Estimates for Mahler’s Measure of a Linear Form

Abstract

Let La(z) = a 1 z 1 + a 2 z 2 + · · · + a N z N be a linear form in N complex variables z 1 , z 2 , . . . , z N with non-zero coefficients. We establish several estimates for the logarithmic Mahler measure of La. In general, we show that the logarithmic Mahler measure of La(z) and the logarithm of the norm of a differ by a bounded amount that is independent of N . We prove a further estimate which is useful for making an approximate numerical evaluation of the logarithmic Mahler measure.

Mahler measures, short walks and log-sine integrals

“…Computer algebra programs do not seem able to shed much light on the form of the solutions to (42) or of (44). Nonetheless, we still hope to find some closed form for p 5 .…”

“…The moments W n (s), generalized to the situation of walks with varying but prescribed step lengths, are also studied in [42] and various presentations are given for them. The reason for their appearance in [42] is a relation with Mahler measures: in light of (1), one observes that (multiple) derivatives of W n evaluated at zero are (multiple) Mahler measures.…”

“…The reason for their appearance in [42] is a relation with Mahler measures: in light of (1), one observes that (multiple) derivatives of W n evaluated at zero are (multiple) Mahler measures. In particular, W n (0) = µ(x 1 + x 2 + .…”

Арифметические Гипергеометрические Ряды

“…Помимо (101), Смит [33] Такие выражения рассматриваются в [122], а также в [31] в связи с короткими случайными блужданиями на плоскости.…”

On the Mahler measure of resultants in small dimensions

“…where γ is the Euler-Mascheroni constant (see [19], and also [18] for more estimates and generalizations).…”