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) ∈ Z[x], determine all those n ∈ N such that Res(f(x), xn − 1) = ±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. 1. Statement of the problem and preliminary results In the study of cyclically presented groups (see [8], [6], [7], [17], and [10, Chapter 16]), the following problem arises. Let f = ∑d i=0 aix i be a polynomial of degree d with integer coefficients. Set ai = 0 for i > d. For each n > d we may form the circulant matrix Mn(f) of size n whose first row is (a0, a1, . . . , an−1). Problem A. Given f ∈ Z[x], determine all n > deg(f) such that detMn(f) = ±1. We see below that this is essentially equivalent to the following problem about integer polynomials. Problem B. Given f ∈ Z[x], determine all n ∈ N such that Res(f(x), x−1) = ±1. In this section we give some elementary preliminary results, most of which may be found in [10, Chapter 16] and [14], starting with the reduction of Problem A to Problem B in Lemma 1. In the subsequent sections we describe two methods for solving the problem: the first only requires the use of approximations to the complex roots of f and is guaranteed to work provided that none of these roots lies on the unit circle; the second uses Strassman’s Theorem on p-adic power series. The latter has been found to work in practice, at least for polynomials whose degree is small; further remarks on its general effectiveness will be made later. We also discuss the question of how to determine for a given integer polynomial whether or not any of its complex roots do lie on the unit circle, and we give a simple method for this. Received by the editor June 6, 2007 and, in revised form, July 26, 2007. 2000 Mathematics Subject Classification. Primary 11C08, 11C20, 15A36.