ESTIMATES FOR MAHLER’S MEASURE OF A LINEAR FORM

• Published in: Proceedings of the Edinburgh Mathematical Society Vol. 47; no. 2; pp. 473 - 494
• Main Authors: Rodriguez-Villegas, Fernando, Toledano, Ricardo, Vaaler, Jeffrey D.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.06.2004
• Subjects: approximation; linear form; Mahler measure; approximation; linear form; Mahler measure
• ISSN: 0013-0915; 1464-3839
• DOI: 10.1017/S0013091503000701

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