Summary A method of using polynomials to describe objects in finite geometries is outlined and the problems where this method has led to a solution are surveyed. These problems concern nuclei, affine blocking sets, maximal arcs and unitals. In the case of nuclei these methods give lower bounds on the number of nuclei to a set of points in PG(n, q), usually dependent on some binomial coefficient not vanishing modulo the characteristic of the field. These lower bounds on nuclei lead directly to lower bounds on affine blocking sets with respect to lines. A short description of how linear polynomials can be used to construct maximal arcs in certain translation planes is included. A proof of the non-existence of maximal arcs in PG(2, q) when q is odd is outlined and some bounds are given as to when a (k, n)-arc can be extended to a maximal arc in PG(2, q). These methods can also be applied to unitals embedded in PG(2, q). One implication of this is that when q is the square of a prime a non-classical unital has a limited number of Baer sublines amongst its secants.

Introduction

The effectiveness of polynomials as a means of studying problems in finite geometries has become increasingly evident in the 1990's, although the first examples seem to date back to R. Jamison [38] in 1977 and A. E. Brouwer and A. A. Bruen and J. C. Fisher described the "Jamison method" as the following: reformulate the problem in terms of points of an affine space and associate suitable polynomials defined over the corresponding finite field; calculate. This is the approach employed in [19] too; in fact the main difference between [38] and [19] is that Jamison viewed the points of an affine space as elements of a finite field. In effect, this has the advantage of reducing the number of variables in the polynomials and allowing one to use simple arguments concerning the degree or the coefficients of a polynomial.