**Tibor Beke**

Hungarian Academy of Sciences, Budapest

Suppose one has an enumerative combinatorial problem that can be evaluated
over the finite fields with , , , ... elements, giving rise to
the sequence of counts , , , ... . When is the associated
generating function

a rational function?

Part of the Weil conjectures (ie the theorem of Grothendieck-Deligne) is
that if one counts the number of common zeroes of a set of polynomials
over bigger and bigger finite fields, then the associated generating
function is rational. We review the cohomological proof and subsequent
extensions: to counting problems that involve first-order quantifiers (due
to Kiefe, Macintyre and others) and field extensions (due to Wan). I
continue with my own work, and mention some open problems.

2005-05-23