Counting $2\times2$ integer matrices with fixed trace and determinant

TL;DR

This paper estimates the number of 2x2 integer matrices with a given characteristic polynomial, providing precise counts with main and error terms by employing bounds on Weyl sums for quadratic roots.

Contribution

It introduces a novel counting method for matrices with fixed characteristic polynomial using bounds on Weyl sums, advancing the understanding of matrix enumeration under algebraic constraints.

Findings

Derived an asymptotic formula for the count of matrices

Established bounds for sums of Weyl sums for quadratic roots

Provided error estimates for the counting problem

Abstract

We count with a smooth weight the number of $2 \times 2$ integer matrices with a fixed characteristic polynomial with a main term and an error term using bounds for sums of Weyl sums for quadratic roots.