Finite-dimensional Mahler measure of a polynomial and Szegö's theorem
If \(P(z) = a_n \prod_{i=1}^n (z- \alpha_i)\) is a polynomial with integer coefficients, then the Mahler measure of \(P\) is defined by \[ M(P) = \exp\Bigl( {1 \over 2\pi}\int_0^{2\pi} \log|P(e^{it})|dt \Bigr) = |a_n|\prod_{i=1}^n \max(1,|\alpha_i|). \leqno (1) \] If \(|P|_2\) is the \(L^2\)-norm of \(P\) on the circle, then a theorem of Szegö states that \(M(P) = \inf \{|PQ|_2: Q \text{monic}\}\). \textit{W. Lawton} [Proc. Am. Math. Soc. 49, 47-50 (1975; Zbl 0306.12002)] proposed the calculation of \(M(P)\) by computing the approximants \[ M_k(P):= \inf \{|PQ|_2: Q \text{monic, degree}(Q) = k\}, \] which can be computed without computing the roots of \(P\). Clearly \(M(P) \leq M_k(P)\), and \(M_k(P) \to M(P)\) as \(k \to \infty\), but Lawton did not give an estimate of the rate of convergence of \(M_k(P)\) to \(M(P)\). This has now been provided by the author's sharp result: \[ M_k(P) \leq \Bigl( \prod_{i=0}^{n-1} \bigl( 1 + {n \over n - i + k}\bigr)\Bigr)^{1/2} M(P). \leqno (2) \] The proof gives a characterization of the optimal polynomial, which can be used to conveniently compute \(M_k(P)\). However, the rate of convergence given by (2) suggests that this may not be a practical method for the accurate computation of \(M(P)\). It would be interesting to (theoretically and experimentally) compare this method with the two methods suggested by the two formulas in (1).
scientific article; zbMATH DE number 992071