An optimal bound on the number of determinants generated by a subset in ${\mathbb{F}}_q^d$

In this short note, we prove that for $\mathcal{E}\subset {\mathbb{F}}_q^d$ with $\mathrm{\mid }\mathcal{E}\mathrm{\mid }\ge q^{d-1}+O(q^2)$ then the set of determinants generated by $\mathcal{E}$ is ${\mathbb{F}}_q\mathrm{.}$ This result is nearly optimal and generalizes the previous results of Vinh (2013) and Iosevich, Rudnev and Zhai (2015).