Event Detail

Event Type: 
Number Theory Seminar
Thursday, May 26, 2022 - 16:00 to 16:50
Zoom only, contact Clayton Petsche for details

Speaker Info

University of New South Wales

We consider the set M_n(Z; H) of n x n matrices with integer elements of size at most H and obtain upper and lower bounds on the number of s-tuples of matrices from M_n(Z; H), satisfying various multiplicative relations, including multiplicative dependence, commutativity and bounded generation of a subgroup of GL_n(Q). 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. As a part of our method, we 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.

Joint work with Igor Shparlinski.