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The structure of the group of permutations induced by Chebyshev polynomial vectors over the ring of integers mod m

Published online by Cambridge University Press:  09 April 2009

Rex Matthews
Rex Matthews
Affiliation:
Department of Mathematics, University of Tasmania, Hobart, Tasmania, Australia
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Abstract

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In an earlier paper the author investigated the properties of a class of multivanable polynomial vectors which generalise the multivariable Chebyshev polynomial vectors. In this paper the behaviour of these polynomials over rings of the typeZ/(m) is investigated, and conditions are determined for such an n-variable polynomial vector to induce a permutation of (Z/(m))n. More detailed results on the Chebyshev polynomial vectors follow. The composition properties of these vectors imply that the permutations induced by certain subsets of them form groups under composition of mappings, and the structure of these groups is investigated.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 32 , Issue 1 , January 1982 , pp. 88 - 103
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Lausch, H., Muller, W. and Nöbauer, W., ‘Über die Struktur einer durch Dicksonpolynome dargesteilten Permutationsgruppe des Restklassenringes modulo n’, J. reine angew. Math. 261 (1973), 88–99.Google Scholar
[2]Lausch, H. and Nöbauer, W., Algebra of polynomials (North-Holland, Amsterdam, 1973).Google Scholar
[3]Lidl, R. and Wells, C., ‘Chebyshev polynomials in several variables’, J. reine angew. Math. 273 (1972), 178–198.Google Scholar
[4]Lidl, R., ‘Tschebyscheffpolynome und die dadurch dargestellten Gruppen,’ Monatsh. Math. 77 (1973), 132–147.CrossRefGoogle Scholar
[5]Lidl, R., ‘Über die Struktur einer durch Tschebyscheffpolynome in 2 Variablen dargestellten Permutationsgruppe,’ Beiträge Algebra Geometrie 3 (1974), 41–48.Google Scholar
[6]Lidl, R., ‘Reguläre Polynome über endlichen Körpern,’ Beiträge Algebra Geometrie 2 (1974), 58–59.Google Scholar
[7]Lidl, R., ‘Tschebyscheffpolynome in mehreren Variablen,’ J. reine angew. Math. 273 (1975), 178–198.Google Scholar
[8]McDonald, B., Finite rings with identity (Dekker, New York, 1974).Google Scholar
[9]Matthews, R., ‘Some generalisations of Chebyshev polynomiaLs and their induced group structure over a finite field’, Acta Arithmetica, to appear.Google Scholar
[10]Narkiewicz, W., Elementary and analytic theory of algebraic numbers (Polish Scientific Publishers, Warsaw 1974).Google Scholar
[11]Nöbauer, W., ‘Uber eine Kiasse von Permutationspolynomen und die dadurch dargestellten Gruppen’, J. reine angew. Math. 231 (1968), 215–219.Google Scholar
[12]Ward, M., ‘The characteristic number of a sequence of integers satisfying a Linear recursion relation’, Trans. Amer. Math. Soc. 33 (1931), 153–165.CrossRefGoogle Scholar
[13]Ward, M., ‘The arithmetical theory of linear recurring series’, Trans. Amer. Math. Soc. 35 (1933), 600–628.CrossRefGoogle Scholar