Towards the classification of scattered binomials

Abstract: Let ( q ) be a prime power and ( n ) an integer. An ( \mathbb{F}_q )-linearized polynomial ( f ) is said to be scattered if it satisfies the condition that for all ( x, y \in \mathbb{F}_q^n \setminus { 0 } ), whenever ( \frac{f(x)}{x} = \frac{f(y)}{y} ), it follows that ( \frac{x}{y} \in \mathbb{F}_q ). In this paper, we focus on scattered binomials. Two families of scattered binomials are currently known: the one from Lunardon and Polverino (LP), given by $f(x) = \delta x^{q^s} + x^{q^{n-s}},$ and the one from Csajb\'ok, Marino, Polverino, and Zanella (CMPZ), given by $f(x) = \delta x^{q^s} + x^{q^{s + n/2}},$ where ( n = 6 ) or ( n = 8 ). Using algebraic varieties as a tool, we prove some necessary conditions for a binomial to be scattered. As a corollary, we obtain that when ( q ) is sufficiently large and ( n ) is prime, a binomial is scattered if and only if it is of the form (LP). Moreover we obtain a complete classification of scattered binomial in $\Fn$ when $n\leq8$ and $q$ is large enough.