Distribution of determinant of matrices with restricted entries over finite fields

Abstract

For a prime power $q$, we study the distribution of determinent of matrices with restricted entries over a finite field $\mathbbm{F}_q$ of $q$ elements. More precisely, let $N_d(\mathcal{A};t)$ be the number of $d\times d$ matrices with entries in $\mathcal{A}$ having determinant $t$. We show that \[ N_d (\mathcal{A}; t) = (1 + o (1)) \frac{|\mathcal{A}|^{d^2}}{q}, \] if $|\mathcal{A}|=\omega (q^{\frac{d}{2d-1}})$, $d⩾4$. When $q$ is a prime and $\mathcal{A}$ is a symmetric interval $[-H,H]$, we get the same result for $d⩾3$. This improves a result of Ahmadi and Shparlinski (2007).

Journal of Combinatorics and Number Theory (to appear)