Let $1<t<n$ be integers, where $t$ is a divisor of $n$. An R-$q^t$-partially scattered polynomial is a $\mathbb F_q$-linearized polynomial $f$ in $\mathbb F_{q^n}[X]$ that satisfies the condition that for all $x,y\in\mathbb F_{q^n}^*$ such that $x/y\in\mathbb F_{q^t}$, if $f(x)/x=f(y)/y$, then $x/y\in\mathbb F_q$; $f$ is called scattered if this implication holds for all $x,y\in\mathbb F_{q^n}^*$. Two polynomials in $\mathbb F_{q^n}[X]$ are said to be equivalent if their graphs are in the same orbit under the action of the group $\Gamma L(2,q^n)$. For $n>8$ only three families of scattered polynomials in $\mathbb F_{q^n}[X]$ are known: $(i)$~monomials of pseudoregulus type, $(ii)$~binomials of Lunardon-Polverino type, and $(iii)$~a family of quadrinomials defined in [1,10] and extended in [8,13]. In this paper we prove that the polynomial $\varphi_{m,q^J}=X^{q^{J(t-1)}}+X^{q^{J(2t-1)}}+m(X^{q^J}-X^{q^{J(t+1)}})\in\mathbb F_{q^{2t}}[X]$, $q$ odd, $t\ge3$ is R-$q^t$-partially scattered for every value of $m\in\mathbb F_{q^t}^*$ and $J$ coprime with $2t$. Moreover, for every $t>4$ and $q>5$ there exist values of $m$ for which $\varphi_{m,q}$ is scattered and new with respect to the polynomials mentioned in $(i)$, $(ii)$ and $(iii)$ above. The related linear sets are of $\Gamma L$-class at least two.

翻译：设 $1<t<n$ 为整数，其中 $t$ 是 $n$ 的因子。若 $\mathbb F_q$-线性化多项式 $f\in\mathbb F_{q^n}[X]$ 满足：对所有满足 $x/y\in\mathbb F_{q^t}$ 的 $x,y\in\mathbb F_{q^n}^*$，当 $f(x)/x=f(y)/y$ 时必有 $x/y\in\mathbb F_q$，则称 $f$ 为 R-$q^t$-部分散射多项式；若此蕴含关系对所有 $x,y\in\mathbb F_{q^n}^*$ 均成立，则称 $f$ 为散射多项式。若两个 $\mathbb F_{q^n}[X]$ 中的多项式在群 $\Gamma L(2,q^n)$ 作用下的图像属于同一轨道，则称它们等价。对于 $n>8$，目前仅知三类 $\mathbb F_{q^n}[X]$ 中的散射多项式族：$(i)$~伪正则型单项式，$(ii)$~Lunardon-Polverino 型二项式，以及 $(iii)$~在文献 [1,10] 中定义并在 [8,13] 中扩展的四项式族。本文证明对于奇素数幂 $q$ 及 $t\ge3$，多项式 $\varphi_{m,q^J}=X^{q^{J(t-1)}}+X^{q^{J(2t-1)}}+m(X^{q^J}-X^{q^{J(t+1)}})\in\mathbb F_{q^{2t}}[X]$ 对任意 $m\in\mathbb F_{q^t}^*$ 及与 $2t$ 互素的 $J$ 均为 R-$q^t$-部分散射多项式。进一步地，对任意 $t>4$ 与 $q>5$，均存在 $m$ 的取值使得 $\varphi_{m,q}$ 成为散射多项式，且相对于上述 $(i)$、$(ii)$、$(iii)$ 类多项式具有新颖性。相应的线性集的 $\Gamma L$-类至少为二。