New scattered linearized quadrinomials

Abstract

Let $1<t<n$ be integers, where t is a divisor of n. An $\mathrm{R-}{q}^t$-partially scattered polynomial is a q-polynomial f in $F_{q^n}\left[X\right]$ that satisfies the condition that for all $x,y\in F_{q^n}^{⁎}$ such that $x/y\in F_{q^t}$, if $f\left(x\right)/x=f\left(y\right)/y$, then $x/y\in F_q$; f is called scattered if this implication holds for all $x,y\in F_{q^n}^{⁎}$. Two polynomials in $F_{q^n}\left[X\right]$ 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 $F_{q^n}\left[X\right]$ are known: (i) monomials of pseudoregulus type, $\left(ii\right)$ binomials of Lunardon-Polverino type, and $\left(iii\right)$ a family of quadrinomials defined in [1], [10] and extended in [8], [13]. In this paper we prove that the polynomial ${\phi }_{m,q^J}=X^{q^{J(t-1)}}+X^{q^{J(2t-1)}}+m(X^{q^J}-X^{q^{J(t+1)}})\in F_{q^{2t}}\left[X\right]$, q odd, $t\ge 3$ is $\mathrm{R-}{q}^t$-partially scattered for every value of $m\in F_{q^t}^{⁎}$ and J coprime with 2t. Moreover, for every $t>4$ and $q>5$ there exist values of m for which ${\phi }_{m,q}$ is scattered and new with respect to the polynomials mentioned in (i), $\left(ii\right)$ and $\left(iii\right)$ above. The related linear sets are of $\Gamma L$-class at least two.