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ESTIMATES FOR MAHLER’S MEASURE OF A LINEAR FORM

Published online by Cambridge University Press:  01 July 2004

Fernando Rodriguez-Villegas ,
Ricardo Toledano  and
Jeffrey D. Vaaler
Fernando Rodriguez-Villegas
Affiliation:
Department of Mathematics, University of Texas, Austin, TX 78712, USA (villegas@math.utexas.edu; vaaler@math.utexas.edu)
Ricardo Toledano
Affiliation:
Instituto de Matematica y Fisica, Universidad de Talca, 2 Norte 685, Talca, Chile (toledano@inst-mat.utalca.cl)
Jeffrey D. Vaaler
Affiliation:
Department of Mathematics, University of Texas, Austin, TX 78712, USA (villegas@math.utexas.edu; vaaler@math.utexas.edu)
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Abstract

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Let $L_{\bm{a}}(\bm{z})=a_1z_1+a_2z_2+\cdots+a_Nz_N$ be a linear form in $N$ complex variables $z_1,z_2,\dots,z_N$ with non-zero coefficients. We establish several estimates for the logarithmic Mahler measure of $L_{\bm{a}}$. In general, we show that the logarithmic Mahler measure of $L_{\bm{a}}(\bm{z})$ and the logarithm of the norm of $\bm{a}$ differ by a bounded amount that is independent of $N$. We prove a further estimate which is useful for making an approximate numerical evaluation of the logarithmic Mahler measure.

AMS 2000 Mathematics subject classification: Primary 11C08; 11Y35; 26D15

Keywords

Mahler measurelinear formapproximation
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 47 , Issue 2 , June 2004 , pp. 473 - 494
Copyright
Copyright © Edinburgh Mathematical Society 2004