Applications of Polynomials Over Finite Fields
Péter Sziklai · REAL-d (Hungarian Academy of Sciences) · 2013
A most efficient way of investigating combinatorially defined point sets in spaces over finite fields is associating polynomials to them. This technique was first used by Rédei, Jamison, Lovász, Schrijver and Bruen, then, followed by several people, became a standard method; nowadays, the contours of a growing theory can be seen already. The polynomials we use should reflect the combinatorial properties of the point set, then we have to be equipped with enough means to handle our polynomials and get an algebraic description about them; finally, we have to translate the information gained back to the original, geometric language. The first investigations in this field examined the coefficients of the poly-nomials, and this idea proved to be very efficient. Then the derivatives of the polynomials came into the play and solving (differential) equations over finite fields; a third branch of results considered the polynomials as algebraic curves. The idea of associating algebraic curves to point sets goes back to Segre, recently a bunch of new applications have shown the strength of this