Permutation polynomials of the form $x+γ\mathrm{Tr}(H(x))$

Abstract: Given a polynomial ( H(x) ) over (\mathbb{F}{q^n}), we study permutation polynomials of the form ( x + \gamma \mathrm{Tr}(H(x)) ) over (\mathbb{F}{q^n}). Let [P_H={\gamma\in \mathbb{F}{q^n} : x+\gamma \mathrm{Tr}(H(x))~\text{is a permutation polynomial}}.] We present some properties of the set (P_H), particularly its relationship with linear translators. Moreover, we obtain an effective upper bound for the cardinality of the set (P_H) and show that the upper bound can reach up to $q^n - q^{n - 1}$. Furthermore, we prove that when the cardinality of the set (P_H) reaches this upper bound, the function (\mathrm{Tr}(H(x))) must be an (\mathbb{F}_q)-linear function. Finally, we study two classes of functions $H(x)$ over (\mathbb{F}{q^2}) and determine the corresponding sets $P_H$. The sizes of these sets $P_H$ are all relatively small, even only including the trivial case.