Measuring the height of a polynomial
This pleasantly written article surveys some of the recent progress in the study of Mahler's measure and its recent generalization, the elliptic Mahler measure. Beginning with the definition of Mahler's measure, he touches on Lehmer's problem, Smyth's theorem for non-reciprocal polynomials, explicit formulas for multivariable Mahler measures and the appearance of the Mahler measure as the entropy of certain \(\mathbb Z^n\) actions on compact abelian groups. He then turns to the elliptic Mahler measure, as recently defined by the author and \textit{B. Fhlathúin} [Math. Proc. Camb. Philos. Soc. 120, 13-25 (1996; Zbl 0865.11068)], and explains in simple terms the analogy with the classical Mahler measure and the connection of this new measure with the canonical height on elliptic curves. He ends with 5 open problems (one of which is Lehmer's problem). Some recent progress on problem number 3 has been made by \textit{Hubert Bornhorn} [Thesis, Univ. Münster (1999)] and \textit{F. Rodriguez Villegas} [Proceedings of the International number theory conference, Penn. State University (August 1997)].
scientific article; zbMATH DE number 1210474
scientific article