Finite-dimensional Mahler measure of a polynomial and Szegö's theorem
type: Item
instance of: scholarly article
title: Finite-dimensional Mahler measure of a polynomial and Szegö's theorem
zbMATH Open document ID: 0870.11066
author: Jérome Dégot
published in: Journal of Number Theory
publication date: 1997-08-03T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 11R09
zbMATH DE Number: 992071
zbMATH Keywords: Mahler measure
zbMATH Keywords: approximants
zbMATH Keywords: optimal polynomial
reviewed by: David W. Boyd
MaRDI profile type: Publication
full work available at URL: jnth.1997.2052
OpenAlex ID: W1998630639
cites work: Computing the measure of a polynomial
cites work: On Mahler's Measure of a Polynomial
cites work: Heights of Algebraic Numbers and Szego's Theorem
cites work: An application of Jensen's formula to polynomials
cites work: On the zeros of the derivative of a polynomial
DOI: 10.1006/JNTH.1997.2052
description: scientific article; zbMATH DE number 992071