Type: Seminar

Distribution: World

Expiry: 1 Jan 2024

CalTitle1: Algebra Seminar: On some arithmetic statistics for integer matrices

Auth: alexs@desktop-h8gjltm.staff.wireless.sydney.edu.au (ashe8718) in SMS-SAML

Alina Ostafe (UNSW) is speaking in the Algebra Seminar this week. We will go out for lunch after the talk. When: Friday, 6 October, 12-1pm Where: Carslaw 173 Title: On some arithmetic statistics for integer matrices Abstract: 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