On the Mahler measure of the Coxeter polynomial of an algebra

Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is a basic connected and triangular algebra with n pairwise non-isomorphic simple modules. We consider the Coxeter transformation ϕA(T) as the automorphism of the Grothendieck group K0(A) induced by the Auslander-Reiten translation τ in the derived category Derb(modA) of the module category modA of finite dimensional left A-modules. We say that A is of cyclotomic type if the characteristic polynomial χA of ϕA is a product of cyclotomic polynomials, equivalently, if the Mahler measure M(χA)=1. In [6] we have considered many examples of algebras of cyclotomic type in the representation theory literature. In this paper we study the Mahler measure of the Coxeter polynomial of accessible algebras. In 1933, D.H. Lehmer found that the polynomial T10+T9−T7−T6−T5−T4−T3+T+1 has Mahler measure μ0=1.176280..., and he asked whether there exist any smaller values exceeding 1. In this paper we prove that for any accessible algebra A either M(χA)=1 or M(χB)≥μ0 for some convex subcategory B of A. We introduce interlaced tower of algebras Am,…,An with m≤n−2 satisfyingχAs+1=(T+1)χAs−TχAs−1 for m+1≤s≤n−1. We prove that, if SpecϕAn⊂S1∪R+ and An is not of cyclotomic type, then M(χAm)