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Unimodular integer circulants associated with trinomials. (English) Zbl 1228.11033

Summary: The \(n \times n\) circulant matrix associated with the polynomial \(f(t) = \sum_{i=0}^d a_it^i\) (with \(d < n)\) is the one with first row (\(a_{0} \dots a_{d} 0 \dots 0)\). The problem as to when such circulants are unimodular arises in the theory of cyclically presented groups and leads to the following question, previously studied by R. W. K. Odoni [Glasg. Math. J. 41, No. 2, 157–165 (1999; Zbl 0932.20036)] and J. E. Cremona [Math. Comput. 77, No. 263, 1639–1652 (2008; Zbl 1217.11028)]: when is \(\text{Res}(f(t), t^n-1) = \pm 1\)? We give a complete answer to this question for trinomials \(f(t) = t^{m} \pm t^{k} \pm 1\). Our main result was conjectured by the author in an earlier paper and (with two exceptions) implies the classification of the finite Cavicchioli-Hegenbarth-Repovš generalized Fibonacci groups, thus giving an almost complete answer to a question of V. G. Bardakov and A. Yu.Vesnin [Algebra Logika 42, No. 2, 131–160 (2003); translation in Algebra Logic 42, No. 2, 73–91 (2003; Zbl 1031.57001)].

MSC:

11C08 Polynomials in number theory
11C20 Matrices, determinants in number theory
15B36 Matrices of integers
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References:

[1] Bardakov V. G., Algebra Logic 42 pp 131–
[2] DOI: 10.1016/j.jalgebra.2008.07.015 · Zbl 1201.20027 · doi:10.1016/j.jalgebra.2008.07.015
[3] DOI: 10.1090/S0025-5718-08-02089-9 · Zbl 1217.11028 · doi:10.1090/S0025-5718-08-02089-9
[4] DOI: 10.1017/CBO9780511629303 · doi:10.1017/CBO9780511629303
[5] Narkiewicz W., Elementary and Analytic Theory of Algebraic Numbers (1990) · Zbl 0717.11045
[6] DOI: 10.1017/S0017089599950383 · Zbl 0932.20036 · doi:10.1017/S0017089599950383
[7] Williams G., J. Group Theory 12 pp 139–
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