Relations Between Roots and Coefficients, Interpolation and Application to System Solving

We propose an algebraic framework to represent zero-dimensional algebraic systems. In this framework, we give new interpolation formulæ. We use this good representation of the algebraic systems to develop a generalization of Weierstrass’s method to the multivariate systems. This method allows us to approximate simultaneously all the roots of an algebraic system. We obtain an effective iteration function with a quadratic convergence in a neighbourhood of the solutions. We use this Weierstrass iteration function in a continuation method to obtain a global method. Experiments are exposed to underline the efficiency of the approach.