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. We will outline these developments and report on some recent progress.

An attempt will be made to keep the prerequisites at a minimum. 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.

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Last updated: 05 Dec 2019