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 $f$.

References in corpus (2)

• A characterization of linearized polynomials with maximum kernel
• A Characterization of the Number of Roots of Linearized and Projective Polynomials in the Field of Coefficients

Cited by in corpus (6)

• Linear sets and MRD-codes arising from a class of scattered linearized polynomials
• Rank-metric codes over arbitrary Galois extensions and rank analogues of Reed-Muller codes
• A new family of maximum scattered linear sets in $\mathrm{PG}(1,q^6)$
• On a conjecture about maximum scattered subspaces of ${\mathbb{F}}_{q^6}\times {\mathbb{F}}_{q^6}$
• On certain linearized polynomials with high degree and kernel of small dimension
• Decoding a class of maximum Hermitian rank metric codes