Let \(T\) and \(U\) be elements of \(\mathrm{GL}(n,\mathbb{Q})\). The rational conjugacy problem asks if there exists \(X\in\mathrm{GL}(n, \mathbb{Q})\) such that \(XTX^{-1}=U\). It is well known that this can be decided effectively by computing and comparing the rational canonical forms of \(T\) and \(U\). More difficult is the integral conjugacy problem: decide whether there exists \(X\in\mathrm{GL}(n,\mathbb{Z})\) with \(XTX^{-1}=U\). Associated to the integral conjugacy problem is the centraliser problem: determine a generating set for \(C_{\mathbb{Z}}(T)=\{X\in\mathrm{GL}(n,\mathbb{Z}) \mid XTX^{-1}=T\}\). Since \(C_{\mathbb{Z}}(T)\) is arithmetic, the work of F. Grunewald and D. Segal [Ann. Math. (2) 112, 531–583 (1980; Zbl 0457.20047)] implies that it is both finitely generated and finitely presented. But no practical algorithm to compute a finite generating set for an arithmetic group is known.
In the article under review, the authors present a new algorithm that, given two matrices in \(\mathrm{GL}(n,\mathbb{Q})\), decides if they are conjugate in \(\mathrm{GL}(n, \mathbb{Z})\) and, if so, determines a conjugating matrix. They also give an algorithm to construct a generating set for \(C_{\mathbb{Z}}(T)\) for \(T \in\mathrm{GL}(n,\mathbb{Q})\). To do this they reduce these problems, respectively, to the isomorphism and automorphism group problems for certain modules over rings of the form \(\mathcal{O}_{K}[y]/(y^{\ell})\), where \(\mathcal{O}_{K}\) is the maximal order of an algebraic number field \(K\) and \(\ell \in \mathbb{N}\), and then provide algorithms to solve the latter. The algorithms are effective and usable in practice, the authors provide their implementation using the MAGMA software.