Author

Tim Browning

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Preprint Oct 2026

Counting matrices with given characteristic polynomial

Let $f$ be a monic polynomial of degree $n$ with integer coefficients. We use Fourier analysis to prove a new upper bound on the number of $n\times n$ matrices with characteristic polynomial $f$ and integer entries in $[-T,T]$, as $T\to \infty$. The implied constant in our bound only depends on $n$ and is otherwise uni...

Tim Browning, Fabian Gundlach, W. Sawin · 0 citations

Preprint Aug 2026

The divisor function for matrices

We introduce a matrix divisor function $\tau_n(T,M)$, counting factorisations $AB=M$ for $n\times n$ integer matrices $A,B$ of height at most $T$. For a fixed non-singular $M$, or for the zero matrix $M=O_n$, we prove an asymptotic formula for $\tau_n(T,M)$, as $T\to \infty$, using lattice point counting. We also prove...

Tim Browning, N. Kalinin, Alina Ostafe et al. · 0 citations