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06-11-2024
The Center of Mathematics and Applications (NOVA Math), promote the Seminar of Algebra and Logic with the title: “Zeros of homogeneous polynomials, linear sections of Veroneseans, and projective Reed-Muller codes”. Sudhir R. Ghorpade (Indian Institute of Technology Bombay) is the speaker.
Abstract: Let F be a nite eld 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 eld 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 armative in the next case of r = 2 by Boguslavsky (1997). Then about two decades later, it was shown that the 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 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.
Monday, November 11, 14h00 to 15h00.
