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Enumerating permutation polynomials over finite fields by degree. (English) Zbl 1029.11067
Every permutation on the elements of $$\mathbb F_q$$ ($$q>2$$) is uniquely represented by a polynomial over $$\mathbb F_q$$ of degree $$\leq q-2$$. The authors deal with the problem of enumerating such permutation polynomials having degree $$<q-2$$. This is equivalent to counting the permutations $$\sigma$$ of $$\mathbb F_q$$ for which $$\sum_{c\in\mathbb F_q} c\sigma(c)=0$$.
Let $$N$$ denote the number of permutations of $$\mathbb F_q$$ satisfying this condition. By inclusion exclusion, $N=\sum_{S\subseteq\mathbb F_q} (-1)^{q-|S|}n_S$ where $$n_S$$ is the number of mappings $$f\colon\mathbb F_q\to S$$ with $$\sum_{c\in S} cf(c)=0$$. Using an expression for $$n_S$$ in terms of exponential sums, the authors then show that $|N-(q-1)!|\leq\sqrt{2e/\pi}q^{q/2}.$ A similar estimate for a prime $$q$$ was determined, using a different method, by P. Das [Finite Fields Appl. 8, 478–490 (2002; Zbl 1029.11066)].

##### MSC:
 11T06 Polynomials over finite fields
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##### References:
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