Skip to main content
SpringerLink
Log in

A Bound for a Typical Differential Dimension of Systems of Linear Differential Equations

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We prove upper and lower bounds for the leading coefficient of Kolchin dimension polynomial of systems of partial linear differential equations in the case of codimension two, quadratic with respect to the orders of the equations in the system. A notion of typical differential dimension plays an important role in differential algebra, some of its estimations were proved by J. Ritt and E. Kolchin; they also advanced several conjectures that were later refuted. Our bound generalizes the analog of the Bézout theorem for one differential indeterminate. It is better than an estimation proved by D. Grigoriev.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. F. Boulier, D. Lazard, F. Ollivier, and M. Petitot, “Computing representations for radicals of finitely generated differential ideals,” Appl. Algebra Eng., Commun. Comput., 20, No. 1, 73–121 (2009).

    Article  MathSciNet  Google Scholar 

  2. A. Chistov and D. Grigoriev, “Complexity of a standard basis of a D-module,” St. Petersburg Math. J., 20, 709–736 (2009).

    Article  MathSciNet  Google Scholar 

  3. T. Dubé, “The structure of polynomial ideals and Gröbner bases,” SIAM J. Comput., 19, No. 4, 750–773 (1990).

    Article  MathSciNet  Google Scholar 

  4. D. Grigoriev, “Weak Bezout inequality for D-modules,” J. Complexity, 21, 532–542 (2005).

    Article  MathSciNet  Google Scholar 

  5. E. Hubert, “Factorization-free decomposition algorithms in differential algebra,” J. Symbolic Comput., 29, No. 4-5, 641–662 (2000).

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  7. M. V. Kondratieva, A. B. Levin, A. V. Mikhalev, and E. V. Pankratiev, Differential and Difference Dimension Polynomials, Kluwer Academic, Dordrecht (1999).

    Book  Google Scholar 

  8. M. V. Kondratieva, “An upper bound for minimizing coefficients of dimension Kolchin polynomial,” Program. Comput. Software, 36, No. 2, 83–86 (2010).

    Article  MathSciNet  Google Scholar 

  9. E. W. Mayr and A. R. Meyer, “The complexity of a word problem for commutative semigroups and polynomial ideals,” Adv. Math., 46, 305–329 (1982).

    Article  MathSciNet  Google Scholar 

  10. J. Ritt, Differential Algebra, Amer. Math. Soc., New York (1950).

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, Russia, 119991

    M. V. Kondratieva

Authors
  1. M. V. Kondratieva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. V. Kondratieva.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 22, No. 5, pp. 259–269, 2019.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kondratieva, M.V. A Bound for a Typical Differential Dimension of Systems of Linear Differential Equations. J Math Sci 259, 563–569 (2021). https://doi.org/10.1007/s10958-021-05647-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-021-05647-1

Search

Navigation