Undergraduate Seminar: "A New Frobenius Template in a Matrix Ring"

Abstract: The classical Frobenius problem is to find the largest integer that cannot be written as a linear combination of a given set of positive, co-prime integers using non-negative integer coefficients. Prior research has been done to generalize this classical Frobenius problem from a topic in number theory to a topic in ring theory; the Frobenius problem has been generalized from the ring of integers to the ring of Gaussian integers as well as to the rings Z[ √ m], where m is a square-free positive integer. 

In this presentation, I will introduce a new generalization of the classical Frobenius problem to the commutative ring of 2 × 2 upper triangular matrices with constant diagonal. We will present answers to two research questions: for which lists of matrices is the Frobenius set non-empty, and for each list such that the Frobenius set is non-empty, what are the matrices in the Frobenius set? For the lists of two matrices, I will explain the construction of every matrix in the Frobenius set.

Host: Adeli Hutton