Factoring zero-dimensional ideals of linear partial differential operators

ABSTRACT

We present an algorithm for factoring a zero-dimensional left ideal in the ring Q(x, y) [∂x, ∂y], i.e. factoring a linear homogeneous partial differential system whose coefficients belong to Q(x, y), and whose solution space is finite-dimensional over Q. The algorithm computes all the zero-dimensional left ideals containing the given ideal. It generalizes the Beke-Schlesinger algorithm for factoring linear ordinary differential operators, and uses an algorithm for finding hyperexponential solutions of such ideals.

REFERENCES

• 1: E. Beke. Die Irreduzibilität der homogenen linearen Differentialgleichungen. Mathematische Annalen, 45:278-294, 1894.
• 2: Manuel Bronstein, An improved algorithm for factoring linear ordinary differential operators, Proceedings of the international symposium on Symbolic and algebraic computation, p.336-340, July 20-22, 1994, Oxford, United Kingdom [doi>10.1145/190347.190436]
• 3: Frédéric Chyzak , Bruno Salvy, Non-commmutative elimination in ore algebras proves multivariate identities, Journal of Symbolic Computation, v.26 n.2, p.187-227, Aug. 1998 [doi>10.1006/jsco.1998.0207]
• 4: M. Janet. Les systéms d'équations aux dérivées partielles. Journal de mathématiques, 83:65-123, 1920.
• 5: A. Kandr-Rody , V. Weispfenning, Non-commutative Gro¨bner bases in algebras of solvable type, Journal of Symbolic Computation, v.9 n.1, p.1-26, Jan. 1990 [doi>10.1016/S0747-7171(08)80003-X]
• 6: E. Kolchin. Differential algebra and algebraic groups. Academic Press., New York, 1973.
• 7: Ziming Li , Fritz Schwarz, Rational solutions of Riccati-like partial differential equations, Journal of Symbolic Computation, v.31 n.6, p.691-716, June 2001 [doi>10.1006/jsco.2001.0461]
• 8: A. Rosenfeld. Specializations in differential algebra. Trans. Amer. Math. Soc., 90:394-407, 1959.
• 9: M. Saito, B. Sturmfels, and N. Takayama. Gröbner deformations of hypergeometric differential equations. Springer-Verlag, Berlin Heidelberg New York, 2000.
• 10: L. Schlesinger. Handbuch der Theorie der linearen Differentialgleichungen. Teubner, Leipzig, 1895.
• 11: F. Schwarz, A factorization algorithm for linear ordinary differential equations, Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation, p.17-25, July 17-19, 1989, Portland, Oregon, United States [doi>10.1145/74540.74544]
• 12: Fritz Schwarz, Reduction and completion algorithms for partial differential equations, Papers from the international symposium on Symbolic and algebraic computation, p.49-56, July 27-29, 1992, Berkeley, California, United States [doi>10.1145/143242.143265]
• 13: F. Schwarz. Janet bases for symmetry groups. In Gröbner bases and applications, London Math. Soc. Lecture Notes Series251., pages 221-234. B. Buchberger, F. Winkler (eds.), Cambridge University Press, 1998.
• 14: S. Tsarev. Factorization of overdetermined systems of linear partial differential equations with finite dimensional solution space. In Proc. 4th Int. Workshop Computer Algebra Scient. Comput. (CASC-2001), pages 529-539. V. Ganzha, E. Mayr, V. Vorozhtsov (eds), Spring-Verlag, 2001.

INDEX TERMS

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

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
          Subjects: Linear systems (direct and iterative methods)
      G.1.5 Roots of Nonlinear Equations
          Subjects: Systems of equations

General Terms:
Algorithms, Documentation