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The number of permutation polynomials of a given degree over a finite field. (English) Zbl 1029.11066
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 author deals with the problem of enumerating such permutation polynomials by their degree.
Let \(N_q(d)\) denote the number of permutation polynomials \(f\in\mathbb F_q[x]\) of degree \(d\) satisfying \(f(0)=0\). Relating \(N_q(d)\) to the number of solutions of a system of linear equations over \(\mathbb F_q\), the author finds an expression for \(N_p(p-2)\) in terms of the permanent of a Vandermonde matrix. Asymptotic bounds then follow from an estimate for the permanent of a complex matrix given by H. Minc [Permanents, Addison-Wesley (1978; Zbl 0401.15005)]. It is shown that \[ |N_p(p-2)-(1-1/p)(p-1)!|\leq\sqrt{(p-1)(1+(p-2)p^{p-1})}/p. \] A similar estimate was determined, using a different method, by S. Konyagin and F. Pappalardi [Finite Fields Appl. 8, 548–553 (2002; Zbl 1029.11067)].
Furthermore, the author indicates how to generalize his result to permutation polynomials of arbitrary degree over prime fields.

MSC:
11T06 Polynomials over finite fields
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[1] S. Konyagin, and, F. Pappalardi, Enumerating permutation polynomials over finite fields by degree, preprint, 2001. · Zbl 1029.11067
[2] Lidl, R.; Mullen, G.L., When does a polynomial over a finite field permute the elements of the field?, Amer. math. monthly, 95, 243-246, (1988) · Zbl 0653.12010
[3] Lidl, R.; Niederreiter, H., Finite fields, (1997), Cambridge University Press Cambridge
[4] Marcus, M.; Minc, H., Permanents, (1978), Addison-Wesley Reading
[5] Mullen, G.L., Permutation polynomials over finite fields, Finite fields, coding theory, and advances in communications and computing, (1993), Marcel Dekker New York, p. 131-151 · Zbl 0808.11069
[6] Shparlinski, I.E., Finite fields: theory and computation, (1999), Kluwer Academic Publishers Dordrecht · Zbl 0967.11052
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