Scalar $q$-subresultants and Dickson matrices

TL;DR

This paper introduces $q$-analogues of scalar subresultants and demonstrates their application in determining the rank of linear transformations over finite fields, with a focus on Dickson matrices.

Contribution

It develops $q$-subresultants and applies them to analyze the rank of $ ext{F}_q$-linear transformations via Dickson matrices, extending classical results.

Findings

$q$-subresultants effectively determine the rank of linear transformations.

Minors of Dickson matrices are linked to the rank of the associated transformation.

The approach generalizes existing methods for rank determination over finite fields.

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 $f$.