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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 Augot and Stepanov (2009) and Dvir et al. (2009) a general version of the Schwartz–Zippel was presented in Geil and Thomsen (2013) 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 Geil and Thomsen (2013) 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 (Geil and Thomsen, 2013). 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 (Høholdt, 1998; Geil and Høholdt, 2000)—the classical footprint bound taking not multiplicity into account. [ABSTRACT FROM AUTHOR]