Scalar $q$-subresultants and Dickson matrices

Following the ideas of Ore and Li we study $q$-analogues of scalar subresultants and show how these results can be applied to determine the rank of an ${\mathbb{F}}_q$-linear transformation $f$ of ${\mathbb{F}}_{q^n}$. As an application we show how certain minors of the Dickson matrix $D(f)$, associated with $f$, determine the rank of $D(f)$ and hence the rank of $f$.