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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
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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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