Unimodular integer circulants

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.