Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

    B. Riemann, BeitrĂ€ge 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 . J. Gray, (a) Fuchs and the theory of differential equations, Bull. Amer. Math. Soc. 10, No. 1 (1984 ), 1–26 . J. Gray, Linear Differential Equations and Group Theory from Riemann to PoincarĂ©, BirkhĂ€user, Boston, Basel, 1986 . 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 .
  • Carl H. FitzGerald, Sheng Gong, and Roger W. Barnard, The growth and $1/4$-theorems for starlike mappings in ${\bf C}^n$, 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
  • A. Treibich Kohn, Un resultat de Plemelj, “Mathematique et Physique ", Sem. Ecole. Norm. Sup. (eds. L. Boutet de Monvel, A. Douady, and J. L. Verdier), BirkhĂ€user, 1983 D. Bertrand, Travaux rĂ©cent sur les points singuliers des Ă©quations diffĂ©rentielle linĂ©aires, Springer Lecture Notes in Mathematics-SĂ©m. Bourbaki, 1978/79, ExposĂ©s 525–542, 770(1980) . D. Bertrand, Groupes algĂ©briques et Ă©quations diffĂ©rentielles linĂ©aires, SĂ©m. Bourbaki, 1991–1992, ExposĂ© n$^\circ$ 750, (c) A. Beauville, Monodromie des systĂšmes diffĂ©rentiels linĂ©aires Ă  pĂŽles simples sur la sphĂšre Riemann, SĂ©m. Bourbaki, 1992–1993, ExposĂ© n$^\circ$ 765 .
  • D. V. Anosov and A. A. Bolibruch, The Riemann-Hilbert problem, Aspects of Mathematics, E22, Friedr. Vieweg & Sohn, Braunschweig, 1994. MR 1276272
  • V. Kostov, Fuchsian linear systems on $\mathbf {CP}^1$ and Riemann–Hilbert’s problem, PrĂ©publication UniversitĂ© de Nice (1991); Fuchsian linear systems on $\mathbf {CP}^1$ and the Riemann–Hilbert problem, C. R. Acad. Sci. Paris. Ser. I, t. 315 (1992), 143–148 .
  • Vladimir Petrov Kostov, Fuchsian linear systems on ${\bf C}{\rm P}^1$ and the Riemann-Hilbert problem, C. R. Acad. Sci. Paris SĂ©r. I Math. 315 (1992), no. 2, 143–148 (English, with English and French summaries). MR 1197226
  • K. Iwasaki, Moduli and deformation for Fuchsian projective connections on a Riemann surface, Jour. of the Fac. of Sci., Univ. of Tokyo, Sec. IA 38 ( 1991), 431–531 .
  • 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
  • M. F. Singer, Moduli of linear differential equations on the Riemann sphere with fixed Galois groups, Pacific. Jour. of Math. 160, 343–395 .
  • Jean-Pierre Ramis, PhĂ©nomĂšne de Stokes et filtration Gevrey sur le groupe de Picard-Vessiot, C. R. Acad. Sci. Paris SĂ©r. I Math. 301 (1985), no. 5, 165–167 (French, with English summary). MR 801953
  • Jean-Pierre Ramis, PhĂ©nomĂšne de Stokes et resommation, C. R. Acad. Sci. Paris SĂ©r. I Math. 301 (1985), no. 4, 99–102 (French, with English summary). MR 799602
  • J. P. Ramis, Filtration Gevrey sur le groupe de Picard–Vessiot d’une Ă©quation diffĂ©rentielle irrĂ©guliere , Preprint IMPA, Rio de Janeiro 45 (1985 ). J. P. Ramis, Irregular connections, savage $\pi _1$, and confluence, Conference in Katata, Taniguchi Foundation, Preprint, 1988 . J. P. Ramis, Les series $k$-summables et leurs applications, Analysis, microlocal analysis and relativistic quantum field theory, Springer Lecture Notes in Physics 126 (1980), 178–197 . J. P. Ramis, Confluence and resurgence, Jour. Fac. Sci. Univ. Tokyo, Sec. IA, 36 (1989), 703–716 . J. P. Ramis, Divergent series and holomorphic dynamical systems, Preprint, 1993 .
