Asymptotic nature of higher Mahler measure (Q2925466)

For a positive integer \(k\), the \(k\)-higher Mahler measure \(m_k(P)\) of a Laurent polynomial \(P(x)\) is the average over the unit circle of \(\log^k|P(x)|\). So the \(1\)-Mahler measure is the familiar Mahler measure. The author obtains asymptotic estimates for \(m_k(P)\) as \(k \to \infty\) but only for the specific degree \(1\) polynomials \(P(x) = x - r\) with \(|r| = 1\). For these \(P\) he shows that \(m_{k+1}(P)/(k+1)! + m_k(P)/k! = O(1/k)\) and that \(\lim_{k \to \infty} |m_k(P)/k!| = 1/\pi\). In spite of the restriction to this small class of polynomials, the proofs require an intricate analysis.