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Permutation polynomials over finite fields from a powerful lemma. (English) Zbl 1258.11100
Finite Fields Appl. 17, No. 6, 560-574 (2011); corrigenda ibid. 30, 153-154 (2014).
The main results of this paper are contained in the following theorem.
Theorem 3.1. Let \(q\) be a prime power, and let \(r \geq 1\) and \(n \geq 1\) be positive integers. Let \(B(x)\), \(L_1(x),\dots,L_r(x) \in\mathbb{F}_q[x]\) be \(q\)-polynomials, \(g(x) \in\mathbb{F}_{q^n}[x]\), \(h_1(x),\dots,h_r(x) \in \mathbb{F}_q[x]\) and \(\delta_1,\dots,\delta_r \in\mathbb{F}_{q^n}\) such that \(B(\delta_i) \in {\mathbb{F}}_q\) and \(h_i(B(\mathbb{F}_{q^n})) \subseteq\mathbb{F}_q\). Then
\[ f(x) = g(B(x)) + \sum_{i=1}^r (L_i(x)+\delta_i) h_i(B(x)) \] is a permutation polynomial of \(\mathbb{F}_{q^n}\) if and only if
(1)
\(B(g(x)) + \sum_{i=1}^r (L_i(x)+B(\delta_i)) h_i(x)\) permutes \(B(\mathbb{F}_{q^n})\); and
(2)
for any \(y \in B\left(\mathbb{F}_{q^n}\right)\) \(\sum_{i=1}^r L_i(x)h_i(y)\) permutes \(\ker(B)\).
Moreover (2) is equivalent to \(\gcd\left(\sum_{i=1}^r l_i(x)h_i(y),b(x)\right)=1\) for any \(y \in\mathbb{F}_q\), where \(l_i(x)\) and \(b(x)\) are the conventional \(q\)-associates of \(L_i(x)\) and \(B(x)\).
This theorem generalizes earlier results on permutation polynomials and gives new classes of such polynomials.
Added in 2014: A condition has been corrected in the corrigenda [ibid. 30, 153–154 (2014; Zbl 1297.11150)].

MSC:
11T06 Polynomials over finite fields
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