有限域上具有交换图表性质的多项式

Polynomials over Finite Fields with the Commuting Diagram Properties

本文给出了有限域Ｆｑ上满足条件ｆ（ｇ（ｘ））≡ｈ（ｆ（ｘ））的多项式ｆ（ｘ）的通解表示公式及在ｄｅｇｆ＜ｑ条件下的ｆ（ｘ）的个数计算公式，此地ｇ（ｘ）和ｈ（ｘ）是Ｆｑ上给定的两个多项式，其中之一是置换多项式．这一结果推广了文献［３］的主要结果，而在ｇ（ｘ）和ｈ（ｘ）均为线性多项式的特殊情况，则分别推广了文献［１］和［２］中的主要结果．

We give explicit expressions for the general solutions of the polynomials f(x) over finite field F q satisfying the condition f(g(x))≡h(f(x)) and give the counting formula for the number of such f(x) with deg f<q , where g(x) and h(x) are two given polynomials over F q and one of them is a permutation polynomial. This generalizes the main results in and also generalizes the main results in and in the special cases where g(x) and h(x) are both linear polynomials.