Solving systems of algebraic equations

Let f1,…,fk be k multivariate polynomials which have a finite number of common zeros in the algebraic closure of the ground field, counting the common zeros at infinity. An algorithm is given and proved which reduces the computations of these zeros to the resolution of a single univariate equation whose degree is the number of common zeros. This algorithm gives the whole algebraic and geometric structure of the set of zeros (multiplicities, conjugate zeros,…). When all the polynomials have the same degree, the complexity of this algorithm is polynomial relative to the generic number of solutions.