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Notes on Triangular Sets and Triangulation-Decomposition Algorithms II: Differential Systems

  • Evelyne Hubert
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2630)

Abstract

This is the second in a series of two tutorial articles devoted to triangulation-decomposition algorithms. The value of these notes resides in the uniform presentation of triangulation-decomposition of polynomial and differential radical ideals with detailed proofs of all the presented results. We emphasize the study of the mathematical objects manipulated by the algorithms and show their properties independently of those. We also detail a selection of algorithms, one for each task. The present article deals with differential systems. It uses results presented in the first article on polynomial systems but can be read independently.

Keywords

Polynomial Ring Singular Solution Determine Equation Polynomial Algebra Characteristic Decomposition 
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References

  1. [1]
    R. L. Anderson, V. A. Baikov, R. K. Gazizov, W. Hereman, N. H. Ibragimov, F. M. Mahomed, S. V. Meleshko, M. C. Nucci, P. J. Olver, M. B. Sheftel, A. V. Turbiner, and E. M. Vorob’ev. CRC handbook of Lie group analysis of differential equations. Vol. 3. CRC Press, Boca Raton, FL, 1996. New trends in theoretical developments and computational methods.Google Scholar
  2. [2]
    J. Apel. Passive Complete Orthonormal Systems of PDEs and Riquier Bases of Polynomial Modules. 2003. In this volume.Google Scholar
  3. [3]
    U. M. Ascher and L. R. Petzold. Computer methods for ordinary differential equations and differential-algebraic equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998.MATHGoogle Scholar
  4. [4]
    P. Aubry and D. Wang. Reasonning about surfaces using differential zero and ideal decomposition. In J. Richet-Gebert and D. Wang, editors, ADG 2000, number 2061 in LNAI, pages 154–174, 2001.Google Scholar
  5. [5]
    T. Becker and V. Weispfenning. Gröbner Bases — A Computational Approach to Commutative Algebra. Springer-Verlag, New York, 1993.MATHGoogle Scholar
  6. [6]
    F. Boulier. Étude et Implantation de Quelques Algorithmes en Algèbre Différentielle. PhD thesis, Université de Lille, 1994.Google Scholar
  7. [7]
    F. Boulier. A new criterion to avoid useless critical pairs in buchberger’s algorithm. Technical Report 2001-07, LIFL, Université de Lille, ftp://ftp.lifl.fr/pub/reports/internal/2001-07.ps, 2001.
  8. [8]
    F. Boulier and E. Hubert. DIFFALG: description, help pages and examples of use. Symbolic Computation Group, University of Waterloo, Ontario, Canada, 1998. Now available at http://www.inria.fr/cafe/Evelyne.Hubert/webdiffalg.Google Scholar
  9. [9]
    F. Boulier, D. Lazard, F. Ollivier, and M. Petitot. Representation for the radical of a finitely generated differential ideal. In A. H. M. Levelt, editor, ISSAC’95. ACM Press, New York, 1995.Google Scholar
  10. [10]
    F. Boulier, D. Lazard, F. Ollivier, and M. Petitot. Computing representations for radicals of finitely generated differential ideals. Technical Report IT-306, LIFL, 1997.Google Scholar
  11. [11]
    F. Boulier and F. Lemaire. Computing canonical representatives of regular differential ideals. In C. Traverso, editor, ISSAC. ACM-SIGSAM, ACM, 2000.Google Scholar
  12. [12]
    F. Boulier, F. Lemaire, and M. Moreno-Maza. Pardi! In ISSAC 2001. pp 38–47, ACM, 2001.Google Scholar
  13. [13]
    D. Bouziane, A. Kandri Rody, and H. Maârouf. Unmixed-dimensional decomposition of a finitely generated perfect differential ideal. Journal of Symbolic Computation, 31(6):631–649, 2001.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    K. E. Brenan, S. L. Campbell, and L. R. Petzold. Numerical solution of initial value problems in differential-algebraic equations. North-Holland, 1989.Google Scholar
  15. [15]
