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

Parameterization of a Set Determined by the Generalized Discriminant of a Polynomial

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

Abstract

A generalization of the classical discriminant of the real polynomial defined using the linear Hahn operator that decreases the degree of the polynomial by one is studied. The structure of the generalized discriminant set of the real polynomial, i.e., the set of values of the polynomial coefficients at which the polynomial and its Hahn operator image have a common root, is investigated. The structure of the generalized discriminant of the polynomial of degree n is described in terms of the partitions of n Algorithms for the construction of a polynomial parameterization of the generalized discriminant set in the space of the polynomial coefficients are proposed. The main steps of these algorithms are implemented in a Maple library. Examples of calculating the discriminant set are discussed.

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

Similar content being viewed by others

Explore related subjects

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

References

  1. 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 

  2. Batkhin, A.B., The boundary of the stability set of a multiparameter Hamiltonian system, Vestn. Volgograd Gos. Univ., Ser. 1 Mat., Fiz., 2014, vol. 24, no. 5, pp. 6–23.

    Article  MathSciNet  Google Scholar 

  3. Kac, V. and Cheung, P., Quantum Calculus, New York: Springer, 2002.

    Book  MATH  Google Scholar 

  4. Ernst, T., A Comprehensive Treatment of q-Calculus, Basel: Springer, 2012.

    Book  MATH  Google Scholar 

  5. Gasper, G. and Rahman, M., Basic Hypergeometric Series, Cambridge: Campridge Univ. Press, 1990.

    MATH  Google Scholar 

  6. Koekoek, R., Lesky, P.A., and Swarttouw, R.F., Hypergeometric Orthogonal Polynomials and Theirq-Analogues, Berlin: Springer, 2010.

    Book  MATH  Google Scholar 

  7. Batkhin, A.B., Parameterization of the discriminant set of a polynomial, Program. Comput. Software, 2016, vol. 42, no. 2, pp. 67–76.

    Article  MathSciNet  MATH  Google Scholar 

  8. Batkhin, A.B., The structure of the resonance set of a real polynomial, Preprint of the Keldysh Institute of Applied Mathematics, Moscow, 2016, no. 29. http://www.keldysh.ru/papers/2015/prep2016_29.pdf.

    Google Scholar 

  9. Magnus, A.P., Associated Askey–Wilson polynomials as Laguerre Hahn orthogonal polynomials, Proc. of an Int. Symp. on Orthogonal Polynomials and their Applications, Segovia, Spain, 1986, ed. by Alfaro, M., Dehesa, J.S., Marcellan, F.J., et al., Berlin: Springer, 1988, vol. 1329 of Lecture Notes in Mathematics, pp. 261–278.

    Chapter  Google Scholar 

  10. Hahn, W., Über Orthogonalpolynome, die q-Differenzengleichungen genügen, Math. Nachrichten, 1949, vol. 2, pp. 4–34.

    Article  MathSciNet  MATH  Google Scholar 

  11. Ismail, M.H.E., q-Discriminants and vertex operators, Adv. Appl. Math., 2001, vol. 27, pp. 482–492.

    Article  MathSciNet  MATH  Google Scholar 

  12. Goulden, I.P. and Jackson, D.M., Combinatorial Enumeration, New York: Wiley, 1983.

    MATH  Google Scholar 

  13. Batkhin, A.B., Structure of the discriminant set of real polynomial, Chebyshevskii Sb., 2015, vol. 16, no. 2, pp. 23–34. http://mi.mathnet.ru/eng/cheb/v16/i2/p23

    MathSciNet  Google Scholar 

  14. Batkhin, A.B., Parametrization of the discriminant set of a real polynomial, Preprint of the Keldysh Institute of Applied Mathematics, Moscow, 2015, no. 76. http://keldysh.ru/papers/2015/prep2015_76.pdf

    Google Scholar 

  15. Batkhin, A.B., On the structure of the resonance set of the real polynomial, Chebyshov Sb. (Tula), 2016, vol. 17, no. 3, pp. 5–17. http://mi.mathnet.ru/cheb494

    MathSciNet  Google Scholar 

  16. Batkhin, A.B., The resonance set of the polynomial and the formal stability problem, Vestn. Volgograd Gos. Univ., Ser. 1 Mat., Fiz., 2016, no. 4 (35), pp. 5–23. http: //mi.mathnet.ru/cheb494

    MathSciNet  Google Scholar 

  17. Kalinina, E.A. and Uteshev, A.Yu., Elimination Theory, St. Petersburg: Naucho-Issledovatel’skii Ist. Khimii, St. Petersburg Univ., 2002 [in Russian].

    Google Scholar 

  18. Basu, S., Pollack, R., and Roy, M.-F., Algorithms in Real Algebraic Geometry, Algorithms and Computations in Mathematics, vol. 10, Berlin: Springer, 2006.

    MATH  Google Scholar 

  19. von zur Gathen, J. and Lücking, T., Subresultants revisited, Theor. Comput. Sci., 2003, vol. 297, pp. 199–239.

    Article  MathSciNet  MATH  Google Scholar 

  20. Jury, E.I., Inners and Stability of Dynamic Systems, New York: Wiley, 1974.

    MATH  Google Scholar 

  21. Sushkevich, A.K., Foundation of Higher Algebra, 4 ed., Moscow: ONTI, 1941 [in Russian].

    Google Scholar 

  22. Andrews, G.E., The Theory of Partitions, Reading: Addison-Wesley, 1976.

    MATH  Google Scholar 

  23. MacDonald, I.G., Symmetric Functions and Hall Polynomials, New York: Oxford Univ. Press, 1995.

    MATH  Google Scholar 

  24. Prasolov, V.V., Polynomials, Berlin: Springer, 2004, vol. 11 of Algorithms and Computation in Mathematics.

    Book  MATH  Google Scholar 

  25. Cigler, J., Operatormethoden für-Identitäten, Monatshefte für Mathematik, 1979, vol. 2, no. 88, pp. 87–105.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    A. B. Batkhin

  2. Moscow Institute of Physics and Technology, Dolgoprudnyi, Moscow oblast, 141701, Russia

    A. B. Batkhin

Authors
  1. A. B. Batkhin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. B. Batkhin.

Additional information

Original Russian Text © A.B. Batkhin, 2018, published in Programmirovanie, 2018, Vol. 44, No. 2.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Batkhin, A.B. Parameterization of a Set Determined by the Generalized Discriminant of a Polynomial. Program Comput Soft 44, 75–85 (2018). https://doi.org/10.1134/S0361768818020032

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0361768818020032