CSUN Algebra, Number Theory, and Discrete Mathematics Seminar

Let $F$ be a finite field. We will discuss a variety of questions concerning the enumeration of certain classes of matrices with entries in $F$, 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 Hankel matrices of a given rank?
4. What is the number of primitive linear recurrence sequences of a given order?

We will attempt to outline some results, both old and new, and some 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 a joint work with Sartaj Ul Hasan and Meena Kumari, with Samrith Ram, and with Mario Garcia Armas and Samrith Ram.