Unimodular integer circulants

Abstract

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) ∈ Z[x], determine all those n ∈ N such that Res(f (x), x n − 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.

Groups of Fibonacci Type Revisited

“…Since k = m we have (by replacing f (t) by f (t −1 ), if necessary) that f is of positive degree and is not equal to ±t. Then Theorem 1 of [35] (or Proposition 1 of [14]) implies that there are at most finitely many n for which Res(f (t), t n −1) = ±1.…”

Unimodular Integer Circulants Associated With Trinomials

“…A number of equivalent characterizations of R n (f ) = ±1 were given in [3], [7]. We only need some of them:…”

“…Well known properties of circulants and resultants give that det(M n ) = Res(f, t n − 1). The question as to when M n is unimodular arises in the theory of cyclically presented groups and has been considered by Odoni [7] and Cremona [3].…”

“…Our approach, as in [3], [7], is to work with R n (f ) rather than with M n (f ). It was shown in [3], [7] that, for n > d, det(M n ) = R n (f ) so it is enough to consider when R n (f ) = ±1.…”

“…It was shown in [3], [7] that, for n > d, det(M n ) = R n (f ) so it is enough to consider when R n (f ) = ±1. We note that R n (f ) is defined for all n ≥ 1 whereas M n (f ) is only defined for n > d. Briefly, the connection with cyclically presented groups is as follows.…”