Unimodular integer circulants
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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 $\operatorname {Res}(f(x),x^n-1)=\pm 1$. 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.References
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Additional Information
- J. E. Cremona
- Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
- MR Author ID: 52705
- ORCID: 0000-0002-7212-0162
- Email: J.E.Cremona@warwick.ac.uk
- Received by editor(s): June 6, 2007
- Received by editor(s) in revised form: July 26, 2007
- Published electronically: February 11, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 77 (2008), 1639-1652
- MSC (2000): Primary 11C08, 11C20, 15A36
- DOI: https://doi.org/10.1090/S0025-5718-08-02089-9
- MathSciNet review: 2398785
Dedicated: Dedicated to the memory of R. W. K. Odoni, 1947–2002