The Mahler measure of a multivariate polynomial family with non-linear degree (Q6620004)

The (logarithmic) Mahler measure \(m(P)\) of a non-zero rational function \(P\in \mathbb{C}(x_{1},\dots,x_{n})^{\times}\) is given by\N\[\Nm(P)=\frac{1}{(2\pi i)^{n}}\int_{\mathbb{T}^{n}}\log \left\vert P(x_{1},\dots,x_{n})\right\vert \frac{dx_{1}}{x_{1}}\dots\frac{dx_{n}}{x_{n}},\N\]\Nwhere the integration is taken over the unit torus \(\mathbb{T}^{n}=\{(x_{1},\dots,x_{n})\in \mathbb{C}^{n}\mid \left\vert x_{1}\right\vert =\cdots =\left\vert x_{n}\right\vert =1\}\) with respect to the Haar measure.\N\NIn the paper under review, the authors investigate the Mahler measure of the family\N\[\NQ_{m,r}(x_{1},\dots,x_{m},x,y,z)=(1+x)z+\left[ \left( \frac{1-x_{1}}{1+x_{1}} \right) \cdots \left( \frac{1-x_{m}}{1+x_{m}}\right) \right] ^{r}(1+y),\N\]\Nwhere \((m,r)\in \mathbb{N}^{2},\) and thus generalize the previous results obtained by \textit{M. Lalin} [J. Number Theory 116, No. 1, 102--139 (2006; Zbl 1162.11388)] for the case \(r=1\).