Integer matrices with a given characteristic polynomial and multiplicative dependence of matrices · arXivDesk
2203.03880Mar 8, 2022In this new version, now jointly with Philipp Habegger, we use a very different approach to counting integer matrices with a given characteristic polynomial, which leads to a much stronger estimate. In turn, this also improves several other results
Integer matrices with a given characteristic polynomial and multiplicative dependence of matrices
Philipp Habegger, Alina Ostafe, Igor E. Shparlinski
We consider the set Mn(Z;H) of n×n-matrices with integer elements of size at most H and obtain a new upper bound on the number of matrices from Mn(Z;H)
Nearby in the stack
with a given characteristic polynomial
f∈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. Using this result, among others, we obtain upper and lower bounds on the number of
s
-tuples of matrices from
Mn(Z;H)
, satisfying various multiplicative relations, including multiplicative dependence and bounded generation of a subgroup of
GLn(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. Motivated by these problems, we also prove various properties of the variety of complex matrices with fixed characteristic polynomial, including computing the degree of this variety.