The number of permutation polynomials of a given degree over a finite field

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Abstract

We relate the number of permutation polynomials in Fq[x] of degree d ≤ q - 2 to the solutions (x1,x2,...,xq) of a system of linear equations over Fq, with the added restriction that xi ≠ 0 and xi ≠ xj whenever i ≠ j. Using this we find an expression for the number of permutation polynomials of degree p - 2 in Fp[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.

Original languageEnglish (US)
Pages (from-to)478-490
Number of pages13
JournalFinite Fields and their Applications
Volume8
Issue number4
DOIs
StatePublished - Oct 2002

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Algebra and Number Theory
  • Engineering(all)
  • Applied Mathematics

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