Let F be a finite field and let m, d and r be positive integers. Consider the following
question: What is the maximum number of common zeros over F that a system of r linearly
independent homogeneous polynomials of degree d in m + 1 variables can have?
Because of homogeneity, we will disregard the trivial zero (viz., the origin) and regard two
zeros as equivalent if they are proportional to each other, i.e., if one is obtained from another
upon multiplying all coordinates with a nonzero scalar. In other words, we look for zeros in
the m -dimensional projective space over the field F.
This question was first raised by M. Tsfasman in the case of a single homogeneous
polynomial, i.e., when r = 1. It was then settled by J.-P. Serre (1991). Later Tsfasman
together with M. Boguslavsky formulated a remarkable conjecture in the general case, and
this was shown to hold in the affirmative in the next case of r = 2 by Boguslavsky (1997).
Then about two decades later, it was shown that the conjecture is valid conjecture is valid if
the number of polynomials is at most the number of variables, i.e., r ≤ m + 1, but the
conjecture can be false in general. Newer conjectures were then formulated and although
there has been considerable progress concerning them, the general case is still open.
These questions are intimately related to the study of maximal sections of Veronese varieties
by linear subvarieties of the ambient projective case, and also to the study of the an important
class of linear error correcting codes, called projective Reed-Muller codes.
In this talk, we will outline these developments and explain the above connections.
An attempt will be made to keep the prerequisites at a minimum.