Fibonacci多项式的置换性质

对多项式置换性的研究在代数学、组合学、数论、编码理论、密码学等领域中均有广泛而又重要的应用. 本文主要研究Fibonacci多项式, 通过计算它们函数值的等幂和得到了判定这些定义在有限域上的Fibonacci多项式为置换多项式的必要条件, 解决了Fernando和Rashid提出的公开问题. 这些条件推广了有关Fibonacci多项式置换性研究的已有结论.

Permutation properties of the Fibonacci polynomials

The study for the permutation of polynomials has wide and important applications in the fields of algebra, combinatorics, number theory, coding theory, cryptography and so on. In this paper, we mainly study Fibonacci polynomials, by calculating the sum of equal powers of their function values, we obtain the necessary conditions for determining these Fibonacci polynomials defined in finite fields as permutation polynomials, solve the open problem raised by Fernando and Rashid. These conditions generalize the existing conclusions on the permutation of Fibonacci polynomials.