Hybrid.

Contact: Pieter Moree (moree@mpim-bonn.mpg.de)

We discuss two questions related to counting $n\times n$-matrices with integer elements of size at most $H$ which satisfy various arithmetic conditions such as:

A. Multiplicative dependence (due to non-commutativity there are two natural and equally interesting definitions), which in turn leads to counting such matrices with a given characteristic polynomial.

B. Avoidance of being a square of another matrix modulo a prime p, which leads us to various questions on character sums with determinants (so far only in dimension 2).

A is a joint project with Alina Ostafe, B is a joint project with Etienne Fouvry, both are in progress and both lead us to counting solutions to various Diophantine equations associated with determinants.

For these we need upper bounds which seems to be beyond the capabilities of the determinant method, so we use different approaches.

Several open problems will be discussed as well.

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