    A. Buium and P. J. Cassidy. Differential algebraic geometry and differential algebraic groups: from algebraic differential equations to diophantine geometry. In Bass et al. [39].Google Scholar
  16. [16]
    G. Carra Ferro. Gröbner bases and differential algebra. In AAECC, volume 356 of Lecture Notes in Computer Science. Springer-Verlag Berlin, 1987.Google Scholar
  17. [17]
    G. Carrà Ferro and W. Y. Sit. On term-orderings and rankings. In Computational algebra (Fairfax, VA, 1993), number 151 in Lecture Notes in Pure and Applied Mathematics, pages 31–77. Dekker, New York, 1994.Google Scholar
  18. [18]
    E. Cartan. Les systèmes Différentiels Extérieurs et leurs Applications Géométrique. Hermann, 1945.Google Scholar
  19. [19]
    J. Chazy. Sur les équations différentielles du troisième ordre et d’ordre supérieur dont l’intégrale générale a ses points critiques fixes. Acta Mathematica, 34:317–385, 1911.CrossRefMATHMathSciNetGoogle Scholar
  20. [20]
    E. S. Cheb-Terrab. Odetools: A maple package for studying and solving ordinary differential equations. http://lie.uwaterloo.ca/description/odetools, 1998. presented for the incoming update of “Handbook of Computer Algebra”.
  21. [21]
    E. S. Cheb-Terrab, L. G. S. Duarte, and L. A. C. P. da Mota. Computer algebra solving of first order ODEs using symmetry methods. Computer Physics Communications. An International Journal and Program Library for Computational Physics and Physical Chemistry, 101(3):254–268, 1997.MATHMathSciNetGoogle Scholar
  22. [22]
    E. S. Cheb-Terrab, L. G. S. Duarte, and L. A. C. P. da Mota. Computer algebra solving of second order ODEs using symmetry methods. Computer Physics Communications. An International Journal and Program Library for Computational Physics and Physical Chemistry, 108(1):90–114, 1998.MATHMathSciNetGoogle Scholar
  23. [23]
    S-C. Chou and X-S. Gao. Automated reasonning in differential geometry and mechanics using the characteristic set method. part II. mechanical theorem proving. Journal of Automated Reasonning, 10:173–189, 1993.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    P. A. Clarkson, D. K. Ludlow, and T. J. Priestley. The classical, direct, and nonclassical methods for symmetry reductions of nonlinear partial differential equations. Methods and Applications of Analysis, 4(2): 173–195, 1997. Dedicated to Martin David Kruskal.MATHMathSciNetGoogle Scholar
  25. [25]
    P. A. Clarkson and E. L. Mansfield. Symmetry reductions and exact solutions of a class of non-linear heat equations. Physica, D70:250–288, 1994.MathSciNetGoogle Scholar
  26. [26]
    T. Cluzeau and E. Hubert. Resolvent representation for regular differential ideals. Applicable Algebra in Engineering, Communication and Computing, 13(5):395–425, 2003.CrossRefMathSciNetGoogle Scholar
  27. [27]
    S. Diop. Differential-algebraic decision methods and some applications to system theory. Theoretical Computer Science, 98(1):137–161, 1992. Second Workshop on Algebraic and Computer-theoretic Aspects of Formal Power Series (Paris, 1990).MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    D. Eisenbud. Commutative Algebra with a View toward Algebraic Geometry. Graduate Texts in Mathematics. Springer-Verlag New York, 1994.Google Scholar
  29. [29]
    M. Fliess and S.T. Glad. An algebraic approach to linear and nonlinear control. In H.L. Trentelman and J.C. Willems, editors, Essays on control: Perspectives in the theory and its applications, volume 14 of PCST, pages 223–265. Birkhäuser, Boston, 1993.Google Scholar
  30. [30]
    L. Guo, W. F. Keigher, P. J. Cassidy, and W. Y. Sit, editors. Differential Algebra and Related Topics. World Scientific Publishing Co., 2002.Google Scholar
  31. [31]
    E. Hairer and G. Wanner. Solving Ordinary Differential Equations II — Stiff and Differential-Algebraic Problems — Second Revised Edition. Springer, 1996.Google Scholar
  32. [32]
