Construction of permutation polynomials with specific cycle structure over finite fields

Abstract

For a finite field of odd order q, and a divisor n of \(q-1\), we construct families of permutation polynomials of n terms with one fixed-point (namely zero) and remaining elements being permuted as disjoint cycles of same length. Our polynomials will all be of same format: that is the degree, the terms are identical. For our polynomials their compositional inverses are also polynomials in the same format and are easy to write down. The special cases of \(n=2,3\) give very simple families of permutation binomials and trinomials. For example, in the field of 121 elements our methods provide 4080 permutation trinomials all decomposing into three disjoint cycles of length 40 along with a unique fixed point.