More results on the number of zeros of multiplicity at least r

We consider multivariate polynomials and investigate how many zeros of multiplicity at least $r$ they can have over a Cartesian product of finite subsets of a field. Here r is any prescribed positive integer and the definition of multiplicity that we use is the one related to Hasse derivatives. As a generalization of material in [2,5] a general version of the Schwartz-Zippel was presented in [8] which from the leading monomial – with respect to a lexicographic ordering – estimates the sum of zeros when counted with multiplicity. The corresponding corollary on the number of zeros of multiplicity at least $r$ is in general not sharp and therefore in [8] a recursively defined function $D$ was introduced using which one can derive improved information. The recursive function being rather complicated, the only known closed formula consequences of it are for the case of two variables [8]. In the present paper we derive closed formula consequences for arbitrary many variables, but for the powers in the leading monomial being not too large. Our bound can be viewed as a generalization of the footprint bound [10,6] – the classical footprint bound taking not multiplicity into account.