The (logarithmic Mahler) measure \(m(P)\) of a non-zero rational function \(P\in \mathbb{C}(x_{1},\dots ,x_{n})\) is defined by \[ m(P)=\frac{1}{(2\pi i)^{n}}\int_{\mathbb{T}^{n}}\log \left\vert P(x_{1},\dots ,x_{n})\right\vert \frac{dx_{1}}{x_{1}}\cdots \frac{dx_{n}}{x_{n}}, \] where 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.
The main result of the paper under review says that the measure of a polynomial \(P(x,y_{1},\dots ,y_{n})\in \mathbb{C}[x,y_{1},\dots ,y_{n}]\) is equal to the measure of the rational function \(\widetilde{P}\in \mathbb{C}(x,y_{1},\dots ,y_{n})\), obtained by replacing in \(P\) the variable \(x\) by the rational fraction \(\lambda x^{k}\overline{g}(x^{-1})/g(x)\), where \(g(x)\in \mathbb{C}[x]\) has all its roots outside the unit disc, \(k\) is an integer greater than the degree of the polynomial \(g\), \(\lambda\) is a complex number of modulus one, and \(\overline{g}\) is the polynomial resulting from the complex conjugation of the coefficients of \(g\).
This change of variables allows the authors to construct highly non-trivial polynomials with given measure and to settle some conjectural numerical formulas due to D. W. Boyd and F. Brunault. For example, by taking \[ P(x,y_{1},y_{2}):=x+1+(x-1)(y_{1}+y_{2}), \] with \[ (g(x),k,\lambda )\in \{(x+2,2,1),\ (x^{2}-2x+2,4,1),\ (x^{4}+x+2,5,1)\}, \] they obtain that the measures of the polynomials \(x^{2}+x+1+(x^{2}-1)(y_{1}+y_{2})\), \(x^{4}-x^{3}+x^{2}-x+1+(x^{4}-x^{3}+x-1)(y_{1}+y_{2})\) and \(x^{5}+x^{4}+x+1+(x^{5}-1)(y_{1}+y_{2})\) are all equal to \(\frac{28}{5\pi ^{2}}\zeta (3)\), where \(\zeta\) is the Riemann zeta function, since according to [J. D. Condon, Mahler measure evaluations in terms of polylogarithms. Austin: University of Texas (PhD Thesis) (2004)] \(m(P)=\frac{28}{5\pi ^{2}}\zeta (3)\).