Mahler’s measure for polynomials in several variables

Abstract

If $P(x\_1, \ldots, x\_k)$ is a polynomial with complex coefficients, the Mahler measure of $P$, $M(P)$, is defined to be the geometric mean of $|P|$ over the $k$-torus, $\mathbb{T}^k$. We briefly describe Mahler's motivation for defining this function and his applications of it to polynomial inequalities. We then describe how this function occurs naturally in the study of Lehmer's problem concerning the set of all measures of one-variable polynomials with integer coefficients. We describe work of Deninger which shows how Mahler measure arises in the study of the far-reaching Beĭlinson conjectures and leads to surprising conjectural explicit formulas for some measures of multivariable polynomials. Finally we describe some of the recent work of many authors proving some of these formulas by a variety of different methods.