Asymptotic expansion of the difference of two Mahler measures (Q432468)

The author gives a remarkable generalization of a result of the reviewer relating the logarithmic Mahler measure \(m(P(x,x^n))\) of the sequence of one-variable Laurent polynomials \(P(x,x^n)\) to the logarithmic Mahler measure \(m(P(x,y))\) of the two-variable Laurent polynomial \(P(x,y)\). His result applies to polynomials which satisfy a natural condition which he calls \(\mathbb T\)-separability. So, for example, his result applies to the polynomial \(1 + x + y\) but not to the polynomial \(1 + x + 1/x + y + 1/y\). Under this condition, he gives an expansion for the difference \(\mu_n(P) = m(P(x,x^n)) - m(P(x,y))\) as an asymptotic series \(\sum_{r=2}^\infty c_r(n)/n^r\). Here the coefficients \(c_r(n)\) depend on \(n\) so the series is not strictly a classical asymptotic expansion but the functions \(c_r(n)\) are always bounded and often periodic in \(n\) and hence the expansion is just as useful in practice as a classical asymptotic expansion. The coefficients are given by explicit (if somewhat complicated) finite sums. The example \(P = 1 + x + y\) is treated in detail. The reviewer had obtained the first term and error term of this expansion in [Can. Math. Bull. 24, 453--469 (1981; Zbl 0474.12005)]. For \(\mathbb T\)-separable polynomials the approximation shows that \(\mu_n(P)\) is exactly of order \(n^{-2}\). For polynomials which are not \(\mathbb T\)-separable, this is no longer true although the reviewer showed that \(\mu_n(n) = O(n^{-1-\delta})\) for some \(\delta > 0\) even in this case using essentially the method of stationary phase. The author refers to numerical experiments which suggest that for the polynomial \(P = 1 + x + 1/x + y + 1/y\), \(\mu_n(P)\) has a main term of the form \(c(n)/n^{3/2}\) where \(c(n)\) has period \(6\) in \(n\). A proof of this would be of considerable interest.