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

#### Abstract

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.

Link: http://rmmc.asu.edu/rmj/rmj.html

$F_n(x)$ 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.

Link: http://rmmc.asu.edu/rmj/rmj.html