A condition for scattered linearized polynomials involving Dickson matrices

TL;DR

This paper introduces a new Dickson matrix-based condition for scattered linearized polynomials over finite fields, applies it to specific binomials, and characterizes when they are scattered.

Contribution

It presents a novel Dickson matrix condition for scattered polynomials and applies it to classify certain binomials as scattered or not.

Findings

If _{q^n/q}(\u03b4)=1, then the Lunardon-Polverino binomial is not scattered.

A necessary and sufficient condition for x^{q^s}+bx^{q^{2s}} to be scattered is derived.

The new condition involves algebraic curves and 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 $x^{q^s}+bx^{q^{2s}}$ to be scattered is shown which is stated in terms of a special plane algebraic curve.