  The polynomial method has been used recently to obtain many striking results
in combinatorial geometry. In this paper, we use affine Hilbert functions to
obtain an estimation theorem in finite field geometry. The most natural way to
state the theorem is via a sort of bounded degree Zariski closure operation:
given a set, we consider all polynomials of some bounded degree vanishing on
that set, and then the common zeros of these polynomials. For example, the
degree d closure of d + 1 points on a line will contain the whole line, as any
polynomial of degree at most d vanishing on the d + 1 points must vanish on the
line. Our result is a bound on the size of a finite degree closure of a given
set. Finally, we adapt our use of Hilbert functions to the method of
multiplicities.
