An optimal condition for determining the exact number of roots of a polynomial system

It was shown in ~er75] that the number of roots in (C”) n of a polynomial system depends only on the Newton polytopes of the system, for almost all specializations of the coefficients. This result, henceforth referred to as the BKK bound, gives an upper bound on the number of roots of a polynomial system. The BKK bound is often much better than the Bezout bound for the same system, but the original theorem gives an exact bound only if all the coefficients corresponding to Newton polytope boundaries are generically chosen. In this paper, we show that the BKK bound is exact under much weaker assumptions: only coefficients corresponding to certain vertices of the Newton polytopes need be generic. This result allows application of the BKK bound to many practical problems.