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 [9] and extended in [7,12]. 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。一个R-$q^t$-部分散项多项式是$\mathbb F_q$线性化多项式$f\in\mathbb F_{q^n}[X]$，它满足条件：对所有满足$x/y\in\mathbb F_{q^t}$的$x,y\in\mathbb F_{q^n}^*$，若$f(x)/x=f(y)/y$，则$x/y\in\mathbb F_q$；若这一蕴含关系对所有$x,y\in\mathbb F_{q^n}^*$成立，则称$f$为散项多项式。$\mathbb F_{q^n}[X]$中的两个多项式若其图像在群$\Gamma L(2,q^n)$作用下属于同一轨道，则称它们等价。当$n>8$时，$\mathbb F_{q^n}[X]$中已知的散项多项式仅有三族：$(i)$拟正则型单项式，$(ii)$Lunardon-Polverino型二项式，$(iii)$文献[9]定义并在[7,12]中扩展的一族四次多项式。本文证明：对于$q$为奇数、$t\ge3$，多项式$\varphi_{m,q^J}=X^{q^{J(t-1)}}+X^{q^{J(2t-1)}}+m(X^{q^J}-X^{q^{J(t+1)}})\in\mathbb F_{q^{2t}}[X]$（其中$m\in\mathbb F_{q^t}^*$且$J$与$2t$互素）对所有参数均为R-$q^t$-部分散项。进一步，对每个$t>4$且$q>5$，存在某些$m$值使得$\varphi_{m,q}$为散项多项式，且相对于上述$(i)$、$(ii)$和$(iii)$中的多项式是新的。相关的线性集具有至少为2的$\Gamma L$-类。