  We study multivariate polynomials over `structured' grids. We begin by
proposing an interpretation as to what it means for a finite subset of a field
to be structured; we do so by means of a numerical parameter, the nullity. We
then extend several results--notably, the Combinatorial Nullstellensatz and the
Coefficient Theorem--to polynomials over structured grids. The main point is
that the structure of a grid allows the degree constraints on polynomials to be
relaxed.
