[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Computing Level Lines of a Polynomial on the Plane

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

Abstract

Application of the method of computing the location of all types of level lines of a real polynomial on the real plane is demonstrated. The theory underlying this method is based on methods of local and global analysis by the means of power geometry and computer algebra. Three nontrivial examples of computing level lines of real polynomials on the real plane are discussed in detail. The following computer algebra algorithms are used: factorization of polynomials, computation of the Gröbner basis, construction of the Newton polygon, and representation of an algebraic curve on a plane. It is shown how computational difficulties can be overcome.

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
£29.99 /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 (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.
Fig. 11.
Fig. 12.
Fig. 13.
Fig. 14.
Fig. 15.
Fig. 16.
Fig. 17.
Fig. 18.
Fig. 19.
Fig. 20.
Fig. 21.

Similar content being viewed by others

Explore related subjects

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

REFERENCES

  1. Bruno, A.D. and Batkhin, A.B., Algorithms and programs for calculating the roots of polynomial of one or two variables, Program. Comput. Software, 2021, vol. 47, no. 5, pp. 353–373.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bruno, A.D. and Batkhin, A.B., Level Lines of a polynomial on a plane, Program. Comput. Software, 2022, vol. 48, no. 1, pp. 19–29.

    Article  MathSciNet  MATH  Google Scholar 

  3. Batkhin, A.B., Bruno, A.D., and Varin, V.P., Stability sets of multiparameter Hamiltonian systems, J. Appl. Math. Mech., 2012, vol. 76, no. 1, pp. 56–92.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bruno, A.D., Local Methods in Nonlinear Differential Equations, Berlin: Springer, 1989.

    Book  Google Scholar 

  5. Kollár, J. Lectures on Resolution of Singularities, Princeton: Princeton Univ. Press, 2007.

    MATH  Google Scholar 

  6. Milnor, J.W., Morse theory. Based on lecture notes by M. Spivak and R. Wells, Princeton, N.J.: Princeton Univ. Press, 1963.

    MATH  Google Scholar 

  7. Cox, D., Little, J., and O’Shea, D., Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, Heidelberg: Springer, 2015, 4th ed.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, Russian Academy of Sciences, 125047, Moscow, Russia

    A. D. Bruno & A. B. Batkhin

  2. Moscow Institute of Physics and Technology (National Research University), 141700, Dolgoprudnyi, Moscow oblast, Russia

    A. B. Batkhin

  3. Samarkand State University, 140104, Samarkand, Uzbekistan

    Z. Kh. Khaidarov

Authors
  1. A. D. Bruno
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. B. Batkhin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Z. Kh. Khaidarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. D. Bruno, A. B. Batkhin or Z. Kh. Khaidarov.

Ethics declarations

The authors declare that they have no conflicts of interest.

Additional information

Translated by A. Klimontovich

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bruno, A.D., Batkhin, A.B. & Khaidarov, Z.K. Computing Level Lines of a Polynomial on the Plane. Program Comput Soft 49, 69–85 (2023). https://doi.org/10.1134/S0361768823020068

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0361768823020068

Search

Navigation