On the Mahler measure of the Coxeter polynomial of tensor products of algebras

Abstract

Let \(A\) be a finite-dimensional algebra over an algebraically closed field \(k\). Assume \(A\) is basic and connected with \(n\) pairwise non-isomorphic simple modules. We consider the Coxeter transformation\(\phi _A\) as the automorphism of the Grothendieck group \(K_0(A)\) induced by the Auslander–Reiten translation \(\tau \) in the derived category \(\mathrm{Der}(\mathrm{mod}_A)\) of the module category \(\mathrm{mod}_A\) of finite-dimensional left \(A\)-modules. We say that \(A\) is an algebra of cyclotomic type if the characteristic polynomial \(\chi _A\) of \(\phi _A\) is a product of cyclotomic polynomials. In this paper we consider algebras of the form \(A\otimes B\) and the Mahler measure of their Coxeter polynomials. We show $$\begin{aligned} M(\chi _A)\,M(\chi _B) \le M(\chi _{A\otimes B}) \le M(\chi _A)^{r(B)+1} \, M(\chi _B)^{r(A)+1}, \end{aligned}$$ where \(r(A)\) (resp. \(r(B)\)) be the number of roots \(\lambda \) of \(\chi _A\) (resp. \(\chi _B\)) with \(|\lambda |\ge 1\). In particular, for \(A\) a connected finite-dimensional algebra with \(n \ge 3\), it is equivalent that \(A\) is of cyclotomic type and that \(M(\chi _{A \otimes A})= M(\chi _A)^2\). We provide examples and applications.