    E. Hubert. Essential components of an algebraic differential equation. Journal of Symbolic Computation, 28(4–5):657–680, 1999.MATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    E. Hubert. Factorisation free decomposition algorithms in differential algebra. Journal of Symbolic Computation, 29(4–5):641–662, 2000.MATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    E. Hubert. Notes on triangular sets and triangulation-decomposition algorithms. I Polynomial systems. In this volume, 2003.Google Scholar
  35. [35]
    E. L. Ince. Ordinary Differential Equations. Dover Publications, Inc., 1956.Google Scholar
  36. [36]
    M. Janet. Sur les systèmes d’équations aux dérivées paritelles. Gauthier-Villars, 1929.Google Scholar
  37. [37]
    M. Kalkbrener. A generalized Euclidean algorithm for computing triangular representations of algebraic varieties. Journal of Symbolic Computation, 15(2):143–167, 1993.MATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    I. Kaplansky. An Introduction to Differential Algebra. Hermann, Paris, 1970.Google Scholar
  39. [39]
    E. Kolchin. Selected works of Ellis Kolchin with commentary. Commentaries by Armand Borel, Michael F. Singer, Bruno Poizat, Alexandru Buium and Phyllis J. Cassidy, Edited and with a preface by Hyman Bass, Buium and Cassidy. American Mathematical Society, Providence, RI, 1999.Google Scholar
  40. [40]
    E. R. Kolchin. Differential Algebra and Algebraic Groups, volume 54 of Pure and Applied Mathematics. Academic Press, New York-London, 1973.MATHGoogle Scholar
  41. [41]
    D. Lazard. Solving zero dimensional algebraic systems. Journal of Symbolic Computation, 15: 117–132, 1992.CrossRefMathSciNetGoogle Scholar
  42. [42]
    F. Lemaire. Contribution à l’algorithmique en algèbre differentielle. PhD thesis, Université des Sciences et Technoligies de Lille, http://www.lifl.fr/~lemaire/pub, 2002.
  43. [43]
    Z. Li. Mechanical theorem proving in the local theory of surfaces. Annals of Mathematics and Artificial Intelligence, 13(1–2):25–46, 1995.MATHCrossRefMathSciNetGoogle Scholar
  44. [44]
    Z. Li and D. Wang. Coherent, regular and simple system in zero decompositions of partial differential systems. Systems Sciences and Mathematical Sciences, 12 suppl, 1999.Google Scholar
  45. [45]
    B. Malgrange. Differential algebra and differential geometry. In Differential Geometry — Kyoto Nara 2000, 2000. to appear.Google Scholar
  46. [46]
    E. L. Mansfield. Differential Gröbner Bases. PhD thesis, University of Sydney, 1991.Google Scholar
  47. [47]
    E. L. Mansfield. DIFFGROB2: a symbolic algebra package for analysing systems of PDE using MAPLE. University of Exeter, 1994. Preprint M/94/4.Google Scholar
  48. [48]
    E. L. Mansfield, G. J. Reid, and P. A. Clarkson. Nonclassical reductions of a 3+1-cubic nonlinear Schrödinger system. Computer Physics Communications, 115:460–488, 1998.MATHCrossRefMathSciNetGoogle Scholar
  49. [49]
    G. Margaria, E. Riccomagno, M. J. Chappell, and H. P. Wynn. Differential algebra methods for the study of the structural identifiability of rational function state-space models in the biosciences. Mathematical Biosciences, 174(1):1–26, 2001.MATHCrossRefMathSciNetGoogle Scholar
  50. [50]
    M. Moreno Maza and R. Rioboo. Polynomial gcd computations over towers of algebraic extensions. In Applied algebra, algebraic algorithms and error-correcting codes (Paris, 1995), pages 365–382. Springer, Berlin, 1995.Google Scholar
  51. [51]
    S. Morrison. The differential ideal [P]: M . Journal of Symbolic Computation, 28(4–5):631–656, 1999.MATHCrossRefMathSciNetGoogle Scholar
  52. [52]
    F. Ollivier. Canonical bases: Relations with standard bases, finiteness conditions and application to tame automorphisms. In Costiglioncello, editor, MEGA’ 90. Birkhauser, August 1990.Google Scholar
  53. [53]
    P. Olver. Applications of Lie Groups to Differential Equations. Number 107 in Graduate texts in Mathematics. Springer-Verlag, New York, 1986.MATHGoogle Scholar
  54. [54]
