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Journal of the London Mathematical Society Advance Access originally published online on August 12, 2009
Journal of the London Mathematical Society 2009 80(2):495-513; doi:10.1112/jlms/jdp037
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© 2009 London Mathematical Society

Multiplicative dependence and isolation II

P. Habegger

Departement Mathematik
ETH Zürich
Rämistrasse 101
8092 Zürich
Switzerland

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.

2000 Mathematics Subject Classification 11G50 (primary), 11D45, 11K36 (secondary).

Received March 11, 2009; revised May 19, 2009; published online August 12, 2009.


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