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Enumerating permutation polynomials over finite fields by degree. II. (English) Zbl 1163.11350
Summary: This note is a continuation of Part I [Finite Fields Appl. 8, 548–553 (2002; Zbl 1029.11067)]. First we extend the method of the previous paper proving an asymptotic formula for the number of permutations for which the associated permutation polynomial has $$d$$ coefficients in specified fixed positions equal to 0. This also applies to the function $$N_{q,d}$$ that counts the number of permutations for which the associated permutation polynomial has degree $$< q-d-1$$. Next we adopt a more precise approach to show that the asymptotic formula $$N_{q,d} \sim q!/q^d$$ holds for $$d\leq\alpha q$$ and $$\alpha=0.03983$$.

##### MSC:
 11T06 Polynomials over finite fields 05A16 Asymptotic enumeration 11T23 Exponential sums
##### Keywords:
Permutation polynomials; Finite fields; Exponential sums
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##### References:
 [1] Das, P., The number of permutation polynomials of a given degree over a finite field, Finite fields appl., 8, 478-490, (2002) · Zbl 1029.11066 [2] Konyagin, S.; Pappalardi, F., Enumerating permutation polynomials over finite fields by degree, Finite fields appl., 8, 548-553, (2002) · Zbl 1029.11067 [3] Lidl, R.; Niederreiter, H., Finite fields, (1997), Cambridge University Press Cambridge
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