Carlitz 定理的一个注记

置换多项式一直是一个热门的研究课题,事实上,研究有限域上的置换多项式相当于研究有限域上的一一映射.所以它在编码密码、组合设计、代数曲线等许多领域有重要的应用.Carlitz曾经对一些置换多项式有一个刻画,证明了如果f(x)是一个系数在F0的多项式满足f(0)=0,f(1)=l,并且对任意a,b ∈Fq有η(f(a)-f(b))=η(a-b),这里η是Fq的乘法群Fq*的二次特征,则存在某个非负整数j使得对任意χ ∈F0,有f(x)=xpj.本文给出了这个结果的推广.

A note on a theorem of Carlitz

The study of permutation polynomials over finite fields has been a hotspot research topic for a long time.In fact,it is equivalent tothe study of one-to-one mapping between finite fields.Therefore,it has many important applications in coding theory,cryptography and algebraic curves,etc.Carlitz had a characterization of permutation polynomials.He proved that if f(x) is a polynomial with coefficients over finite field Fq satisfying f(0) = f(1) and η(f(a)-f(b) = η(a-b) for every a,b∈Fq,where η is the quadratic character of Fq*.Then f(x) = xpj for some integer.In this note,we proved that the above result is also true for any multiplicative character of Fq*.