Scalar $q$-subresultants and Dickson matrices

Abstract

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 $f$.