Characteristic classes for irregular singularities

ABSTRACT

For an endomorphism in a finite dimensional vector space, one can define its characteristic polynomial and rational Jordan normal form. In this article something analogous is done for differential operators in a finite dimensional vector space. An overview of (partial) algorithms to compute these invariants is also given. Proofs and more results and details can be found in [8] on which this article is based.

REFERENCES

• 1: Donald G. Babbitt and V. S. Varadarajan, Formal reduction of meromorphic differential equations: a group theoretic view, Pacific Journal of Mathematics 109 (1983), no. 1, 1-80.
• 2: Arno van den Essen and A. H. M. Levelt, An explicit description of all simple k{{x}}{O}-modules, Abelian Groups and Noncommutative Rings (Providence, Rhode Island) (L. Fuchs et al., ed.), Contemporary Mathematics, vol. 130, American Mathematical Society, Providence, Rhode Island, 1992, pp. 121-131.
• 3: A. Hilali, On the algebraic and differential Newton–Puiseux polygons, Journal of Symbolic Computation, v.4 n.3, p.335-349, Dec. 1988 [doi>10.1016/S0747-7171(87)80011-1]
• 4: ----, Soution formues de sustems differemtles lindaires au voisinage d'un singulier, Ph.D. thesis, l'Universit~ Scientifique, Technologique et M~dicale de Grenoble, 1987.
• 5: A. Hilali and A. Wazner, Un algorithme de calcul de l'invariant de Katz d'un syst~me diffdrentiel lindaire, Anhales de l'Institut Fourier 36 (1986), no. 3, 67-81.
• 6: A. Hilali , A. Wazner , Pamela B. Lawhead, Super-irreducible form of linear differential systems, Numerische Mathematik, v.50 n.4, p.429-449, FEB. 1987 [doi>10.1007/BF01396663]
• 7: A. H. M. Levelt, Jordan decomposition for a class of singular differential operators, Arkiv fSr matematik 13 (1975), no. 1, 1-27.
• 8: Ron Sommeling, Characteristic classes for irregular singularities, Ph.D. thesis, University of Nijmegen, June 1993.
• 9: Wolfgang Wasow, Asymptotic expansions for ordinary differential equatwns, Pure and Applied Mathematics (Interscience), vol. 14, Interscience Publishers, New York, London, Sydney, 1965.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Algebraic algorithms

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computations on polynomials; Computations on matrices

General Terms:
Algorithms, Theory, Verification