We show that if a system of polynomials f_1, f_2,..., f_r in n variables with deg(f_i) <= d over the rational
numbers has only finitely many affine zeros, then, ail the affine zeros can be determined in time polynomial
in d^n by a Las Vegas type randomized algorithm. We then describe single exponential time algorithms
to compute reduced Grobner bases for the radical of the ideal generated by f_i and for all the prime ideals
containing the radical.