, J. Inst. Math. Jussieu, Tome 12 (2013) no. 1, pp. 199-224 | Article | MR 3001738

[65] Andrey Minchenko; Alexey Ovchinnikov; Michael F. Singer Unipotent differential algebraic groups as parameterized differential Galois groups, J. Inst. Math. Jussieu, Tome 13 (2014) no. 4, pp. 671-700 | Article | MR 3249687

[66] K. Mitchi Some applications of parameterized Picard-Vessiot theory, Izv. Ross. Akad. Nauk, Ser. Mat., Tome 80 (2016) no. 1, pp. 177-200 | Article | MR 3462679

[67] Claude Mitschi; Michael F. Singer Monodromy groups of parameterized linear differential equations with regular singularities, Bull. Lond. Math. Soc., Tome 44 (2012) no. 5, pp. 913-930 | Article | MR 2975151

[68] Claude Mitschi; Michael F. Singer Projective isomonodromy and Galois groups, Proc. Am. Math. Soc., Tome 141 (2013) no. 2, pp. 605-617 | Article | MR 2996965

[69] Juan J. Morales-Ruiz Picard-Vessiot theory and integrability, J. Geom. Phys., Tome 87 (2015), pp. 314-343 | Article | MR 3282377

[70] Shuji Morikawa On a general difference Galois theory. I, Ann. Inst. Fourier, Tome 59 (2009) no. 7, pp. 2709-2732 | MR 2649331

[71] Pierre Nguyen Hypertranscedance de fonctions de Mahler du premier ordre, C. R. Math. Acad. Sci. Paris, Tome 349 (2011) no. 17-18, pp. 943-946 | Article | MR 2838240

[72] Hiroshi Ogawara Differential transcendency of a formal Laurent series satisfying a rational linear q-difference equation, Funkc. Ekvacioj, Tome 57 (2014) no. 3, pp. 477-488 | Article | MR 3308705

[73] Alexey Ovchinnikov Tannakian approach to linear differential algebraic groups, Transform. Groups, Tome 13 (2008) no. 2, pp. 413-446 | Article | MR 2426137

[74] Alexey Ovchinnikov Differential Tannakian categories, J. Algebra, Tome 321 (2009) no. 10, pp. 3043-3062 | Article | MR 2512641

[75] Alexey Ovchinnikov Tannakian categories, linear differential algebraic groups, and parametrized linear differential equations, Transform. Groups, Tome 14 (2009) no. 1, pp. 195-223 | Article | MR 2480859

[76] Alexey Ovchinnikov Difference integrability conditions for parameterized linear difference and differential equations, Adv. Appl. Math., Tome 53 (2014), pp. 61-71 | Article | MR 3149694

[77] Alexey Ovchinnikov; Michael Wibmer Tannakian categories with semigroup actions, Can. J. Math., Tome 69 (2017) no. 3, pp. 687-720

[78] Ana Peón Nieto On σδ-Picard-Vessiot extensions, Commun. Algebra, Tome 39 (2011) no. 4, pp. 1242-1249 | Article | MR 2782602

[79] Michael F. Singer Linear algebraic groups as parameterized Picard-Vessiot Galois groups, J. Algebra, Tome 373 (2013), pp. 153-161 | Article | MR 2995020

[80] Ivan Tomašić Twisted Galois stratification, Nagoya Math. J., Tome 222 (2016) no. 1, pp. 1-60 | Article | MR 3509221

[81] Dmitry Trushin Splitting fields and general differential Galois theory, Mat. Sb., Tome 201 (2010) no. 9, pp. 77-110 | Article | MR 2760461

[82] Hiroshi Umemura On the definition of the Galois groupoid, Differential equations and singularities, Société Mathématique de France (Astérisque) Tome 323 (2009), pp. 441-452 | MR 2647982 | Zbl 1205.12005

[83] Michael Wibmer A Chevalley theorem for difference equations, Math. Ann., Tome 354 (2012) no. 4, pp. 1369-1396 | Article | MR 2992999

[84] Michael Wibmer Existence of -parameterized Picard-Vessiot extensions over fields with algebraically closed constants, J. Algebra, Tome 361 (2012), pp. 163-171 | Article | MR 2921616