Advertisement

Functional Analysis and Its Applications

, Volume 46, Issue 1, pp 33–40 | Cite as

Resultants and contour integrals

  • A. Morozov
  • Sh. Shakirov
Article

Abstract

Resultants are important special functions used to describe nonlinear phenomena. The resultant \(R_{r_1 \ldots r_n }\) determines a consistency condition for a system of n homogeneous polynomials of degrees r 1, ..., r n in n variables in precisely the same way as the determinant does for a system of linear equations. Unfortunately, there is a lack of convenient formulas for resultants in the case of a large number of variables. In this paper we use Cauchy contour integrals to obtain a polynomial formula for resultants, which is expected to be useful in applications.

Key words

rezultant algebraic equation contour integral 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, Boston, MA, 1994.MATHCrossRefGoogle Scholar
  2. [2]
    V. Dolotin and A. Morozov, Introduction to Non-Linear Algebra, World Scientific, Hackensack, NJ, 2007; http://arxiv.org/abs/hep-th/0609022.Google Scholar
  3. [3]
    A. Miyake and M. Wadati, “Multipartite entaglement and hyperdeterminants,” Quantum Inf. Comput., 2(Special) (2002), 540–555; http://arxiv.org/abs/quant-ph/0212146.MathSciNetMATHGoogle Scholar
  4. [4]
    M. Duff, String triality, black hole entropy and Cayley’s hyperdeterminant, http://arxiv.org/abs/hep-th/0601134.
  5. [5]
    M. Duff, Hidden symmetries of the Nambu-Goto action, http://arxiv.org/abs/hep-th/0602160.
  6. [6]
    R. Kallosh and A. Linde, “Strings, Black holes and quantum information,” Phys. Rev. D, 73:10 (2006), 104033; http://arxiv.org/abs/hep-th/0602061.MathSciNetCrossRefGoogle Scholar
  7. [7]
    S. Kachru, A. Klemm, W. Lerche, P. Mayr, and C. Vafa, “Nonperturbative results on the point particle limit of N = 2 heterotic string compactifications,” Nucl. Phys. B, 459:3 (1996), 537–555; http://arxiv.org/abs/hep-th/9508155.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    T. Eguchi and Y. Tachikawa, “Rigid limit in N = 2 supergravity and weak-gravity conjecture,” J. High Energy Phys., 2007:8, 068; http://arxiv.org/abs/0706.2114.
  9. [9]
    P. Aspinwall, B. Greene, and D. Morrison, “Measuring small distances in N = 2 sigma models,” Nucl. Phys. B, 420:1–2 (1994), 184–242; http://arxiv.org/abs/hep-th/9311042.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    A. Morozov and A. Niemi, “Can renormalization group flow end in a Big Mess?,” Nucl. Phys. B, 666:3 (2003), 311–336; http://arxiv.org/abs/hep-th/0304178.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    V. Dolotin and A. Morozov, Algebraic geometry of discrete dynamics. The case of one variable, http://arxiv.org/abs/hep-th/0501235.
  12. [12]
    And. Morozov, “Universal Mandelbrot set as a model of phase transition theory,” JETP Lett., 86 (2007), 745–748; http://arxiv.org/abs/0710.2315.CrossRefGoogle Scholar
  13. [13]
    C. Andrea and A. Dickenstein, Explicit formulas for the multivariate resultant, http://arxiv.org/abs/math/0007036.
  14. [14]
    M. Chardin, “Formules à la Macaulay pour les sous-résultants en plusieurs variables,” C. R. Acad. Sci. Paris, 319:5 (1994), 433–436.MathSciNetMATHGoogle Scholar
  15. [15]
    D. Manocha and J. Canny, “Multipolynomial resultant algorithms,” J. Symb. Comp., 15:5 (1993), 99–122.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    J. Canny, E. Kaltofen, and L. Yagati, “Solving systems of non-linear equations faster,” in: Proc. Internat. Symp. Symbolic Algebraic Comput., ISSAC’89, ACM Press, New York, 1989, 121–128.Google Scholar
  17. [17]
    B. Gustafsson and V. Tkachev, “The resultant on compact Riemann surfaces,” Comm. Math. Phys., 286:1 (2009), 313–358; http://arxiv.org/abs/0710.2326.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    A. Morozov and Sh. Shakirov, “Analogue of the identity Log Det=Trace Log for resultants,” J. Geom. Phys., 61:3 (2011), 708–726; http://arxiv.org/abs/0804.4632.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    A. K. Tsikh, Multidimensional Residues and Their Applications, Transl. Math. Monographs, vol. 103, Amer. Math. Soc., Providence, RI, 1992.MATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institute for Theoretical and Experimental PhysicsMoscowRussia

Personalised recommendations