The Number of Permutation Polynomials of a Given Degree Over a Finite Field
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We relate the number of permutation polynomials in F"q[x] of degree [email protected]?q-2 to the solutions (x"1,x"2,...,x"q) of a system of linear equations over F"q, with the added restriction that x"i 0 and x"i x"j whenever i j. Using this we find an expression for the number of permutation polynomials of degree p-2 in F"p[x] in terms of the permanent of a Vandermonde matrix whose entries are the primitive pth roots of unity. This leads to nontrivial bounds for the number of such permutation polynomials. We provide numerical examples to illustrate our method and indicate how our results can be generalised to polynomials of other degrees.
[1] Igor E. Shparlinski. Finite Fields: Theory and Computation , 1999 .
[2] Rudolf Lide,et al. Finite fields , 1983 .
[3] Gary L. Mullen,et al. Finite fields, coding theory, and advances in communications and computing , 1993 .
[4] Wang Daqing,et al. Permutation polynomials over finite fields , 1987 .
[5] R. Lidl,et al. When does a polynomial over a finite field permute the elements of the fields , 1988 .