标题： 朝向散落二项式的分类 显示英文标题

标题： Towards the classification of scattered binomials

摘要： 设\( q \)为一个素数幂，\( n \)为一个整数。 一个\( \mathbb{F}_q \)-线性化多项式\( f \)被称为分散的，如果满足以下条件：对于所有\( x, y \in \mathbb{F}_q^n \setminus \{ 0 \} \)，当\( \frac{f(x)}{x} = \frac{f(y)}{y} \)时，可以得出\( \frac{x}{y} \in \mathbb{F}_q \)。 在本文中，我们关注分散的二项式。 已知两种散列二项式族：Lunardon和Polverino（LP）给出的$f(x) = \delta x^{q^s} + x^{q^{n-s}},$，以及Csajbók、Marino、Polverino和Zanella（CMPZ）给出的$f(x) = \delta x^{q^s} + x^{q^{s + n/2}},$，其中\( n = 6 \)或\( n = 8 \)。 使用代数簇作为工具，我们证明了二项式为散列的一些必要条件。 作为推论，当\( q \)足够大且\( n \)为素数时，二项式是散列的当且仅当它是（LP）的形式。 此外，当$n\leq8$和$q$足够大时，我们得到了$\Fn$中分散二项式的完整分类。 显示英文摘要

摘要： 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.