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Permutation polynomials in one and several variables

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Matthews, RW (1982) Permutation polynomials in one and several variables. PhD thesis, University of Tasmania.

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Abstract

Various 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 → 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-Műller-Nőbauer [18].
Chapter 6 settles some questions concerning the conjecture
of Schur on polynomials f(x) ϵ 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.

Item Type: Thesis (PhD)
Keywords: Polynomials
Copyright Holders: The Author
Additional Information:

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

Date Deposited: 19 Dec 2014 02:25
Last Modified: 11 Mar 2016 05:56
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