Combinatorics seminar
Speaker: Dr. Prasant Singh (IIT Jammu)
Host: Sudhir R. Ghorpade
Title: Hyperplane Sections of Determinantal Varieties of Symmetric Matrices over Finite Fields
Time, day and date: 4:00:00 PM, Tuesday, May 27
Venue: Ramanujan Hall
Abstract: Let X = (Xij ) be a m × m generic symmetric matrix whose entries are independent indeterminates over a field F. The symmetric determinantal variety St = St(m) is given by vanishing of (t + 1) × (t + 1) minors of X. This variety is defined over any finite field Fq and has many Fq-rational points. This makes it a useful object from the point of view of applications to coding theory. An explicit formula for the number of Fq-rational points St was determined by Carlitz (1954) in a special case and by MacWilliams (1969) in the general case. Partly from the viewpoint of applications, one is also interested in the following questions:

(i) What are the possible values of |St ∩ H(Fq)|, where H is a Fq-rational hyperplane in the projective space P ( m+1/2 )−1?

(ii) What is the maximum possible value of |St ∩ H(Fq)|, where H varies over the hyperplanes as in (i) above?

In this talk, we will address both the problems above and answer them in the cases when the Fq is of odd characteristic case. This is a joint work with Peter Beelen and Trygve Johnsen.

Link to the abstract: https://drive.google.com/open?id=11m-HFafFkmCOTv6oyozxsL2em81TKJYn