  We upper bound the number of common zeros over a finite grid of multivariate
polynomials and an arbitrary finite collection of their consecutive Hasse
derivatives (in a coordinate-wise sense). To that end, we make use of the tool
from Gr\"obner basis theory known as footprint. Then we establish and prove
extensions in this context of a family of well-known results in algebra and
combinatorics. These include Alon's combinatorial Nullstellensatz, existence
and uniqueness of Hermite interpolating polynomials over a grid, estimations on
the parameters of evaluation codes with consecutive derivatives, and bounds on
the number of zeros of a polynomial by DeMillo and Lipton, Schwartz, Zippel,
and Alon and F\"uredi. As an alternative, we also extend the Schwartz-Zippel
bound to weighted multiplicities and discuss its connection with our extension
of the footprint bound.
