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arXiv 1309.3530math.NTmath.CO
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有限域上一类置换三项式的判定

Determination of a Type of Permutation Trinomials over Finite Fields

Xiang-dong Hou

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28 pages
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摘要：

令$f=a{\tt x} +b{\tt x}^q+{\tt x}^{2q-1}\in\Bbb F_q[{\tt x}]$。我们发现$a$和$b$的显式条件是必要的和充分的$f$是一个置换多项式的$\Bbb F_{q^2}$。这个结果使我们能够解决一个相关的问题。令$g_{n,q}\in\Bbb F_p[{\tt x}]$（$n\ge 0$，$p=\text{char}\,\Bbb F_q$）为由函数方程$\sum_{c\in\Bbb F_q}({\tt x}+c)^n=g_{n,q}({\tt x}^q-{\tt x})$定义的多项式。我们确定所有的$n$的形式$n=q^α-q^β-1$，$α>β\ge 0$，其中$g_{n,q}$是一个置换多项式的$\Bbb F_{q^2}$。

英文摘要：

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 $g_{n,q}\in\Bbb F_p[{\tt x}]$ ($n\ge 0$, $p=\text{char}\,\Bbb F_q$) 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 $n=q^α-q^β-1$, $α>β\ge 0$, for which $g_{n,q}$ is a permutation polynomial of $\Bbb F_{q^2}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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