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, fP(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) = a0 + a1x +···+ amxm, with ai ∈ R, 0 ≤ i ≤ m, and a real number u so that P (u) = 0, we have got an analytic function fP(y) that satisfies x = fP(P(x)) around x = u. Besides, we also introduce a new proof (completely different) of the theorems involves in the construction of fP(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 fP(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.