The Mahler Measure of Parametrizable Polynomials

Our aim is to explain instances in which the value of the logarithmic Mahler measure m(P ) of a polynomial P ∈ Z[x, y] can be written in an unexpectedly neat manner. To this end we examine polynomials defining rational curves, which allows their zero-locus to be parametrized via x = f(t), y = g(t) for f, g ∈ C(t). As an illustration of this phenomenon, we prove the equality πm(y2 + y(x+ 1) + x2 + x+ 1) = 2D(i) − 3 4 D(ω), where ω = e and D(z) is the Bloch-Wigner dilogarithm. As we shall see, formulas of this sort are a consequence of the Galois descent property for Bloch groups. This principle enables one to explain why the arguments of the dilogarithm function depend only on the points where the rational curve defined by P intersects the torus |x| = |y| = 1. In the process we also present a general method for computing the Mahler measure of any parametrizable polynomial.