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On standard bases in rings of differential polynomials

Abstract

We consider Ollivier’s standard bases (also known as differential Gröbner bases) in an ordinary ring of differential polynomials in one indeterminate. We establish a link between these bases and Levi’s reduction process. We prove that the ideal [x p] has a finite standard basis (w.r.t. the so-called β-orderings) that contains only one element. Various properties of admissible orderings on differential monomials are studied. We bring up the question of whether there is a finitely generated differential ideal that does not admit a finite standard basis w.r.t. any ordering.

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References

  1. 1.

    F. Boulier, D. Lazard, F. Ollivier, and M. Petitot, “Representation for the radical of a finitely generated differential ideal,” in: Proceedings of 1995 International Symposium on Symbolic and Algebraic Computation, ACM Press (1995), pp. 158–166.

  2. 2.

    G. Carrà Ferro, “Gröbner bases and differential algebra,” in: Lecture Notes in Computer Science, Vol. 356, 1989, pp. 141–150.

    MATH  Google Scholar 

  3. 3.

    G. Carrà Ferro, “Differential Gröbner bases in one variable and in the partial case,” in: Math. Comput. Modelling, Vol. 25, Pergamon Press (1997), pp. 1–10.

    Article  MATH  Google Scholar 

  4. 4.

    G. Gallo, B. Mishra, and F. Ollivier, “Some constructions in rings of differential polynomials,” in: Lecture Notes in Computer Science, Vol. 539, 1991, pp. 171–182.

    MathSciNet  Google Scholar 

  5. 5.

    E. Hubert, “Factorization-free decomposition algorithms in differential algebra,” J. Symb. Comp., 29, 641–662 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    E. R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press (1973).

  7. 7.

    H. Levi, “On the structure of differential polynomials and on their theory of ideals,” Trans. Amer. Math. Soc., 51, 532–568 (1942).

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    D. G. Mead, “A necessary and sufficient condition for membership in [uv],” Proc. Amer. Math. Soc., 17, 470–473 (1966).

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    D. G. Mead and M. E. Newton, “Syzygies in [ypz],” Proc. Amer. Math. Soc., 43, No. 2, 301–305 (1974).

    Article  MathSciNet  Google Scholar 

  10. 10.

    K. B. O’Keefe, “A property of the differential ideal [yp],” Trans. Amer. Math. Soc., 94, 483–497 (1960).

    Article  MathSciNet  Google Scholar 

  11. 11.

    F. Ollivier, Le problème de l’identifiabilité structurelle globale, Doctoral Dissertation, Paris (1990).

  12. 12.

    F. Ollivier, “Standard bases of differential ideals,” in: Lecture Notes in Computer Science, Vol. 508, 304–321 (1990).

    MathSciNet  Google Scholar 

  13. 13.

    A. Ovchinnikov and A. Zobnin, “Classification and applications of monomial orderings and the properties of differential orderings,” in: V. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov, eds., Proceedings of the Fifth International Workshop on Computer Algebra in Scientific Computing (CASC-2002), Technische Universität München, Garching, Germany (2002), pp. 237–252.

    Google Scholar 

  14. 14.

    E. V. Pankratiev, “Some approaches to construction of standard bases in commutative and differential algebra,” in: V. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov, eds., Proceedings of the Fifth International Workshop on Computer Algebra in Scientific Computing (CASC-2002), Technische Universität München, Garching, Germany (2002), pp. 265–268.

    Google Scholar 

  15. 15.

    E. V. Pankratiev, “Some approaches to construction of the differential Gröbner bases,” in: Calculemus 2002. 10th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning. Marseille, France, July 3–5, 2002. Work in Progress Papers, Univ. Saarlandes (2002), pp. 50–55.

  16. 16.

    J. F. Ritt, Differential Algebra, Volume XXXIII of Colloquium Publications, American Mathematical Society, New York (1950).

    Google Scholar 

  17. 17.

    C. Rust and G. J. Reid, “Rankings of partial derivatives,” in: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, ACM Press, New York (1997), pp. 9–16.

    Google Scholar 

  18. 18.

    V. Weispfenning, “Differential term-orders,” in: Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation, ACM Press, Kiev (1993), pp. 245–253.

    Google Scholar 

  19. 19.

    A. Zobnin, “Essential properties of admissible orderings and rankings,” to appear in Contributions to General Algebra, 14 (2004). Available at http://shade.msu.ru/~difalg/Articles/Our/Zobnin/ess-properties.ps.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 89–102, 2003.

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Zobnin, A.I. On standard bases in rings of differential polynomials. J Math Sci 135, 3327–3335 (2006). https://doi.org/10.1007/s10958-006-0161-3

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Keywords

  • Reduction Process
  • Standard Basis
  • Differential Polynomial
  • Differential Ideal
  • Admissible Ordering