Towards the classification of scattered binomials

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.