Geodesic
Rational expressions for multiple roots of algebraic equations
Sbornik. Mathematics, Tome 209 (2018) no. 10, pp. 1419-1444

Voir la notice de l'article provenant de la source Math-Net.Ru

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
Mots-clés : logarithmic Gauss map. 
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     author = {I. A. Antipova and E. N. Mikhalkin and A. K. Tsikh},
     title = {Rational expressions for multiple roots of algebraic equations},
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I. A. Antipova; E. N. Mikhalkin; A. K. Tsikh. Rational expressions for multiple roots of algebraic equations. Sbornik. Mathematics, Tome 209 (2018) no. 10, pp. 1419-1444. http://geodesic.mathdoc.fr/item/SM_2018_209_10_a0/