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Linear Meromorphic Differential Equations:
a Modern Point of View

Author: V. S. Varadarajan
Journal: Bull. Amer. Math. Soc. 33 (1996), 1-42
MSC (1991): Primary 34A20, 13N05
MathSciNet review: 1339809
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Abstract: A large part of the modern theory of differential equations in the complex domain is concerned with regular singularities and holonomic systems. However the theory of differential equations with irregular singularities has a long history and has become very active in recent years. Substantial links of this theory to the theory of algebraic groups, commutative algebra, resurgent functions, and Galois differential methods have been discovered. This survey attempts a general introduction to some of these aspects, with emphasis on reduction theory, asymptotic analysis, Stokes phenomena, and certain moduli problems.

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  • 1 B. Riemann, Beiträge zur Theorie der durch die Gauss'sche Reihe $F(\alpha , \beta , \gamma , x)$ darstellbaren Funktionen, Abh. Kon. Ges. d. Wiss. zu Göttingen VII Math. Classe, A-22 (1857); Collected Papers (Raghavan Narasimhan, ed.), Springer-Verlag, Berlin, 1990, pp. 99--119.
  • 2 J. Gray, (a) Fuchs and the theory of differential equations, Bull. Amer. Math. Soc. 10, No. 1 (1984), 1--26, MR 85h:0102b; (b) Linear Differential Equations and Group Theory from Riemann to Poincaré, Birkhäuser, Boston, Basel, 1986, MR 85h:0102a.
  • 3 V.I. Arnold and Yu.S. Il'yashenko, Dynamical Systems I, vol. 1 of Encyclopaedia of Mathematical Sciences (D.V. Anosov and V.I. Arnold, eds.), Springer, New York, 1988.
  • 4 Carl H. FitzGerald, Sheng Gong, and Roger W. Barnard, The growth and 1/4-theorems for starlike mappings in 𝐶ⁿ, Chinese Sci. Bull. 35 (1990), no. 5, 357–359. MR 1057229
    V. S. Varadarajan, Some remarks on meromorphic differential equations with simple singularities, Calcutta Mathematical Society. Diamond-cum-platinum jubilee commemoration volume (1908–1983), Part I, Calcutta Math. Soc., Calcutta, 1984, pp. 49–61. MR 845039
  • 5 (a) D. Bertrand, Travaux récent sur les points singuliers des équations différentielle linéaires, Springer Lecture Notes in Mathematics-Sém. Bourbaki, 1978/79, Exposés 525--542, 770(1980); (b) Groupes algébriques et équations différentielles linéaires, Sém. Bourbaki, 1991--1992, Exposé n$^\circ $ 750, (c) A. Beauville, Monodromie des systèmes différentiels linéaires à pôles simples sur la sphère Riemann, Sém. Bourbaki, 1992--1993, Exposé n$^\circ $ 765.
  • 6 D. V. Anosov and A. A. Bolibruch, The Riemann-Hilbert problem, Aspects of Mathematics, E22, Friedr. Vieweg & Sohn, Braunschweig, 1994. MR 1276272
    Vladimir Petrov Kostov, Fuchsian linear systems on 𝐶𝑃¹ and the Riemann-Hilbert problem, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 2, 143–148 (English, with English and French summaries). MR 1197226
    Toshiaki Yokoyama, A system of total differential equations of two variables and its monodromy group, Funkcial. Ekvac. 35 (1992), no. 1, 65–93. MR 1172422
    Michael F. Singer, An outline of differential Galois theory, Computer algebra and differential equations, Comput. Math. Appl., Academic Press, London, 1990, pp. 3–57. MR 1038057
  • 7 Jean-Pierre Ramis, Phénomène de Stokes et filtration Gevrey sur le groupe de Picard-Vessiot, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 5, 165–167 (French, with English summary). MR 801953
    Jean-Pierre Ramis, Phénomène de Stokes et resommation, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 4, 99–102 (French, with English summary). MR 799602
  • 8 Donald G. Babbitt and V. S. Varadarajan, Formal reduction theory of meromorphic differential equations: a group theoretic view, Pacific J. Math. 109 (1983), no. 1, 1–80. MR 716289
