Solving the Polynomial Equations by Algorithms of Power Geometry

New methods for computation of solutions of an algebraic equation of three variables near a critical point are proposed. These methods are: Newton polyhedron, power transformations, new versions of the implicit function theorem [Bruno A. D. Power Geometry in Algebraic and Differential Equations. Elsevier, Amsterdam, 2000] and uniformization of a planar algebraic curve. We begin from a survey of the new methods of computation of solutions of an algebraic equation of one and of two variables by means of the Hadamard polygon [Hadamard J., Etude sur les proprietes des fonctions entieres et en particulier d'une fonction consideree par Riemann// Journal de mathematiques pures et appliquees, (1893) tome 9, 171–216 ; Bruno A. D., Local Methods in Nonlinear Differential Equations. Springer-Verlag, Berlin, 1989, P. I, Ch. IV, Sec. 2.1] and Hadamard polyhedron [Bruno A. D., On solution of an algebraic equation. Preprint of KIAM, No. 70, Moscow, 2016 (in Russian)].
That approach works for differential equations (ordinary and partial) as well. In the survey [Bruno A. D., Asymptotic solution of nonlinear algebraic and differential equations// International Mathematical Forum, (2015) 10:11, 535–564, http://dx.doi.org/10.12988/imf.2015.5974] there are several nontrivial applications.