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Unimodular integer circulants

Author: J. E. Cremona
Journal: Math. Comp. 77 (2008), 1639-1652
MSC (2000): Primary 11C08, 11C20, 15A36
Published electronically: February 11, 2008
MathSciNet review: 2398785
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Abstract: We study families of integer circulant matrices and methods for determining which are unimodular. This problem arises in the study of cyclically presented groups, and leads to the following problem concerning polynomials with integer coefficients: given a polynomial $ f(x)\in\mathbb{Z}[x]$, determine all those $ n\in\mathbb{N}$ such that $ \operatorname{Res}(f(x),x^n-1)=\pm1$. In this paper we describe methods for resolving this problem, including a method based on the use of Strassman's Theorem on $ p$-adic power series, which are effective in many cases. The methods are illustrated with examples arising in the study of cyclically presented groups and further examples which illustrate the strengths and weaknesses of the methods for polynomials of higher degree.

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Additional Information

J. E. Cremona
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom

Keywords: Unimodular matrices, circulants
Received by editor(s): June 6, 2007
Received by editor(s) in revised form: July 26, 2007
Published electronically: February 11, 2008
Dedicated: Dedicated to the memory of R. W. K. Odoni, 1947–2002
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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