  • 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, DOI
  • D. G. Babbitt and V. S. Varadarajan, Local moduli for meromorphic differential equations, AstĂ©risque 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
  • D.G. Babbitt and V.S. Varadarajan, Meromorphic connections with irregular singularities : some problems, Katata conference and workshop, Taniguchi Foundation (1987 ), Preprint. D.G. Babbitt and V.S. Varadarajan, Local moduli for meromorphic differential equations. I. The Stokes sheaf and its cohomology, UCLA, preprint (1985 ). D.G. Babbitt and V.S. Varadarajan, Local isoformal deformation theory for meromorphic differential equations near an irregular singularity (M. Hazewinkel and M. Gerstenhaber, eds.), Deformation theory of algebras and structures and applications, NATO ASI Series C. Mathematical and Physical Sciences, Vol. 247, Kluwer Academic Publishers, 1988 , pp. 583–700. D. G. Babbitt, Groupes algĂ©briques et rĂ©duction formelle de systĂšmes diffĂ©rentielles linĂ©aires , Publ. Math. Univ. Pierre et Marie Curie 84 (1986–87), II.1–II.16 .
  • Pierre Deligne, Équations diffĂ©rentielles Ă  points singuliers rĂ©guliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
  • P. Deligne, Letters to Malgrange, December 12, 1976; 22 August, 1977; April, 1978; Letters to Varadarajan, 4 January, 1986, 2 February, 1986; Letters to Ramis, January 1, 1986; February 25, 1986; February 28, 1986 . P. Deligne, CatĂ©gories tannakiennes, Grothendieck Festschrift (P. Cartier et al., eds.), BirkhĂ€user, 1991, pp. 111–195.
  • 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
  • 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
  • M. Hukuhara, (a) Sur les points singuliers des Ă©quations diffĂ©rentielles linĂ©aires, II, Jour. Fac. Sci. Hokkaido Univ. 5 (1937), 123–166.(b) Sur les points singuliers des Ă©quations diffĂ©rentielles linĂ©aires, III, Jour. Fac. Sci. Kyushu Univ. 2 (1942), 125–137 . H. Turrittin, Convergent solutions of ordinary differential equations in the neighborhood of an irregular singular point , Acta Math. 93 (1955), 27–66 .
  • A. H. M. Levelt, Jordan decomposition for a class of singular differential operators, Ark. Mat. 13 (1975), 1–27. MR 500294, DOI
  • E. Fabry, Sur les intĂ©grales des Ă©quations diffĂ©rentielles linĂ©aires Ă  coefficients rationnels, ThĂ©se, Paris , 1885 . H. PoincarĂ©, Sur les intĂ©grales des Ă©quations linĂ©aires, Acta Math. 8 (1986), 295–344 .
  • Werner Balser, Zum Einzigkeitssatz in der Invariantentheorie meromorpher Differentialgleichungen, J. Reine Angew. Math. 318 (1980), 51–82 (German). MR 579383, DOI
  • 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
  • G. Appleby, Thesis, UCLA, 1993 . W. Wasow, Asymptotic expansions for ordinary differential equations, Dover, 1987 .
  • Wolfgang Wasow, Linear turning point theory, Applied Mathematical Sciences, vol. 54, Springer-Verlag, New York, 1985. MR 771669
  • 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, DOI
  • F. Beukers and G. Heckman, Monodromy for the hypergeometric function $_nF_{n-1}$, Invent. Math. 95 (1989), no. 2, 325–354. MR 974906, DOI
  • 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, DOI
  • 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, DOI
  • 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
  • Harish–Chandra, Some results on differential equations and their applications, Proc. Nat. Acad. Sci. USA 45 (1959), 1763–1764 .
