Commutative Algebra Seminar

Title

Number of zeros of polynomials over finite fields

Abstract

It is elementary and well-known that a polynomial in one variable of degree d with coefficients in a field F has at most d zeros in F. An analogue of this for homogeneous polynomials is that a homogeneous polynomial in two variables of degree d with coefficients in F has at most d non-proportional zeros in F^2 (excluding the origin), or in other words, d projective zeros. We note that d is a “good” bound in the sense that if F has at least d elements, then there are polynomials of degree d that attain this bound.

When F is a finite field with q elements, it makes sense to ask similar questions for the number of common zeros of systems of multivariable polynomials of a given degree. A remarkable answer for the general case of r linearly independent polynomials of degree d in m variables over the field with q elements was given by Heijnen and Pellikaan in 1998. The analogous problem for systems of multivariable homogeneous polynomials turned out to be more challenging, and answers were known only in the case of a single polynomial or a system of two linearly independent polynomials, thanks to the work of Serre (1991) and Boguslavsky (1997). For the general case, there was an elaborate conjecture by Boguslavsky and Tsfasman that remained open for almost two decades. In this talk we will outline some recent progress on this conjecture as well as some newer developments.

This is based on a joint work with M. Datta, and also with P. Beelen and M. Datta