A condition for scattered linearized polynomials involving Dickson matrices

Abstract

A linearized polynomial over ${\mathbb{F}}_{q^n}$ is called scattered when for any $t,x\in {\mathbb{F}}_{q^n}$, the condition $xf(t)-tf(x)=0$ holds if and only if $x$ and $t$ are ${\mathbb{F}}_q$-linearly dependent. General conditions for linearized polynomials over ${\mathbb{F}}_{q^n}$ to be scattered can be deduced from the recent results in [4,7,15,19]. Some of them are based on the Dickson matrix associated with a linearized polynomial. Here a new condition involving Dickson matrices is stated. This condition is then applied to the Lunardon-Polverino binomial $x^{q^s}+\delta x^{q^{n-s}}$, allowing to prove that for any $n$ and $s$, if ${\mathbb{N}}_{q^n/q}(\delta )=1$, then the binomial is not scattered. Also, a necessary and sufficient condition for $x^{q^s}+b{x}^{q^{2s}}$ to be scattered is shown which is stated in terms of a special plane algebraic curve.