Leírás
Let $f$ and $g$ be two $q$-polynomials over $GF(q^n)$.
Following the ideas of Ore and Li we study $q$-analogous of scalar subresultants and show how these results can be applied to determine the rank of the common kernel of $f$ and $g$. If we put $g(x)=x^{q^n}-x$, then we obtain conditions on the rank of $f$. As an application we show how certain minors of the Dickson matrix $D(f)$, a $q$-circulant matrix associated with $f$, determine the rank of $D(f)$ and hence the rank of $f$. To determine the rank of $q$-polynomials from their coefficients has been crucial in recent studies of MRD-codes.
Introduce the notion $D_{m(f)}$ to denote the $(n-m)\times (n-m)$ matrix obtained from $D(f)$ after removing its first m columns and last m rows. Then our result is as follows.
Theorem:
$dim_q (ker f)=k$ if and only if $det D_0(f)= det D_1(f)=...=det D_{k-1}(f)=0$ and $det D_k(f)\neq 0$.