The Mahler measure of linear forms as special values of solutions of algebraic differential equations

Ricardo D. Toledano


We prove that for each $n\geq 4$ there is an analytic function
$F_n(x)$ satisfying an algebraic differential equation of degree
$n+1$ such that the logarithmic Mahler measure of the linear form
$\lf_n=x_1+\cdots + x_n$ can be essentially computed as the evaluation of $F_n(z)$ at $z=n^{-1}$. We
show that the coefficients of the series representing $F_n(z)$ can
be computed recursively using the $n$-th. symmetric power of a
second order linear algebraic differential equation and we give
an estimate on the growth of these coefficients.

Published: Rocky Mountain Journal of Mathematics. Vol.39, 4, 2009, 1323-1338.


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