Lehmer’s conjecture asserts that the smallest Mahler measure is the only real zero \(\mu_0\sim 1.176280\ldots\) of the polynomial \[ f(T)=T^{10}+T^9-T^7-T^6-T^5-T^4-T^3+T+1. \] This polynomial is also a Coxeter polynomial \(\chi_A\), i.e. the characteristic polynomial of the Coxeter transformation defined by a hereditary algebra \(A\).
Let \(M_{(\chi_A)}\) be the Mahler measure of the Coxeter polynomial \(\chi_A\) of a finite dimensional algebra \(A\) over an algebraically closed field \(k\). Suppose, \(A\) is a basic connected and triangular algebra with \(n\) pairwise non-isomorphic simple modules.
The purpose of this paper is to study the Mahler measure of Coxeter polynomials of accessible algebras. An algebra \(A\) is called accessible, if there is a sequence \(k=A_1,A_2,\ldots,A_n=A\) of algebras such that each \( A_{i+1}\) is a one-point extension of \(A_i\) by some exceptional \(A_i\)-module \(M_i\) for \(i=1,2,\ldots,n-1\).
It is proved that for any accessible algebra \(A\) the Mahler measure of Coxeter transformation is either \(M_{\chi_A}=1\) or \(M_{\chi_B}\geq\mu_0\) for some convex subcategory \(B\) of \(A\).
A sequence of algebras, called interlaced tower of algebras \(A_m,\ldots,A_n\) with \(m\leq n-2\) is introduced. It is supposed for the corresponding Coxeter polynomials and \(m+1\leq s\leq n-1\) that \[ \chi_{A_{s+1}}=(T+1)\chi_{A_s}-T\chi_{A_{s-1}}. \] It is proved that if the zeros of \(\chi_{A_n}\) are either on the unit circle or they are positive real numbers and \(M_{(\chi_{A_n})}>1\) then \(M_{(\chi_{A_m})}<M_{(\chi_{A_n})}\).
Several examples are also constructed.