Matrices, Polynomials, and Recurrences over Finite Fields

Abstract: Let F be a finite field. We will discuss a variety of questions concerning the enumeration of certain classes of matrices with entries F, polynomials with coefficients in F, and homogeneous linear recurrence sequences of elements of F. Here is a sample of the questions, in their simplest form, that we will consider.

1. What is the number of nonsingular matrices of a given size whose order is maximum in the corresponding general linear group?
2. What is the probability that two polynomials of positive degrees are relatively prime?
3. What is the number of primitive linear recurrence sequences of a given order?
4. What is the number of Hankel matrices of a given rank?

We will attempt to outline some results and conjectures concerning these questions and their extensions and generalisations. Connection of these with combinatorics, cryptography, and algebra may also be indicated. Parts of this talk are joint works with Sartaj Ul Hasan and Meena Kumari, with Samrith Ram, and with Mario Garcia Armas and Samrith Ram.