Polynomials in finite geometries and combinatorics

Abstract

Abstract It is illustrated how elementary properties of polynomials can be used to attack extremal problems in finite and euclidean geometry, and in combinatorics. Also a new result, related to the problem of neighbourly cylinders is presented. Introduction In this paper I will present a number of problems and for the most part recent results in combinatorics in general and finite geometry in particular. The property these problems share is the fact that they can all be attacked using some kind of polynomial trick. Unfortunately I am not able to give a characterization of the kind of problem that can be solved with these methods, but I am convinced that seeing how the polynomials work in a number of cases should give an impression of the type of problems that might be attacked. The starting point is usually a combinatorial problem of the following form: Given a set of points (or vectors, or sets) that satisfy some property, we want to say something about the size or the structure of this set. The approach is then to associate to this set a polynomial, or a collection of polynomials, and use properties of polynomials to obtain information on the size or structure of the set. The setup of this paper is roughly as follows. We consider a particular property of polynomials and give examples where this property can be used to attack the problem.