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 $t$ 整除 $n$。若 $\mathbb F_q$-线性化多项式 $f\in\mathbb F_{q^n}[X]$ 满足：对所有使得 $x/y\in\mathbb F_{q^t}$ 的 $x,y\in\mathbb F_{q^n}^*$，若 $f(x)/x=f(y)/y$，则 $x/y\in\mathbb F_q$，则称 $f$ 为 R-$q^t$-部分散射多项式；若该蕴涵关系对所有 $x,y\in\mathbb F_{q^n}^*$ 成立，则称 $f$ 为散射的。若两个多项式 $f,g\in\mathbb F_{q^n}[X]$ 在群 $\Gamma L(2,q^n)$ 作用下其图位于同一轨道中，则称它们等价。当 $n>8$ 时，$\mathbb F_{q^n}[X]$ 中仅已知三类散射多项式：(i) 伪正则型单项式，(ii) Lunardon-Polverino 型二项式，以及 (iii) 文献 [1,10] 中定义并在 [8,13] 中推广的四项式族。本文证明：当 $q$ 为奇数、$t\ge3$ 时，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-$q^t$-部分散射的。进一步地，对任意 $t>4$ 与 $q>5$，存在某些 $m$ 值使得 $\varphi_{m,q}$ 是散射的，且相对于上述 (i)、(ii)、(iii) 类多项式是新的。相关的线性集至少具有 $\Gamma L$-类数二。