Siberian Mathematical Journal

, Volume 58, Issue 3, pp 493–499 | Cite as

A criterion for the convergence of the Mellin–Barnes integral for solutions to simultaneous algebraic equations

  • V. R. KulikovEmail author


We obtain a criterion for the convergence of the Mellin–Barnes integral representing the solution to a general system of algebraic equations. This yields a criterion for a nonnegative matrix to have positive principal minors. The proof rests on the Nilsson–Passare–Tsikh Theorem about the convergence domain of the general Mellin–Barnes integral, as well as some theorem of a linear algebra on a subdivision of the real space into polyhedral cones.


algebraic equations Mellin–Barnes integral hypergeometric functions convergence domain 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Mellin H. J., “Résolution de l’équation algébrique générale à l’aide de la fonction gamma,” C. R. Acad. Sci. Paris Sér. I. Math., vol. 172, 658–661 (1921).zbMATHGoogle Scholar
  2. 2.
    Antipova I. A., “Inversion of many-dimensional Mellin transforms and solutions of algebraic equations,” Sb. Math., vol. 198, no. 4, 447–463 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Antipova I. A. and Zykova T. V., “Mellin transforms and algebraic functions,” Integral Transform. Spec. Funct., vol. 26, no. 10, 753–767 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Sadykov T. M., “On a multidimensional system of hypergeometric differential equations,” Sib. Math. J., vol. 39, no. 5, 986–997 (1998).CrossRefzbMATHGoogle Scholar
  5. 5.
    Dickenstein A. and Sadykov T. M., “Bases in the solution space of the Mellin system,” Sb. Math., vol. 198, no. 9, 1277–1298 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Antipova I. A., “An expression for the superposition of general algebraic functions in terms of hypergeometric series,” Sib. Math. J., vol. 44, no. 5, 757–764 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Stepanenko V. A., “Solution of the system of n algebraic equations in n unknowns by hypergeometric functions,” Vestn. KrasGU, vol. 1, 35–48 (2003).Google Scholar
  8. 8.
    Nilsson L., Amoebas, Discriminants, and Hypergeometric Functions, Doctoral Thes., Department of Mathematics, Stockholm University, Sweden (2009).Google Scholar
  9. 9.
    Sadykov T. M. and Tsikh A. K., Hypergeometric and Algebraic Functions in Several Variables [Russian], Nauka, Moscow (2014).zbMATHGoogle Scholar
  10. 10.
    Samelson H., Thrall R. M., and Wesler O., “A partition theorem for Euclidean n-space,” Proc. Amer. Math. Soc., vol. 9, 805–807 (1958).MathSciNetzbMATHGoogle Scholar
  11. 11.
    Prasolov V. V., Problems and Theorems in Linear Algebra, Amer. Math. Soc., Providence (1994).zbMATHGoogle Scholar
  12. 12.
    Antipova I. A. and Tsikh A. K., “The discriminant locus of a system of n Laurent polynomials in n variables,” Izv. Math., vol. 76, no. 5, 881–906 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kulikov V. R., “Conditions for convergence of the Mellin–Barnes integral for solution to system of algebraic equations,” J. Sib. Fed. Univ. Math. Phys., vol. 7, no. 3, 339–346 (2014).Google Scholar
  14. 14.
    Zhdanov O. N. and Tsikh A. K., “Studying the multiple Mellin–Barnes integrals by means of multidimensional residues,” Sib. Math. J., vol. 39, no. 2, 245–260 (1998).CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Siberian Federal UniversityKrasnoyarskRussia

Personalised recommendations