On Computing the Resultant of a Polynomial and an Entire Function | Programming and Computer Software Skip to main content
Springer Nature Link
Log in

On Computing the Resultant of a Polynomial and an Entire Function

  • Published:
Programming and Computer Software Aims and scope Submit manuscript

Abstract

Based on Newton’s recurrence formulas on the complex plane, the sums of values of one entire function at zeros of another entire function are found. This allows one to determine whether these functions have common zeros or not. An algorithm for computing an approximation to the resultant of a polynomial (or an entire function with a finite number of zeros) and an entire function is presented. The algorithm is implemented in the Maple computer algebra system. Examples that illustrate the operation of the algorithm are considered.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

REFERENCES

  1. Aizenberg, L.A., On a formula of the generalized multidimensional logarithmic residue and the solution of systems of nonlinear equations, Dokl. Akad. Nauk SSSR, 1977, vol. 234, no. 3, pp. 505–508.

    MathSciNet  Google Scholar 

  2. Bykov, V.I., Kytmanov, A.M., and Lazman, M.Z., Elimination Methods in Polynomial Computer Algebra, Boston: Kluwer Academic Publishers, 1998.

    Book  Google Scholar 

  3. Bykov, V.I. and Tsybenova, S.B., Nelineinye modeli khimicheskoi kinetiki (Nonlinear Models of Chemical Kinetics), Moscow: KRASAND, 2011.

  4. Kytmanov, A.M. and Naprienko, Ya.M., An approach to define the resultant of two entire functions, J. Complex Var. Elliptic Equations, 2017, vol. 62, no. 2, pp. 269–286.

    Article  MathSciNet  Google Scholar 

  5. Kytmanov, A.M. and Myshkina, E.K., On some approach for finding the resultant of two entire functions, J. Sib. Fed. Univ. Math. Phys., 2019, vol. 12, no. 4, pp. 434–438.

    Article  MathSciNet  Google Scholar 

  6. Kytmanov, A.M. and Myslivets, S.G., On the zeta-function of systems of nonlinear equations, Sib. Math. J., 2007, vol. 48, no. 5, pp. 863–870.

    Article  MathSciNet  Google Scholar 

  7. Kuzovatov, V.I. and Kytmanov, A.A., On the zeta-function of zeros of some class of entire functions, J. Sib. Fed. Univ. Math. Phys., 2014, vol. 7, no. 4, pp. 489–499.

    Article  Google Scholar 

  8. Burbaki, N., Algebra. Mnogochleny i polya, uporyadochennye gruppy (Algebra: Polynomials and Fields, Ordered Groups), Moscow: Nauka, 1965.

  9. Krein, M.G. and Naimark, M.A., The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equation, Linear Multilinear Algebra, 1981, vol. 10, no. 4, pp. 265–308.

    Article  MathSciNet  Google Scholar 

  10. Gohberg, I.C. and Heinig, G., Resultant matrix and its generalization: I. Resultant operator of matrix polynomial, Acta Sci. Math., 1975, vol. 72, pp. 41–61.

    MathSciNet  Google Scholar 

  11. Gohberg, I.C. and Heinig, G., Resultant matrix and its generalization: II. Continual analog of resultant matrix, Acta Math. Acad. Sci. Hungar., 1976, vol. 28, pp. 189–209.

    Article  MathSciNet  Google Scholar 

  12. Gohberg, I.C. and Lerer, L.E., Resultant operators of a pair of analytic functions, Proc. Amer. Math. Soc., 1978, vol. 72, no. 1, pp. 65–73.

    Article  MathSciNet  Google Scholar 

  13. Gustafsson, B. and Tkachev, V.G., The resultant on compact Riemann surfaces, Comm. Math. Phys., 2009, vol. 10, pp. 265–308.

    MathSciNet  MATH  Google Scholar 

  14. Morozov, A.Yu. and Shakirov, Sh.R., New and old results in resultant theory, Theor. Math. Phys., 2010, vol. 163, no. 2, pp. 587–617.

    Article  Google Scholar 

  15. Kytmanov, A.M. and Khodos, O.V., An approach to the determination of the resultant of two entire functions, Russ. Math., 2018, vol. 62, no. 4, pp. 42–51.

    Article  MathSciNet  Google Scholar 

  16. Kytmanov, A.M. and Myshkina, E.K., On finding the resultant of two entire functions, Probl. Anal. Issues Anal., 2020, vol. 9, no. 3, pp. 119–130.

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work was carried out with the support of the following institutions: the first author was supported by the Russian Foundation for Basic Research, the Government of the Krasnoyarsk Krai, and the Krasnoyarsk Regional Science Foundation, grant no. 20-41-243002 (algorithm development and testing); the second author was supported by the Krasnoyarsk Mathematical Center and was financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education, agreement no. 075-02-2021-1388 (development and software implementation of the algorithm); and the third author was supported by the Russian Foundation for Basic Research, grant no. 19-31-60012 (formulation of the requirement specifications for the algorithm and computation of the examples).

Author information

Authors and Affiliations

  1. Siberian Federal University, pr. Svobodnyi 79, 660041, Krasnoyarsk, Russia

    V. I. Kuzovatov & A. A. Kytmanov

  2. Institute of Computational Modeling, Siberian Branch, Russian Academy of Sciences, Akademgorodok 50/44, 660036, Krasnoyarsk, Russia

    E. K. Myshkina

Authors
  1. V. I. Kuzovatov
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. A. A. Kytmanov
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. E. K. Myshkina
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding authors

Correspondence to V. I. Kuzovatov, A. A. Kytmanov or E. K. Myshkina.

Ethics declarations

The authors declare that they have no conflicts of interest.

Additional information

Translated by Yu. Kornienko

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kuzovatov, V.I., Kytmanov, A.A. & Myshkina, E.K. On Computing the Resultant of a Polynomial and an Entire Function. Program Comput Soft 48, 59–64 (2022). https://doi.org/10.1134/S0361768822010066

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0361768822010066