ESTIMATES FOR MAHLER’S MEASURE OF A LINEAR FORM

[摘要] Let $L_{m{a}}(m{z})=a_1z_1+a_2z_2+cdots+a_Nz_N$ 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 $L_{m{a}}$. In general, we show that the logarithmic Mahler measure of $L_{m{a}}(m{z})$ and the logarithm of the norm of $m{a}$ 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

[学科分类] 数学（综合）

[关键词] Mahler measure;linear form;approximation