Inverse functions of polynomials and its applications to initialize the search of solutions of polynomials and polynomial systems

In this paper we present a new algorithm for solving polynomial equations based on the Taylor series of the inverse function of a polynomial, f _P (y). The foundations of the computing of such series have been previously developed by the authors in some recent papers, proceeding as follows: given a polynomial function (y=P(x)=a_0+a_1x+cdots+a_mx^m), with (a_i in mathcal{R}, 0 leq i leq m), and a real number u so that P′(u)?≠?0, we have got an analytic function f _P (y) that satisfies x?=?f _P (P(x)) around x?=?u. Besides, we also introduce a new proof (completely different) of the theorems involves in the construction of f _P (y), which provide a better radius of convergence of its Taylor series, and a more general perspective that could allow its application to other kinds of equations, not only polynomials. Finally, we illustrate with some examples how f _P (y) could be used for solving polynomial systems. This question has been already treated by the authors in preceding works in a very complex and hard way, that we want to overcome by using the introduced algorithm in this paper.