An explicit formule for the Mahler measure of a family of 3-variable polynomials
Mahler's measure for a Laurent polynomial \(P(x_1,\dots,x_n)\) is defined by \[ m(P)=(2\pi)^{-n}\int_0^{2\pi}\cdots\int_0^{2\pi}\log(| P(\exp(it_1),\dots,\exp(it_n)| )\,dt_1\cdots dt_n. \] The author presents an explicit formula for \(m(P)\) in the case \(P(x,y,z)=a+b/x+cy+\)\break \((a+bx+cy)z\), where \(a,b,c\) are arbitrary real numbers, in terms of modified \(2\)- and \(3\)-polylogaritm functions \(L_2,L_3\), defined by \[ L_n(x)=\sum_{r=0}^{n-1}(-\log| x| )^r\text{Li}_{n-r}(x)/r!, \] \(\text{Li}_r(x)\) being the usual \(r\)th polylogarithm function. Previously only one non-trivial example of Mahler's measure of a polynomial in \(3\) variables was known, namely \(m(1+x+y+z)=7\zeta(3)/2\pi^2\) [\textit{D. W. Boyd}, Can. Math. Bull. 24, 453--469 (1981; Zbl 0479.12005)].
scientific article; zbMATH DE number 2184607
scientific article