Measuring the height of a polynomial

Conclusions Mahler’s measure is alive and well in several quite diverse contexts. The differing points of view seem to generate a healthy friction. If the general level of health is measured by the quantity and quality of unsolved problems, then it may help to list these. Lehmer’s Problem. The elliptic analogue of Lehmer, at least in tractable special cases. An explanation of Boyd’s remarkable formulae. It seems that K -theory should provide the conceptual framework. More generally, perhaps values of the elliptic Mahler measure will arise as values of L-functions of higher-dimensional varieties. It looks almost certain that the elliptic Mahler measure should arise as an entropy. This would form a fascinating bridge between two large areas of interest. Ward and I have begun to write about this [10]. At the very least, this would show that the global canonical height of an algebraic point on an elliptic curve arises as an entropy. But of what, and what does this mean? There are many other pretty results about the classical Mahler measure which could be lifted to the elliptic setting.