Abstract
We study the local analytic classification of Fuchsian singular points. The resonance formal normal form (FNF) of a system with a Fuchsian singular point, as well as the local analytic equivalence of a system to its resonance FNF, is well known. However, there are distinct resonance FNFs locally analytically equivalent to each other. The main theorem of the paper reduces the problem of local analytic equivalence of resonance FNFs to a problem about conjugacy of certain matrices associated to two FNFs (which are nil-triangular) by a block upper triangular matrix. As a consequence, the local analytic classification of Fuchsian singular points reduces to the study of the orbits of the group of block upper triangular matrices on nil-triangular matrices by conjugation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
V. I. Arnol'd and Yu. S. Il'ashenko, "Ordinary differential equation," in: Encyclopedia Math. Sci. I, Dynamical Systems I, Springer-Verlag, Berlin, 1988, pp. 1–148.
A. H. M. Levelt, "Hypergeometric functions, I, II, III, IV," Nederl. Acad. Wetensch. Proc. Ser. A, 64 (1961), no. 4, 361–403.
F. R. Gantmakher, The Theory of Matrices, AMS Chelsea Publ., Providence, RI, 1998.
Yu. S. Il'yashenko, "The nonlinear Riemann-Hilbert problem," Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 213 (1996), 6–29.
A. A. Bolibrukh, Fuchsian Differential Equations and Holomorphic Bundles [in Russian], MCCME, Moscow, 2000.
D. V. Anosovand and A. A. Bolibrukh, The Riemann-Hilbert Problem, Aspects of Math, Vieweg, Braunschweig, 1994.
V. V. Kashin, "Orbits of an adjoint and co-adjoint action of Borel subgroups of a semisimple algebraic group," in: Problems in Group Theory and Homological Algebra [in Russian], Matematika, Yaroslav. Gos. Univ., Yaroslavl, 1990, pp. 141–159.
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.
B. A. Rabinovich, Analytic Classification of Fuchsian Singular Points [in Russian], Graduate Diploma, Moscow State Univ., Moscow, 1997.
A. A. Bolibrukh, Hilbert's Twenty-First Problem for Linear Fuchsian Systems, Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], vol. 206, Nauka, Moscow, 1994.
I. B. Brodskii, "The invariants of unipotent groups," Uspekhi Mat. Nauk [Russian Math. Surveys], 31 (1976), no. 1, 243–244.
I. B. Brodskii, "Orbits of unipotent groups," Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 3 (1969), no. 2, 19–23.
A. A. Kirillov, "The orbit method and finite groups," in: Student Lectures at the College of Mathematics, Independent University of Moscow, no. 1 [in Russian], MCCME, Moscow, 2000, pp. 37–73.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kleptsyn, V.A., Rabinovich, B.A. Analytic Classification of Fuchsian Singular Points. Mathematical Notes 76, 348–357 (2004). https://doi.org/10.1023/B:MATN.0000043462.06397.ab
Issue Date:
DOI: https://doi.org/10.1023/B:MATN.0000043462.06397.ab