turkmath.org

We will consider the following question: Given a system of a fixed number of linearly independent homogeneous polynomial equations of a fixed degree with coefficients in a finite field F, what is the maximum number of common solutions they can have in the corresponding protective space over F? The case of a single homogeneous polynomial (i.e., 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, connections to coding theory or to the problem of counting points of sections of Veronese varieties by linear subvarieties of a fixed dimension will also be outlined. Parts of this talk involve joint works with Mrinmoy Datta and with Peter Beelen and Mrinmoy Datta.