A Gröbner Free Alternative for Polynomial System Solving

Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers.

We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton's iterator.

We present our implementation in the Magma system which is called Kronecker in homage to his method for solving systems of polynomial equations. We exhibit some cases for which our program is more efficient than the other available softwares.

This is joint work with Marc Giusti and Bruno Salvy.