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

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