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The Number of Permutation Polynomials of a Given Degree Over a Finite Field

✍ Scribed by Pinaki Das

Publisher: Elsevier Science
Year: 2002
Tongue: English
Weight: 139 KB
Volume: 8
Category: Article
ISSN: 1071-5797
DOI: 10.1006/ffta.2002.0355

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✦ Synopsis

We relate the number of permutation polynomials in F q ½x of degree d q À 2 to the solutions ðx 1 ; x 2 ; . . . ; x q Þ 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 À 2 in F p ½x 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. # 2002 Elsevier Science (USA)

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