Generalized Mahler measures of Laurent polynomials

Following the work of Lalín and Mittal on the Mahler measure over arbitrary tori, we investigate the definition of the generalized Mahler measure for all Laurent polynomials in two variables when they do not vanish on the integration torus. We establish certain relations between the standard Mahler measure and the generalized Mahler measure of such polynomials. Later we focus our investigation on a tempered family of polynomials originally studied by Boyd, namely \(Q_{r}(x, y) = x + \frac{1}{x} + y + \frac{1}{y} + r\) $Q_r(x,y)=x+\frac{1}{x}+y+\frac{1}{y}+r$ with \(r \in \mathbb {C},\) $r\in \mathbb{C},$ and apply our results to this family. For the \(r = 4\) $r=4$ case, we explicitly calculate the generalized Mahler measure of \(Q_4\) $Q_4$ over any arbitrary torus in terms of special values of the Bloch–Wigner dilogarithm. Finally, we extend our results to the several variable setting.