Description
Commutative algebra seminar
Thursday 16 August, 2.00-3.30 pm
Venue: Room 215
Speaker: Sudhir R Ghorpade
Title:
A Finite Field Nullstellensatz and the Number of Zeros of Polynomials over
Finite Fields.
Abstract:
In this series of two talks, we will begin by discussing 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 number 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 recent joint work with Peter Beelen and Mrinmoy Datta.