University of Sydney Algebra Seminar
Alina Ostafe
Friday 6 October, 12-1pm, Place: Carslaw 173
On some arithmetic statistics for integer matrices
We consider the set M_n(Z; H) of n by n matrices with integer elements of size at most H and obtain a new upper bound on the number of matrices from M_n(Z; H) with a given characteristic polynomial f \in Z[X], which is uniform with respect to f. This complements the asymptotic formula of A. Eskin, S. Mozes and N. Shah (1996) in which f has to be fixed and irreducible. We use our result to address various other questions of arithmetic statistics for matrices from M_n(Z; H), e.g. satisfying certain multiplicative relations. Some of these problems generalise those studied in the scalar case n=1 by F. Pappalardi, M. Sha, I. E. Shparlinski and C. L. Stewart (2018) with an obvious distinction due to the non-commutativity of matrices.
Joint works with Kamil Bulinski and Igor Shparlinski.