Igor Shparlinski, University of New South Wales
“Effective Hilbert’s Nullstellensatz and Finite Fields”
We give an overview of recent applications of effective versions of Hilbert’s Nullstellensatz to various problems in the theory of finite fields. In particular, we resent some results about the size of the set generated by s-fold products of some rational fractions in a finite field. This result has some algorithmic applications.
We also show that almost all points on algebraic varieties over finite fields avoid Cartesian products of small order groups. This result is a step towards Poonen’s conjecture.
We finish with an outline of some open problems.