Relations between roots and coefficients, interpolation and application to system solving

摘要

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