Determination of a type of permutation trinomials over finite fields

Xiang-dong Hou¹

¹ Department of Mathematics and Statistics University of South Florida Tampa, FL 33620, U.S.A.

Acta Arithmetica, Tome 166 (2014) no. 3, pp. 253-278

Résumé

Let $f=a{\tt x} +b{\tt x}^q+{\tt x}^{2q-1}\in\Bbb F_q[{\tt x}]$. We find explicit conditions on $a$ and $b$ that are necessary and sufficient for $f$ to be a permutation polynomial of $\Bbb F_{q^2}$. This result allows us to solve a related problem: Let $g_{n,q}\in\Bbb F_p[{\tt x}]$ ($n\ge 0$, $p={\rm char}\,\Bbb F_q$) be the polynomial defined by the functional equation $\sum_{c\in\Bbb F_q}({\tt x}+c)^n=g_{n,q}({\tt x}^q-{\tt x})$. We determine all $n$ of the form $n=q^\alpha-q^\beta-1$, $\alpha>\beta\ge 0$, for which $g_{n,q}$ is a permutation polynomial of $\Bbb F_{q^2}$.

Détail

DOI : 10.4064/aa166-3-3

Keywords: q bbb explicit conditions necessary sufficient permutation polynomial bbb result allows solve related problem bbb char bbb polynomial defined functional equation sum bbb q determine form alpha q beta alpha beta which permutation polynomial bbb