Programming and Computer Software

, Volume 44, Issue 2, pp 75–85 | Cite as

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

  • A. B. Batkhin


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.


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  1. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Kac, V. and Cheung, P., Quantum Calculus, New York: Springer, 2002.CrossRefzbMATHGoogle Scholar
  4. 4.
    Ernst, T., A Comprehensive Treatment of q-Calculus, Basel: Springer, 2012.CrossRefzbMATHGoogle Scholar
  5. 5.
    Gasper, G. and Rahman, M., Basic Hypergeometric Series, Cambridge: Campridge Univ. Press, 1990.zbMATHGoogle Scholar
  6. 6.
    Koekoek, R., Lesky, P.A., and Swarttouw, R.F., Hypergeometric Orthogonal Polynomials and Theirq-Analogues, Berlin: Springer, 2010.CrossRefzbMATHGoogle Scholar
  7. 7.
    Batkhin, A.B., Parameterization of the discriminant set of a polynomial, Program. Comput. Software, 2016, vol. 42, no. 2, pp. 67–76.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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. Scholar
  9. 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.CrossRefGoogle Scholar
  10. 10.
    Hahn, W., Über Orthogonalpolynome, die q-Differenzengleichungen genügen, Math. Nachrichten, 1949, vol. 2, pp. 4–34.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ismail, M.H.E., q-Discriminants and vertex operators, Adv. Appl. Math., 2001, vol. 27, pp. 482–492.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Goulden, I.P. and Jackson, D.M., Combinatorial Enumeration, New York: Wiley, 1983.zbMATHGoogle Scholar
  13. 13.
    Batkhin, A.B., Structure of the discriminant set of real polynomial, Chebyshevskii Sb., 2015, vol. 16, no. 2, pp. 23–34. Scholar
  14. 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. Scholar
  15. 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. Scholar
  16. 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: // Scholar
  17. 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. 18.
    Basu, S., Pollack, R., and Roy, M.-F., Algorithms in Real Algebraic Geometry, Algorithms and Computations in Mathematics, vol. 10, Berlin: Springer, 2006.zbMATHGoogle Scholar
  19. 19.
    von zur Gathen, J. and Lücking, T., Subresultants revisited, Theor. Comput. Sci., 2003, vol. 297, pp. 199–239.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jury, E.I., Inners and Stability of Dynamic Systems, New York: Wiley, 1974.zbMATHGoogle Scholar
  21. 21.
    Sushkevich, A.K., Foundation of Higher Algebra, 4 ed., Moscow: ONTI, 1941 [in Russian].Google Scholar
  22. 22.
    Andrews, G.E., The Theory of Partitions, Reading: Addison-Wesley, 1976.zbMATHGoogle Scholar
  23. 23.
    MacDonald, I.G., Symmetric Functions and Hall Polynomials, New York: Oxford Univ. Press, 1995.zbMATHGoogle Scholar
  24. 24.
    Prasolov, V.V., Polynomials, Berlin: Springer, 2004, vol. 11 of Algorithms and Computation in Mathematics.CrossRefzbMATHGoogle Scholar
  25. 25.
    Cigler, J., Operatormethoden für-Identitäten, Monatshefte für Mathematik, 1979, vol. 2, no. 88, pp. 87–105.CrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Keldysh Institute of Applied MathematicsRussian Academy of SciencesMoscowRussia
  2. 2.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia

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