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Sufficient Set of Integrability Conditions of an Orthonomic System

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Abstract

Every orthonomic system of partial differential equations is known to possess a finite number of integrability conditions sufficient to ensure the validity of them all. Here we show that a redundancy-free sufficient set of integrability conditions can be constructed in a time proportional to the number of equations cubed.

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References

  1. J. Apel, R. Hemmecke, Detecting unnecessary reductions in an involutive basis computation, J. Symb. Comput. 40, 1131–1149 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  2. G. Birkhoff, Lattice Theory, 3rd edn. (Am. Math. Soc., Providence, 1967).

    MATH  Google Scholar 

  3. A.V. Bocharov, V.N. Chetverikov, S.V. Duzhin, N.G. Khor’kova, I.S. Krasil’shchik, A.V. Samokhin, Yu.N. Torkhov, A.M. Verbovetsky, A.M. Vinogradov, Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Translations of Mathematical Monographs, vol. 182 (Am. Math. Soc., Providence, 1999).

    Google Scholar 

  4. F. Boulier, An optimization of Seidenberg’s elimination algorithm in differential algebra, Math. Comput. Simul. 42, 439–448 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  5. B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. Thesis, Univ-Innsbruck, 1965.

  6. B. Buchberger, A criterion for detecting unnecessary reductions in the construction of Gröbner bases, in Proc. EUROSAM ’79, ed. by E.W. Ng. Lecture Notes in Computer Science, vol. 72 (Springer, Berlin, 1979), pp. 3–21.

    Google Scholar 

  7. M. Caboara, M. Kreuzer, L. Robbiano, Efficiently computing minimal sets of critical pairs, J. Symb. Comput. 38, 1169–1190 (2004).

    Article  MathSciNet  Google Scholar 

  8. J.-C. Faugère, A new efficient algorithm for computing Gröbner bases (F4), J. Pure Appl. Algebra 139, 61–88 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  9. R. Gebauer, M. Möller, On an instalation of Buchberger’s algorithm, J. Symb. Comput. 6, 275–286 (1988).

    Article  MATH  Google Scholar 

  10. V.P. Gerdt, Gröbner bases and involutive methods for algebraic and differential equations, Math. Comput. Model. 25, 75–90 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  11. V.P. Gerdt, Yu.A. Blinkov, Involutive bases of polynomial ideals, Math. Comput. Simul. 45, 519–541 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Grätzer, General Lattice Theory (Akademie, Berlin, 1978).

    Google Scholar 

  13. W. Hereman, Review of symbolic software for Lie symmetry analysis, Math. Comput. Model. 25, 115–132 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  14. E. Hubert, Notes on triangular sets and triangulation-decomposition algorithms II: Differential systems, in Symbolic and Numerical Scientific Computation, Proc. Conf. Hagenberg, ed. by F. Winkler, U. Langer, Austria, 2001. Lecture Notes in Computer Science, vol. 2630 (Springer, Berlin, 2003), pp. 40–87.

    Chapter  Google Scholar 

  15. M. Janet, Leçons sur les Systèmes d’Èquations aux Derivées Partielles (Gauthier-Villars, Paris, 1929).

    MATH  Google Scholar 

  16. R.M. Karp, R.E. Tarjan, Linear expected-time algorithms for connectivity problems, J. Algorithms 1, 374–393 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  17. B. Kruglikov, V. Lychagin, Mayer brackets and solvability of PDEs. I, Differ. Geom. Appl. 17, 251–272 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  18. B. Kruglikov, V. Lychagin, Mayer brackets and solvability of PDEs. II, Trans. Am. Math. Soc. 358, 1077–1103 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  19. W. Küchlin, A confluence criterion based on the generalised Newman lemma, in EUROCAL ’85, Proc. Conf. Linz, Vol. 2, ed. by B.F. Caviness, April 1–3, 1985. Lecture Notes in Comput. Sci., vol. 204 (Springer, Berlin, 1985), pp. 390–399.

    Google Scholar 

  20. M. Marvan, Sufficient set of integrability conditions of an orthonomic system: Extended abstract, in Global Integrability of Field Theories, Proc. GIFT 2006, Cockcroft Inst., Daresbury, ed. by J. Calmet, W.M. Seiler, R.W. Tucker, November 1–3, 2006 (Universitätsverlag, Karlsruhe, 2006), pp. 243–247. http://www.uvka.de/univerlag/volltexte/2006/164/pdf/Tagungsband_GIFT.pdf.

