KTH/SU Mathematics Colloquium
07-05-09
Aart Blokhuis, Eindhoven University of Technology
Polynomials in (finite) Geometry and
Combinatoric
It is illustrated how elementary properties of polynomials can be
used to attack extremal problems in finite and euclidean geometry
and in combinatorics. The problems involved are of the following
type: Given a set of points (or vectors, or objects) that
satisfy some (combinatorial) property, we want to say something
about the size
or the structure of this set. We associate to this set a
polynomial, or a collection of polynomials, and use properties
of polynomials to obtain information about the size or structure
of our set.
One of the more spectacular examples is a result of Frankl
and Wilson from 1981, that led to a counterexample to Borsuk's
Conjecture by Kahn and Kalai in 1993.
In finite geometry the so called R\'edei Polynomial has been a
very useful tool to obtain results on small blocking sets, and
maximal arcs. We'll explain how and why it works, and discuss
some recent developments.