PolSys seminar
Effective Hilbert's Nullstellensatz and Finite Fields
Igor Shparlinski (School of Mathematics and Statistics, The University of New South Wales, Sydney,
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 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 also present 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 finish with an outlineof some open problems.
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