    Donald G. Babbitt and V. S. Varadarajan, Deformations of nilpotent matrices over rings and reduction of analytic families of meromorphic differential equations, Mem. Amer. Math. Soc. 55 (1985), no. 325, iv+147. MR 787539,
    D. G. Babbitt and V. S. Varadarajan, Local moduli for meromorphic differential equations, Astérisque 169-170 (1989), 217 (English, with French summary). MR 1014083
    Donald G. Babbitt and V. S. Varadarajan, Some remarks on the asymptotic existence theorem for meromorphic differential equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), no. 2, 247–262. MR 1014999
  • 9 Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
  • 10 Juri I. Manin, Moduli fuchsiani, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 19 (1965), 113–126 (Italian). MR 180581
    Nicholas M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 175–232. MR 291177
  • 11 A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers, Algebraic 𝐷-modules, Perspectives in Mathematics, vol. 2, Academic Press, Inc., Boston, MA, 1987. MR 882000
  • 12 M. Hukuhara, (a) Sur les points singuliers des équations différentielles linéaires, II, Jour. Fac. Sci. Hokkaido Univ. 5 (1937), 123--166.(b) Sur les points singuliers des équations différentielles linéaires, III, Jour. Fac. Sci. Kyushu Univ. 2 (1942), 125--137, MR 9:92i.
  • 13 H. Turrittin, Convergent solutions of ordinary differential equations in the neighborhood of an irregular singular point, Acta Math. 93 (1955), 27--66, MR 16:925a.
  • 14 A. H. M. Levelt, Jordan decomposition for a class of singular differential operators, Ark. Mat. 13 (1975), 1–27. MR 500294,
  • 15 (a) E. Fabry, Sur les intégrales des équations différentielles linéaires à coefficients rationnels, Thése, Paris, 1885; (b) H. Poincaré, Sur les intégrales des équations linéaires, Acta Math. 8 (1986), 295--344.
  • 16 Werner Balser, Zum Einzigkeitssatz in der Invariantentheorie meromorpher Differentialgleichungen, J. Reine Angew. Math. 318 (1980), 51–82 (German). MR 579383,
    W. Balser, W. B. Jurkat, and D. A. Lutz, A general theory of invariants for meromorphic differential equations. I. Formal invariants, Funkcial. Ekvac. 22 (1979), no. 2, 197–221. MR 556577
    W. B. Jurkat, Meromorphe Differentialgleichungen, Lecture Notes in Mathematics, vol. 637, Springer, Berlin, 1978 (German). MR 494886
  • 17 G. Appleby, Nilpotent matrices over discrete valuation rings, Thesis, UCLA, 1993.
  • 18 Wolfgang Wasow, Linear turning point theory, Applied Mathematical Sciences, vol. 54, Springer-Verlag, New York, 1985. MR 771669
  • 19 Winfried Bruns, E. Graham Evans Jr., and Phillip A. Griffith, Syzygies, ideals of height two, and vector bundles, J. Algebra 67 (1980), no. 1, 143–162. MR 595025,
  • 20 F. Beukers and G. Heckman, Monodromy for the hypergeometric function _{𝑛}𝐹_{𝑛-1}, Invent. Math. 95 (1989), no. 2, 325–354. MR 974906,
  • 21 Philip Candelas, Xenia C. de la Ossa, Paul S. Green, and Linda Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), no. 1, 21–74. MR 1115626,
    Ana Cristina Cadavid and Sergio Ferrara, Picard-Fuchs equations and the moduli space of superconformal field theories, Phys. Lett. B 267 (1991), no. 2, 193–199. MR 1128291,
    David R. Morrison, Picard-Fuchs equations and mirror maps for hypersurfaces, Essays on mirror manifolds, Int. Press, Hong Kong, 1992, pp. 241–264. MR 1191426
  • 22 Harish-Chandra, Collected papers. Vol. I, Springer-Verlag, New York, 1984. 1944–1954; Edited and with an introduction by V. S. Varadarajan; With introductory essays by Nolan R. Wallach and Roger Howe. MR 726025
  • 23 Izrail M. Gelfand, Collected papers. Vol. III, Springer-Verlag, Berlin, 1989. Edited by S. G. Gindikin, V. W. Guillemin, A. A. Kirillov, B. Kostant and S. Sternberg; With a foreword by Gindikin; With a contribution by Kostant. MR 997939
