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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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