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Solving systems of nonlinear polynomial equations faster

Finding the solution to a system of n non-linear polynomial equations in n unknowns over a given field, say the algebraic closure of the coefficient field, is a classical and fundamental problem in computational algebra. For algebraic reasons (refer to footnote 1 in van der Waerden (1953, §80)) one considers projective problems, that is, the polynomials are homogeneous and the solutions are sought in n-dimensional projective space. Note also that the solutions of an affine system are specializations of the solution rays of its homogenized projective version. Going back to Cayley and Bezout in the last century, solvability of such a projective system is determined by the vanishing of a certain invariant, its resultant. This invariant generalizes the Sylvester resultant of two polynomials in a single variable (Knuth 1981) and the determinant of the coefficient matrix on a homogeneous linear system. In 1916 Macaulay (1916) showed that the resultant can be expressed by a quotient of two determinants whose correspond-