Finding all real zeros of polynomial systems using multi-resultant

We present a new practicable method for approximating all real zeros of polynomial systems using the resultants method. It is based on the theory of multi-resultants. We build a sparse linear system. Then, we solve it by the quasi-minimal residual method. Once our test function changes its sign, we apply the secant method to approximate the root. The unstable calculation of the determinant of the large sparse matrix is replaced by solving a sparse linear system. This technique will be able to take advantage of the sparseness of the resultant matrix. Theoretical and numerical results are presented.