Let F be a finite field with q elements. A (projective) algebraic set over F is the set of common zeros in the projective m-space over F of a bunch of homogeneous polynomials in m+1 variables with coefficients in F. Fix positive integers r, m and d with d < q.  We consider the following question:

What is the maximum number of points in an algebraic set in the projective m-s[space over F given by the vanishing of r linearly independent homogeneous polynomials of degree d with coefficients in F?

The case of a single homogeneous polynomial (or in geometric terms, a projective hypersurface) corresponds to a classical inequality proved by Serre in 1989. For the general case, an elaborate conjecture was made by Tsfasman and Boguslavsky, which was open for almost two decades. Recently significant progress in this direction has been made, and it is shown that while the Tsfasman-Boguslavsky Conjecture is true in certain cases, it can be false in general. Some new conjectures have also been proposed. We will give a motivated outline of these developments. If there is time and interest, we will also explain the close connections of these questions to the problem of counting points of sections of Veronese varieties by linear subvarieties of a fixed dimension, and also to coding theory.

This talk is mainly based on joint works with Mrinmoy Datta and with Peter Beelen and Mrinmoy Datta.