Geodesic
Rational expressions for multiple roots of algebraic equations
Sbornik. Mathematics, Tome 209 (2018) no. 10, pp. 1419-1444 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{SM_2018_209_10_a0,
     author = {I. A. Antipova and E. N. Mikhalkin and A. K. Tsikh},
     title = {Rational expressions for multiple roots of algebraic equations},
     journal = {Sbornik. Mathematics},
     pages = {1419--1444},
     year = {2018},
     volume = {209},
     number = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2018_209_10_a0/}
}
TY  - JOUR
AU  - I. A. Antipova
AU  - E. N. Mikhalkin
AU  - A. K. Tsikh
TI  - Rational expressions for multiple roots of algebraic equations
JO  - Sbornik. Mathematics
PY  - 2018
SP  - 1419
EP  - 1444
VL  - 209
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/SM_2018_209_10_a0/
LA  - en
ID  - SM_2018_209_10_a0
ER  - 
%0 Journal Article
%A I. A. Antipova
%A E. N. Mikhalkin
%A A. K. Tsikh
%T Rational expressions for multiple roots of algebraic equations
%J Sbornik. Mathematics
%D 2018
%P 1419-1444
%V 209
%N 10
%U http://geodesic.mathdoc.fr/item/SM_2018_209_10_a0/
%G en
%F SM_2018_209_10_a0
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/

[1] I. M. Gel'fand, M. M. Kapranov, A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Math. Theory Appl., Birkhäuser Boston, Inc., Boston, MA, 1994, x+523 pp. | DOI | MR | Zbl

[2] D. Hilbert, “Ueber die Singularitäten der Diskriminantenfläche”, Math. Ann., 30:4 (1887), 437–441 | DOI | MR | Zbl

[3] S. Kurmann, “Some remarks on equations defining coincident root loci”, J. Algebra, 352 (2012), 223–231 | DOI | MR | Zbl

[4] J. Weyman, “The equations of strata for binary forms”, J. Algebra, 122:1 (1989), 244–249 | DOI | MR | Zbl

[5] H. Lee, B. Sturmfels, “Duality of multiple root loci”, J. Algebra, 446 (2016), 499–526 | DOI | MR | Zbl

[6] G. Katz, “How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties”, Expo. Math., 21:3 (2003), 219–261 | DOI | MR | Zbl

[7] E. N. Mikhalkin, A. K. Tsikh, “Singular strata of cuspidal type for the classical discriminant”, Sb. Math., 206:2 (2015), 282–310 | DOI | DOI | MR | Zbl

[8] K. Oka, “Sur les fonctions analytiques de plusieurs variables. VIII. Lemme fundamental”, J. Math. Soc. Japan, 1951, no. 3, 204–214 | DOI | MR | Zbl

[9] I. A. Antipova, A. K. Tsikh, “The discriminant locus of a system of $n$ Laurent polynomials in $n$ variables”, Izv. Math., 76:5 (2012), 881–906 | DOI | DOI | MR | Zbl

[10] A. Esterov, “The discriminant of a system of equations”, Adv. Math., 245 (2013), 534–572 | DOI | MR | Zbl

[11] T. M. Sadykov, A. K. Tsikh, Gipergeometricheskie i algebraicheskie funktsii mnogikh peremennykh, Nauka, M., 2014, 408 pp.

[12] Ch. W. Curtis, I. Reiner, Representation theory of finite groups and associative algebras, Pure Appl. Math., 11, Interscience Publishers (John Wiley Sons), New York–London, 1962, xiv+685 pp. | MR | MR | Zbl

[13] G. Mikhalkin, “Real algebraic curves, the moment map and amoebas”, Ann. of Math. (2), 151:1 (2000), 309–326 | DOI | MR | Zbl

[14] M. M. Kapranov, “A characterization of $A$-discriminantal hypersurfaces in terms of the logarithmic Gauss map”, Math. Ann., 290:2 (1991), 277–285 | DOI | MR | Zbl

[15] M. Passare, A. Tsikh, “Algebraic equations and hypergeometric series”, The legacy of Niels Henrik Abel, Springer, Berlin, 2004, 653–672 | MR | Zbl

[16] R. Birkeland, “Über die Auflösung algebraischer Gleichungen durch hypergeometrische Funktionen”, Math. Z., 1927, no. 26, 566–578 | DOI | MR | Zbl

[17] E. N. Mikhalkin, “The monodromy of a general algebraic function”, Siberian Math. J., 56:2 (2015), 330–338 | DOI | MR | Zbl

[18] E. N. Mikhalkin, “On solving general algebraic equations by integrals of elementary functions”, Siberian Math. J., 47:2 (2006), 301–306 | DOI | MR | Zbl

[19] V. R. Kulikov, V. A. Stepanenko, “On solutions and Waring's formulas for the systems of $n$ algebraic equations for $n$ unknowns”, St. Petersburg Math. J., 26:5 (2015), 839–848 | DOI | MR | Zbl

[20] Hj. Mellin, “Résolution de l'équation algébrique générale à l'aide de la fonction gamma”, C. R. Acad. Sci. Paris Sér. I Math., 172 (1921), 658–661 | Zbl