Permutation polynomials of the form \(X + \gamma \mathrm{Tr}(X^k)\) (Q2829805)

Let \(\mathbb F_q\) denote the finite field with \(q\) elements. A polynomial \(f\in\mathbb F_q[X]\) is called a \textit{permutation polynomial} (PP) of \(\mathbb F_q\) if it induces a permutation of \(\mathbb F_q\). The paper under review focuses on PPs of the form \(X+\gamma\text{Tr}_{q^n/q}(X^k)\), where \(n\) and \(k\) are positive integers, \(\gamma\in\mathbb F_{q^n}^*\) and \(\text{Tr}_{q^n/q}(X)=X+X^q+\cdots+X^{q^{n-2}}+X^{q^{n-1}}\).NEWLINENEWLINE The main contribution of the paper is Theorem 1 which gives nine classes of PPs of this form: NEWLINE{\parindent=6mm NEWLINE\begin{itemize}\item[(a)] \(n=2,\quad q\equiv\pm 1\pmod 6,\quad \gamma=-1/3\), \(k=2q-1\), NEWLINE\item[(b)] \(n=2,\quad q\equiv 5\pmod 6, \quad \gamma^3=-1/27\), \(k=2q-1\), NEWLINE\item[(c)] \(n=2,\quad q\equiv 1\pmod 3, \quad \gamma=1\), \(k=(q^2+q+1)/3\), NEWLINE\item[(d)] \(n=2,\quad q\equiv 1\pmod 4, \quad (2\gamma)^{(q+1)/2}=1\), \(k=(q+1)^2/4\), NEWLINE\item[(e)] \(n=2,\quad q=Q^2\), \(Q\) odd, \(\gamma=-1\), \(k=Q^3-Q+1\), NEWLINE\item[(f)] \(n=2,\quad q=Q^2\), \(Q\) odd, \(\gamma=-1\), \(k=Q^3+Q^2-Q\), NEWLINE\item[(g)] \(n=3,\quad q\) odd, \(\quad \gamma=1\), \(k=(q^2+1)/2\), NEWLINE\item[(h)] \(n=3,\quad q\) odd, \(\quad \gamma=-1/2\), \(k=q^2-q+1\), NEWLINE\item[(i)] \(n=2lr, \quad {} \quad\gamma^{q^{2l}-1}=-1\), \(k=q^l+1\). NEWLINENEWLINE\end{itemize}} NEWLINESeveral general criteria are given for \(X+\gamma\text{Tr}_{q^n/q}(X^k)\) to be a PP of \(\mathbb F_{q^n}\). The proofs of the constructions of the above nine classes of PPs use a variety of methods and vary in the level of difficulty. Most notably, the proofs of (e) and (f) use a ``multivariable'' method that can be traced back to H. Dobbertin. The authors suggest, based on their computer search, that the above PPs account for the bulk of all PPs of the form \(X+\gamma\text{Tr}_{q^n/q}(X^k)\) modulo some simple cases.NEWLINENEWLINEFor the entire collection see [Zbl 1345.11003].