Functional Analysis and Its Applications

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

Resultants and contour integrals



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 


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  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; Scholar
  3. [3]
    A. Miyake and M. Wadati, “Multipartite entaglement and hyperdeterminants,” Quantum Inf. Comput., 2(Special) (2002), 540–555; Scholar
  4. [4]
    M. Duff, String triality, black hole entropy and Cayley’s hyperdeterminant,
  5. [5]
    M. Duff, Hidden symmetries of the Nambu-Goto action,
  6. [6]
    R. Kallosh and A. Linde, “Strings, Black holes and quantum information,” Phys. Rev. D, 73:10 (2006), 104033; 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; 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;
  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; 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; Scholar
  11. [11]
    V. Dolotin and A. Morozov, Algebraic geometry of discrete dynamics. The case of one variable,
  12. [12]
    And. Morozov, “Universal Mandelbrot set as a model of phase transition theory,” JETP Lett., 86 (2007), 745–748; Scholar
  13. [13]
    C. Andrea and A. Dickenstein, Explicit formulas for the multivariate resultant,
  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; 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; 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

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