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Dickson polynomials over finite fields and complete mappings. (English) Zbl 0576.12020
Let $${\mathbb{F}}_ q$$ denote the finite field of order q. A polynomial f(x) over $${\mathbb{F}}_ q$$ is a permutation polynomial on $${\mathbb{F}}_ q$$ if f induces a bijection on $${\mathbb{F}}_ q$$ and we say that f(x) is a complete mapping polynomial of $${\mathbb{F}}_ q$$ if both f(x) and $$f(x)+x$$ are permutation polynomials of $${\mathbb{F}}_ q$$. If k is a positive integer and $$a\in {\mathbb{F}}_ q$$, the Dickson polynomial $$g_ k(x,a)$$ is defined by $g_ k(x,a)=\sum^{h}_{j=0}\frac{k}{k-j} \left( \begin{matrix} k-j\\ j\end{matrix} \right) (-a)^ j \quad x^{k-2j}$ where h is the greatest integer $$\leq k/2$$. We assume that $$a\neq 0$$ so that $$g_ k(x,a)$$ is a permutation polynomial of $${\mathbb{F}}_ q$$ if and only if $$\gcd (k,q^ 2- 1)=1.$$
We prove that if $$k\geq 2$$ and $$a,b,c\in {\mathbb{F}}_ q$$ with abc$$\neq 0$$, then b $$g_ k(x,a)+c x$$ can be a permutation polynomial of $${\mathbb{F}}_ q$$ only in one of the cases (i) $$k=3$$, $$c=3ab$$, and $$q\equiv 2 (mod 3)$$; (ii) $$k\geq 3$$ and the characteristic of $${\mathbb{F}}_ q$$ divides k; (iii) $$k\geq 4$$, the characteristic of $${\mathbb{F}}_ q$$ does not divide k, and $$q<(9k^ 2-27k+22)^ 2$$. As a corollary we show that if $$k\geq 2$$ and $$a,b,c\in {\mathbb{F}}_ q$$ with ab$$\neq 0$$, then b $$g_ k(x,a)+c x$$ can be a complete mapping polynomial of $${\mathbb{F}}_ q$$ only in one of the cases (ii) or (iii) above. It is also pointed out that in both cases (ii) and (iii), complete mapping polynomials of the form b $$g_ k(x,a)+c x$$ do indeed exist. The case $$a=0$$ was studied by the second author and K. H. Robinson [J. Aust. Math. Soc., Ser. A 33, 197-212 (1982; Zbl 0495.12018)].

##### MSC:
 11T06 Polynomials over finite fields 05A05 Permutations, words, matrices
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