    Google Scholar 

  21. E. Miller, B. Sturmfels, Combinatorial Commutative Algebra, Springer Graduate Texts in Math., vol. 227 (Springer, New York, 2004).

    Google Scholar 

  22. P.J. Olver, Equivalence, Invariants, and Symmetry (Cambridge University Press, Cambridge, 1999).

    Google Scholar 

  23. J.F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups (Gordon and Breach, New York, 1978).

    MATH  Google Scholar 

  24. G.J. Reid, Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution, Eur. J. Appl. Math. 2, 293–318 (1991).

    Article  MATH  Google Scholar 

  25. G.J. Reid, P. Lin, A.D. Wittkopf, Differential elimination-completion algorithms for DAE and PDAE, Stud. Appl. Math. 106, 1–45 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  26. G.J. Reid, A.D. Wittkopf, A. Boulton, Reduction of systems of nonlinear partial differential equations to simplified involutive forms, Eur. J. Appl. Math. 7, 604–635 (1996).

    Article  MathSciNet  Google Scholar 

  27. C. Riquier, Les Systèmes d’Èquations aux Derivées Partielles (Gauthier-Villars, Paris, 1910).

    Google Scholar 

  28. C.J. Rust, Rankings of derivatives for elimination algorithms, and formal solvability of analytic PDE, Ph.D. Thesis, University of Chicago, 1998.

  29. C.J. Rust, G.J. Reid, Rankings of partial derivatives, in Proc. ISSAC 1997 (ACM, New York, 1997), pp. 9–16.

    Chapter  Google Scholar 

  30. C.J. Rust, G.J. Reid, A.D. Wittkopf, Existence and uniqueness theorems for formal power series solutions of analytic differential systems, in Proc. ISSAC 1999 (ACM, New York, 1999), pp. 105–112.

    Chapter  Google Scholar 

  31. D.J. Saunders, The Geometry of Jet Bundles, London Math. Soc. Lect. Notes Series, vol. 142, (Cambridge Univ. Press, Cambridge, 1989–1929).

    MATH  Google Scholar 

  32. J.M. Thomas, Riquier’s existence theorems, Ann. Math. 30, 285–310 (1928).

    Article  Google Scholar 

  33. A. Tresse, Sur les invariants différentiels des groupes de transformations, Acta Math. 18, 1–88 (1894).

    Article  MathSciNet  Google Scholar 

  34. A.M. Verbovetsky, I.S. Krasil’shchik, P. Kersten, M. Marvan, Homological Methods in Geometry of Partial Differential Equations (MCCME, Moscow, under preparation) (in Russian).

  35. F. Winkler, Reducing the complexity of the Knuth–Bendix completion algorithm: a “unification” of different approaches, in EUROCAL ’85, Proc. Conf. Linz, Vol. 2, ed. by B.F. Caviness, April 1–3, 1985. Lecture Notes in Comput. Sci., vol. 204 (Springer, Berlin, 1985), pp. 378–389.

    Google Scholar 

  36. F. Winkler, B. Buchberger, A criterion for eliminating unnecessary reductions in the Knuth–Bendix algorithm, in Algebra, Combinatorics and Logic in Computer Science, Vol. II, ed. by J. Demetrovics et al. Coll. Math. Soc. J. Bolyai, vol. 42 (J. Bolyai Math. Soc., North-Holland, 1986), pp. 849–869.

    Google Scholar 

  37. F. Winkler, B. Buchberger, F. Lichtenberger, H. Rolletschek, Algorithm 628: an algorithm for constructing canonical bases of polynomial ideals, ACM Trans. Math. Softw. 11(1), 66–78 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  38. A.D. Wittkopf, Algorithms and implementations for differential elimination, Ph.D. Thesis, Simon Fraser Univ., 2004.

  39. T. Wolf, The integration of systems of linear PDEs using conservation laws of syzygies, J. Symb. Comput. 35, 499–526 (2003).

    Article  MATH  Google Scholar 

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Correspondence to Michal Marvan.

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Communicated by Elizabeth Mansfield.

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Marvan, M. Sufficient Set of Integrability Conditions of an Orthonomic System. Found Comput Math 9, 651–674 (2009). https://doi.org/10.1007/s10208-008-9039-8

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