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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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