  We present a new effective Nullstellensatz with bounds for the degrees which
depend not only on the number of variables and on the degrees of the input
polynomials but also on an additional parameter called the {\it geometric
degree of the system of equations}. The obtained bound is polynomial in these
parameters. It is essentially optimal in the general case, and it substantially
improves the existent bounds in some special cases. The proof of this result is
combinatorial, and it relies on global estimations for the Hilbert function of
homogeneous polynomial ideals. In this direction, we obtain a lower bound for
the Hilbert function of an arbitrary homogeneous polynomial ideal, and an upper
bound for the Hilbert function of a generic hypersurface section of an unmixed
radical polynomial ideal.
