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Izv. RAN. Ser. Mat., 2012, Volume 76, Issue 5, Pages 29–56 (Mi izv6990)  

This article is cited in 10 scientific papers (total in 10 papers)

The discriminant locus of a system of $n$ Laurent polynomials in $n$ variables

I. A. Antipovaa, A. K. Tsikhb

a Institute of Space and Information Technologies, Siberian Federal University
b Institute of Mathematics, Siberian Federal University

Abstract: We consider a system of $n$ algebraic equations in $n$ variables, where the exponents of the monomials in each equation are fixed while all the coefficients vary. The discriminant locus of such a system is the closure of the set of all coefficients for which the system has multiple roots with non-zero coordinates. For dehomogenized discriminant loci, we give parametrizations of those irreducible components that depend on the coefficients of all the equations. We prove that if such a component has codimension 1, then the parametrization is inverse to the logarithmic Gauss map of the component (an analogue of Kapranov's result for the $A$-discriminant). Our argument is based on the linearization of algebraic systems and the parametrization of the set of its critical values.

Keywords: discriminant locus, linearization of an algebraic system, logarithmic Gauss map.

DOI: https://doi.org/10.4213/im6990

Full text: PDF file (691 kB)
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English version:
Izvestiya: Mathematics, 2012, 76:5, 881–906

Bibliographic databases:

UDC: 517.55+512.7
MSC: 32B10, 32S05, 32S70, 33C70
Received: 03.02.2011
Revised: 21.11.2011

Citation: I. A. Antipova, A. K. Tsikh, “The discriminant locus of a system of $n$ Laurent polynomials in $n$ variables”, Izv. RAN. Ser. Mat., 76:5 (2012), 29–56; Izv. Math., 76:5 (2012), 881–906

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Irina A. Antipova, Tatyana V. Zykova, “Mellin transform for monomial functions of the solution to the general polynomial system”, Zhurn. SFU. Ser. Matem. i fiz., 6:2 (2013), 150–156  mathnet
    2. Vladimir R. Kulikov, “Conditions for convergence of the Mellin–Barnes integral for solution to system of algebraic equations”, Zhurn. SFU. Ser. Matem. i fiz., 7:3 (2014), 339–346  mathnet
    3. V. R. Kulikov, V. A. Stepanenko, “On solutions and Waring's formulae for the system of $n$ algebraic equations with $n$ unknowns”, St. Petersburg Math. J., 26:5 (2015), 839–848  mathnet  crossref  mathscinet  isi  elib  elib
    4. E. N. Mikhalkin, A. K. Tsikh, “Singular strata of cuspidal type for the classical discriminant”, Sb. Math., 206:2 (2015), 282–310  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. Evgeny N. Mikhalkin, Avgust K. Tsikh, “On the structure of the classical discriminant”, Zhurn. SFU. Ser. Matem. i fiz., 8:4 (2015), 426–436  mathnet  crossref
    6. Mikhalkin E.N., Shchuplev A.V., Tsikh A.K., “Amoebas of Cuspidal Strata for Classical Discriminant”, Complex Analysis and Geometry, Springer Proceedings in Mathematics & Statistics, Springer Proceedings in Mathematics & Statistics, 144, eds. Bracci F., Byun J., Gaussier H., Hirachi K., Kim K., Shcherbina N., Springer, 2015, 257–272  crossref  mathscinet  zmath  isi  scopus
    7. V. R. Kulikov, “A criterion for the convergence of the Mellin–Barnes integral for solutions to simultaneous algebraic equations”, Siberian Math. J., 58:3 (2017), 493–499  mathnet  crossref  crossref  isi  elib  elib
    8. Esterov A., “Characteristic Classes of Affine Varieties and Plucker Formulas For Affine Morphisms”, J. Eur. Math. Soc., 20:1 (2018), 15–59  crossref  mathscinet  zmath  isi  scopus
    9. I. A. Antipova, E. N. Mikhalkin, A. K. Tsikh, “Rational expressions for multiple roots of algebraic equations”, Sb. Math., 209:10 (2018), 1419–1444  mathnet  crossref  crossref  isi  elib
    10. Irina A. Antipova, Evgeny N. Mikhalkin, Avgust K. Tsikh, “Singular points of complex algebraic hypersurfaces”, Zhurn. SFU. Ser. Matem. i fiz., 11:6 (2018), 670–679  mathnet  crossref
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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