Small Salem numbers, exceptional units, and Lehmer's conjecture (Q1359154)

This mainly expository article discusses some interesting recent work on the theme of Lehmer's conjecture about Mahler's measure. The first topic concerns exceptional units and is based on the idea that if \(\alpha\) is an algebraic unit with small Mahler measure then one would expect \(1- \alpha^n\) to be a unit for many values of \(n\). Some experimental evidence is presented for this in case \(\alpha\) is a small Salem number. More details on this can be found in the author's recent article [\textit{J. H. Silverman}, Exp. Math. 4, No. 1, 69-83 (1995; Zbl 0851.11064)]. The absolute canonical height of an algebraic number and the Mahler measure are closely related. Observing that the height satisfies \(H(\alpha^n)= H(\alpha)^n\), i.e. that \(H(\phi(\alpha))= H(\alpha)^n\) for the rational function \(\phi(x)= x^n\), motivates a definition essentially due to Tate of a height \(\widehat H_\phi\) for any rational function \(\phi\) of degree \(n \geq 2\) which has the property that \(\widehat H_\phi(\phi(\alpha))= \widehat H_\phi(\alpha)^n\). There is a natural generalization of the Lehmer conjecture (or more appropriately the Schinzel-Zassenhaus conjecture) to this context. A number of interesting examples are discussed, for example the case where \(\phi\) is the duplication map on an elliptic curve \(E\) in which case \(\widehat H_\phi\) is the exponential of the standard canonical height on \(E\). If \(\phi\) is a polynomial with integer coefficients, then a consideration of the Julia set of \(\phi\) leads naturally to the definition of \(\phi\)-Pisot and \(\phi\)-Salem numbers and an interesting question about the existence of such numbers. (Also submitted to MR).