Hilbert Functions and the Finite Degree Zariski Closure in Finite Field Combinatorial Geometry

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.