  Let S be a finite subset of a field. For multivariate polynomials the
generalized Schwartz-Zippel bound [2], [4] estimates the number of zeros over
Sx...xS counted with multiplicity. It does this in terms of the total degree,
the number of variables and |S|. In the present work we take into account what
is the leading monomial. This allows us to consider more general point
ensembles and most importantly it allows us to produce much more detailed
information about the number of zeros of multiplicity r than can be deduced
from the generalized Schwartz-Zippel bound. We present both upper and lower
bounds.
