Progress on Olga Taussky-Todd's circulant problem
Olga Taussky-Todd's circulant problem, see \textit{M.~Newman} [Ill. J. Math.~24, 156--158 (1980; Zbl 0414.15007)], is to determine the set~\({\mathcal D}(n)\) of all integer circulant \(n\times n\) determinants. This problem has been solved in the following cases: (1)~\(n=p\in\mathbb{P}\) by \textit{H.~T.~Laquer} [The Fibonacci sequence, Collect. Manuscr., 18th anniv. Vol., The Fibonacci Assoc., 212--217 (1980; Zbl 0524.15007)] and Newman [loc. cit.], (2)~\(n=2p\), \(p\geq 3\) by Laquer [loc. cit.], (3)~\(n=p^2\), \(p\geq 3\) by \textit{M. Newman} [Linear Multilinear Algebra 9, 187--191 (1980; Zbl 0453.12002)]. Newman also found a subset and a superset of \({\mathcal D}(p^k)\), \(k\geq 2\). Improving his results, the present author proves that \({\mathcal D}(4)=\mathbb{Z}_2^*\cup 16\mathbb{Z}\), \({\mathcal D}(8)=\mathbb{Z}_2^*\cup 32\mathbb{Z}\), and \[ \mathbb{Z}_2^*\cup 2^{2k-1}\mathbb{Z}\subseteq{\mathcal D}(2^k)\subseteq\mathbb{Z}_2^*\cup 2^{k+2}\mathbb{Z},\,\,k\geq 4. \] Here \(\mathbb{Z}_n^*=\{d\in\mathbb{Z}\,|\,\gcd{(d,n)}=1\}\). Applying certain properties of resultants of polynomials in~\(\mathbb{Z}[x]\), with special attention to cyclotomic polynomials, he also refines these bounds as follows: \[ \mathbb{Z}_{2q}^*\cup 32q^2\mathbb{Z}\subseteq{\mathcal D}(8q),\,q\geq 1, \] \[ \mathbb{Z}_{2q}^*\cup 4\mathbb{Z}_q^*\cup q^2\mathbb{Z}_2^*\cup 4q^2\mathbb{Z}\subseteq{\mathcal D}(2q),\,q\geq 1\,\mathrm{odd}, \] \[ {\mathcal D}(2^kq)\subseteq\mathbb{Z}_2^*\cup2^{k+2}\mathbb{Z},\,\, q\geq 1,\,k\geq 2. \]
scientific article; zbMATH DE number 6084539
scientific article