  • 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
  • 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
  • Masaaki Yoshida, Fuchsian differential equations, Aspects of Mathematics, E11, Friedr. Vieweg & Sohn, Braunschweig, 1987. With special emphasis on the Gauss-Schwarz theory. MR 986252
  • G. J. Heckman and E. M. Opdam, Root systems and hypergeometric functions, I. Comp. Math. 64 (1988 ), 329–352 .
  • G. J. Heckman and E. M. Opdam, Root systems and hypergeometric functions. I, Compositio Math. 64 (1987), no. 3, 329–352. MR 918416
  • G. J. Heckman and E. M. Opdam, Root systems and hypergeometric functions. I, Compositio Math. 64 (1987), no. 3, 329–352. MR 918416
  • R. Sommeling, Characteristic classes for irregular singularities, Thesis, University of Nijmegen, 1993 .
  • 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
  • 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, DOI
  • B. Malgrange, Remarques sur les Ă©quations diffĂ©rentielles Ă  points singuliers irrĂ©guliers, Équations diffĂ©rentielles et systĂšmes de Pfaff dans le champ complexe (Sem., Inst. Rech. Math. AvancĂ©e, Strasbourg, 1975) Lecture Notes in Math., vol. 712, Springer, Berlin, 1979, pp. 77–86 (French). MR 548145
  • Yasutaka Sibuya, Stokes phenomena, Bull. Amer. Math. Soc. 83 (1977), no. 5, 1075–1077. MR 442337, DOI
  • M. Loday–Richaud, PrĂ©publications, UniversitĂ© de Paris–Sud Mathematiques, Orsay, 1992 .
  • 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
  • Anne Duval and Claude Mitschi, Matrices de Stokes et groupe de Galois des Ă©quations hypergĂ©omĂ©triques confluentes gĂ©nĂ©ralisĂ©es, Pacific J. Math. 138 (1989), no. 1, 25–56 (French). MR 992173
  • Claude Mitschi, Groupe de Galois diffĂ©rentiel des Ă©quations hypergĂ©omĂ©triques confluentes gĂ©nĂ©ralisĂ©es, C. R. Acad. Sci. Paris SĂ©r. I Math. 309 (1989), no. 4, 217–220 (French, with English summary). MR 1006733
  • 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, DOI
  • J. Ecalle, Les fonctions rĂ©surgents, Publ. Math. Orsay, I, II. (1981) .
  • Jean Écalle, Les fonctions rĂ©surgentes. Tome III, Publications MathĂ©matiques d’Orsay [Mathematical Publications of Orsay], vol. 85, UniversitĂ© de Paris-Sud, DĂ©partement de MathĂ©matiques, 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
  • W. Balser, From divergent power series to analytic functions: theory and applications of multisummable power series, Lecture Notes in Mathematics, Springer, vol. 1582, 1994 . Y. Sibuya, Gevrey asymptotics and Stokes multipliers, Differential equations and computer algebra (M. F. Singer, ed.), Academic Press, 1991, pp. 131–147 .
  • B. Candelpergher, J.-C. Nosmas, and F. Pham, Approche de la rĂ©surgence, ActualitĂ©s MathĂ©matiques. [Current Mathematical Topics], Hermann, Paris, 1993 (French, with French summary). MR 1250603
  • J.-P. Ramis and J. Martinet, ThĂ©orie de Galois diffĂ©rentielle et resommation, Computer algebra and differential equations, Comput. Math. Appl., Academic Press, London, 1990, pp. 117–214 (French). MR 1038060
  • J. P. Ramis and J. Martinet, Elementary acceleration and multisummability , Publ. Univ. Louis Pasteur, 1990 .

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1991): 34A20, 13N05

Retrieve articles in all journals with MSC (1991): 34A20, 13N05

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