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Mat. Sb., 2018, Volume 209, Number 10, Pages 3–30 (Mi msb8950)  

This article is cited in 1 scientific paper (total in 1 paper)

Rational expressions for multiple roots of algebraic equations

I. A. Antipova, E. N. Mikhalkin, A. K. Tsikh

Siberian Federal University, Krasnoyarsk

Abstract: The general polynomial with variable coefficients is considered. In terms of the resultants of this polynomials and its derivatives simple rational expressions in the coefficients of the polynomial are found for its multiple zeros. Similar results are extended to systems of $n$ polynomial equations with $n$ unknowns. Justifications of the formulae for multiple roots thus obtained are based on the properties of the logarithmic Gauss map of the discriminant variety of a system of equations and on a linearization procedure for the system. The resulting formulae are of interest not only for theoretical aspects of the algebra of polynomials, but also for numerical mathematics and various areas of applied mathematics connected with finding critical points of polynomial maps.
Bibliography: 20 titles.

Keywords: general algebraic equation, system of algebraic equations, discriminant variety, multiple root, logarithmic Gauss map.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 1.1462.2014/k
14.Y26.31.0006
НШ-9149.2016.1
1.2604.2017/пч
The research of I. A. Antipova was carried out as part of the implementation of a state assignment of the Ministry of Education and Science of the Russian Federation (project no. 1.1462.2014/k). The research of E. N. Mikhalkin was carried out with the support of a grant for state support of scientific research conducted under the auspices of leading scientists of the Government of the Russian Federation (project no. 14.Y26.31.0006) and in the framework of the Programme for State Support of Leading Scientific Schools of the President of the Russian Federation (grant no. НШ-9149.2016.1). The research of A. K. Tsikh was carried out as part of the implementation of a state assignment of the Ministry of Education and Science of the Russian Federation (project no. 1.2604.2017/пч).

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8950

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English version:
Sbornik: Mathematics, 2018, 209:10, 1419–1444

Bibliographic databases:

UDC: 517.55+512.761
MSC: Primary 13P15, 26C10; Secondary 14M12
Received: 29.03.2017 and 31.12.2017

Citation: I. A. Antipova, E. N. Mikhalkin, A. K. Tsikh, “Rational expressions for multiple roots of algebraic equations”, Mat. Sb., 209:10 (2018), 3–30; Sb. Math., 209:10 (2018), 1419–1444

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. 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
  • Математический сборник Sbornik: Mathematics (from 1967)
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