  Alon's combinatorial Nullstellensatz, and in particular the resulting
nonvanishing criterion is one of the most powerful algebraic tools in
combinatorics, with many important applications. In this paper we extend the
nonvanishing theorem in two directions. We prove a version allowing multiple
points. Also, we establish a variant which is valid over arbitrary commutative
rings, not merely over subrings of fields. As an application, we prove
extensions of the theorem of Alon and F\"uredi on hyperplane coverings of
discrete cubes.
