Multiplicative dependence and isolation II
Philipp Habegger · Journal of the London Mathematical Society · 2009
In this paper we study the set of algebraic x ≠ 0, 1 such that x and 1 − x are multiplicatively dependent. Cohen and Zannier proved that log 2 is a sharp and isolated upper bound for the height max{h(x), h(1 − x)}. Working with a slightly different height, we show that the set of height values has precisely one limit point equal to the Mahler measure of the two-variable polynomial X + Y − 1. Moreover, we prove a conjecture of Masser on an asymptotic estimate for the number of such x of bounded degree. Our results are based on a new, complete factorization statement for certain trinomials with roots of unity as coefficients over a Kroneckerian number field.