# Permutation polynomials in one and several variables

thesis

posted on 2023-05-26, 20:06 authored by Matthews, RWVarious authors have dealt with problems relating to permutation polynomials over finite systems ([4], [8], [10], [18], [20]-[25],[29]-[33], etc.). In this thesis various known results are extended and several questions are resolved. Chapter 2 begins by considering the problem of finding those permutation polynomials in a single variable amongst some given classes of polynomials. Previously, this question was settled only for cyclic polynomials and Chebyshev polynomials of the first kind. Here we consider the Chebyshev polynomials of the second kind and polynomials of the form (x n- 1)/(x - I). Certain questions on multivariable polynomials are then considered. Chapter 3 deals with questions involving polynomials whose coefficients lie in a subfield of the given field, and considers some combinatorial questions. Chapter 4 resolves the structure of the group of maps of F nq ‚Äövúv¿ F nq induced by the extended Chebyshev polynomials of Lidl and Wells [26]. Chapter 5 extends this further to finite rings -Z/(pe), thus generalising results of Lausch-M‚âà¬±ller-N‚âàv´bauer [18]. Chapter 6 settles some questions concerning the conjecture of Schur on polynomials f(x) ˜ì¬µ Z[x] which permute infinitely many residue fields Fp. It is known ([10]) that these are compositions of cyclic and Chebyshev polynomials of the first kind. In chapter 6 it is determined which of these polynomials have the required property.

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Chapter 3 appears to be the equivalent of a pre or post print of an article first published in Contemporary mathematics in volume 9 1982, published by the American Mathematical Society. Chapter 4 appears to be, in part, the equivalent of a post print of an article first published as: Matthews, R., 1982, Some generalisations of Chebyshev polynomials and their induced group structure over a finite field, Acta arithmetica, 41(4), 323-335 Chapter 5 appears to be, in part, the equivalent of a post print of an article first published as: Matthews, R., 1982, The structure of the group of permutations induced by Chebyshev polynomial vectors over the ring of integers mod m, Journal of the Australian Mathematical Society (series A), 32(1), 88-103## Repository Status

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