Algebra Seminar: Ostafe -- On some arithmetic statistics for integer matrices

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