Abstract :

The Nullstellensatz or the Zero-Point Theorem is a result of fundamental importance proved by David Hilbert in 1893. It holds when the base field is algebraically closed. In this talk we will first discuss some Nullstellensatz-like results when the base field is finite, and outline the proofs. Next, we will discuss a combinatorial approach to determining or estimating the number of common zeros of a system of multivariate polynomials with coefficients in a finite field. Here we will outline a remarkable result of Heijnen and Pellikaan about the maximum number of zeros that a given umber of linearly independent multivariate polynomials of a given degree can have over a finite field. A projective analogue of this result about multivariate homogeneous polynomials has been open for quite some time, although there has been considerable progress in the last two decades, and especially in the last few years. We will outline some results and conjectures here, including a joint work with Peter Beelen and Mrinmoy Datta.