  • 24 Masaaki Yoshida, Fuchsian differential equations, Aspects of Mathematics, E11, Friedr. Vieweg & Sohn, Braunschweig, 1987. With special emphasis on the Gauss-Schwarz theory. MR 986252
  • 25 G. J. Heckman and E. M. Opdam, Root systems and hypergeometric functions. I, Compositio Math. 64 (1987), no. 3, 329–352. MR 918416
  • 26 A. H. M. Levelt, Stabilizing differential operators. A method for computing invariants at irregular singularities, Differential equations and computer algebra, Comput. Math. Appl., Academic Press, London, 1991, pp. 181–228. MR 1115234
    D. G. Babbitt and V. S. Varadarajan, Formal reduction of meromorphic differential equations containing a parameter, Differential equations and computer algebra, Comput. Math. Appl., Academic Press, London, 1991, pp. 77–111. MR 1115230
  • 27 Hideyuki Majima, Asymptotic analysis for integrable connections with irregular singular points, Lecture Notes in Mathematics, vol. 1075, Springer-Verlag, Berlin, 1984. MR 757897
    A. R. P. van den Essen and A. H. M. Levelt, Irregular singularities in several variables, Mem. Amer. Math. Soc. 40 (1982), no. 270, iv+43. MR 677092,
  • 28 B. Malgrange, Remarques sur les équations différentielles à points singuliers irréguliers, Équations différentielles et systèmes de Pfaff dans le champ complexe (Sem., Inst. Rech. Math. Avancée, Strasbourg, 1975) Lecture Notes in Math., vol. 712, Springer, Berlin, 1979, pp. 77–86 (French). MR 548145
  • 29 Yasutaka Sibuya, Stokes phenomena, Bull. Amer. Math. Soc. 83 (1977), no. 5, 1075–1077. MR 442337,
  • 30 M. Loday--Richaud, Stokes phenomenon, multisummability, and differential Galois groups, Prépublications, Université de Paris--Sud Mathematiques, Orsay, 1992.
  • 31 J.-P. Ramis and Y. Sibuya, Hukuhara domains and fundamental existence and uniqueness theorems for asymptotic solutions of Gevrey type, Asymptotic Anal. 2 (1989), no. 1, 39–94. MR 991416
  • 32 Anne Duval and Claude Mitschi, Matrices de Stokes et groupe de Galois des équations hypergéométriques confluentes généralisées, Pacific J. Math. 138 (1989), no. 1, 25–56 (French). MR 992173
  • 33 Claude Mitschi, Groupe de Galois différentiel des équations hypergéométriques confluentes généralisées, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 4, 217–220 (French, with English summary). MR 1006733
  • 34 W. Balser, B. L. J. Braaksma, J.-P. Ramis, and Y. Sibuya, Multisummability of formal power series solutions of linear ordinary differential equations, Asymptotic Anal. 5 (1991), no. 1, 27–45. MR 1132079
    B. L. J. Braaksma, Multisummability and Stokes multipliers of linear meromorphic differential equations, J. Differential Equations 92 (1991), no. 1, 45–75. MR 1113588,
  • 35 Jean Écalle, Les fonctions résurgentes. Tome III, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 85, Université de Paris-Sud, Département de Mathématiques, Orsay, 1985 (French). L’équation du pont et la classification analytique des objects locaux. [The bridge equation and analytic classification of local objects]. MR 852210
    B. Candelpergher, J.-C. Nosmas, and F. Pham, Approche de la résurgence, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1993 (French, with French summary). MR 1250603
    J.-P. Ramis and J. Martinet, Théorie de Galois différentielle et resommation, Computer algebra and differential equations, Comput. Math. Appl., Academic Press, London, 1990, pp. 117–214 (French). MR 1038060

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Additional Information

V. S. Varadarajan
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095-1555

Received by editor(s): October 24, 1994
Received by editor(s) in revised form: June 22, 1995
Additional Notes: This is a revised and expanded version of an invited hour talk at the AMS meeting in Portland, Oregon, June 15, 1991. Due to various personal circumstances its preparation has been delayed till now.
Article copyright: © Copyright 1996 American Mathematical Society