    G. J. Reid. Algorithms for reducing a system of pde to standard form, determining the dimension of its solution space and calculating its taylor series solution. European Journal of Applied Mathematics, 2:293–318, 1991.MATHMathSciNetCrossRefGoogle Scholar
  55. [55]
    G. J. Reid. Finding abstract Lie symmetry algebras of differential equations without integrating determining equations. European Journal of Applied Mathematics, 2(4):319–340, 1991.MATHMathSciNetGoogle Scholar
  56. [56]
    G. J. Reid, A. D. Wittkopf, and A. Boulton. Reduction of systems of nonlinear partial differential equations to simplified involutive forms. Eur. J. of Appl. Math., 7:604–635, 1996.MathSciNetGoogle Scholar
  57. [57]
    C. Riquier. Les systèmes d’équations aux dérivés partielles. Gauthier-Villars, Paris, 1910.Google Scholar
  58. [58]
    J. F. Ritt. Differential Equations from the Algebraic Standpoint. Amer. Math. Soc. Colloq. Publ., 1932.Google Scholar
  59. [59]
    J. F. Ritt. On the singular solutions of algebraic differential equations. Annals of Mathematics, 37(3):552–617, 1936.CrossRefMathSciNetGoogle Scholar
  60. [60]
    J. F. Ritt. Differential Algebra, volume XXXIII of Colloquium publications. American Mathematical Society, 1950. Reprinted by Dover Publications, Inc (1966).Google Scholar
  61. [61]
    A. Rosenfeld. Specializations in differential algebra. Transaction of the American Mathematical Society, 90:394–407, 1959.MATHCrossRefMathSciNetGoogle Scholar
  62. [62]
    C. J. Rust and G. J. Reid. Rankings of partial derivatives. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI), pages 9–16, New York, 1997. ACM.Google Scholar
  63. [63]
    C. J. Rust, G. J. Reid, and A. D. Wittkopf. Existence and uniqueness theorems for formal power series solutions of analytic differential systems. In Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation (Vancouver, BC), pages 105–112, New York, 1999 ACM.Google Scholar
  64. [64]
    F. Schwarz. Algorithmic Lie theory for solving ordinary differential equations. 2003. In this volume.Google Scholar
  65. [65]
    A. Sedoglavic. A probabilistic algorithm to test local algebraic observability in polynomial time. In B. Mourrain, editor, Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation (London, Canada, July 22–25 2001), pages 309–316, 2001.Google Scholar
  66. [66]
    A. Seidenberg. An elimination theory for differential algebra. University of California Publications in Mathematics, 3(2):31–66, 1956.MathSciNetGoogle Scholar
  67. [67]
    W. Seiler. Computer algebra and differential equations — an overview. mathPAD7, 7:34–49, 1997.Google Scholar
  68. [68]
    W. Sit. The Ritt-Kolchin theory for differential polynomials. In Guo et al. [30].Google Scholar
  69. [69]
    G. Thomas. Contributions Théoriques et Algorithmiques à l’Étude des Équations Différentielles-Algébriques. Approche par le Calcul Formel. PhD thesis, Institut National Polytechnique de Grenoble, Juillet 1997.Google Scholar
  70. [70]
    J. Visconti. HIDAES (Higher Index Differential Algebraic Equations Solver). Software and test set. Technical report, LMC-IMAG, http://www-lmc.imag.fr/CF/LOGICIELS/page_dae.html, 1999.
  71. [71]
    J. Visconti. Résolution numérique des équations algébro-différentielles, Estimation de l’erreur globale et réduction formel de l’indice. PhD thesis, Institut National Polytechnique de Grenoble, 1999. http://www-lmc.imag.fr/CF/publi/theses/visconti.ps.gz.
  72. [72]
    D. Wang. An elimination method for differential polynomial systems. I. Systems Science and Mathematical Sciences, 9(3):216–228, 1996.MATHMathSciNetGoogle Scholar
  73. [73]
    A. Witkopf and G. Reid. The RIF package. CECM — Simon Fraser University — Vancouver, http://www.cecm.sfu.ca/wittkopf/rif.html.
  74. [74]
    W. T. Wu. Mechanical theorem proving of differential geometries and some of its applications in mechanics. Journal of Automated Reasoning, 7(2):171–191, 1991.MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Evelyne Hubert
    • 1
  1. 1.INRIA - Projet CAFESophia Antipolis

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