Past Event: Oden Institute Seminar

Zeros of homogeneous polynomials over finite fields with applications to linear error correcting codes

Dr. Sudhir Ghorpade, Indian Institute of Technology Bombay, Dept. of Mathematics

Abstract

Let F be a finite field with q elements, 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. Note also that the condition on linear independence forces that r  is at most (m+d/d).

This question was first raised by M. Tsfasman in the case of a single homogeneous polynomial, that is, 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 the 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 linear error correcting codes, or more specifically, to the so called projective Reed-Muller codes.

In this talk, we will outline these developments and explain the connection with coding theory. An attempt will be made to keep the prerequisites at a minimum.

Biography

Sudhir R. Ghorpade received the B.Sc., M.Sc. and Ph.D. degrees in Mathematics from the University of Bombay, Indian Institute of Technology (IIT) Bombay, and Purdue University, West Lafayette in 1982, 1984, and 1989 respectively. Since December 1989, he has been on the faculty of the IIT, Bombay,where he is currently a Professor. He has held short-term visiting positions at the Institut de Mathématiques de Luminy, Marseille, France, Tata Institute of Fundamental Research, Mumbai, India, Université de la Mediterrannée, Aix-Marseille, France, Christian-Alberchts-Universität zu Kiel, Germany, Purdue University, West Lafayette, USA, and the University of Tennessee, Knoxville, USA, and the Technical University of Denmark, Kgs. Lyngby. His research interests include algebraic geometry, coding theory, combinatorics, and commutative algebra. He is a Fellow of the National Academy of Sciences, India since October 2010 and is on the editorial board of the International Journal of Information and Coding Theory.