The equation x+y=α in multiplicatively dependent unknowns

Abstract

When is a non-zero algebraic number we give essentially optimal upper bounds for the height of multiplicatively dependent algebraic numbers x and y with x +y = . Furthermore we improve these bounds for special values of and we give also a strong isolation result. We deduce that when is rational, this equation has at most 10 9 unit solutions x;y in the union of all number elds with rank 1 unit group. 1. Introduction and results. In this paper we give essentially optimal upper bounds for the height of multiplicatively dependent algebraic solutions of the inhomogeneous linear equation x +y = in two unknowns x and y where is any non-zero algebraic number. Furthermore we will study the case where 2 is a rational power of a non- zero integer and derive a better bound for the height of the solution. We will also see that this bound is best possible in the case where 2 is a rational power of 2 and even that the maximal height value is isolated if is also assumed to be an integer. Furthermore for non-zero rational we give a bound independent of for the number of solutions of x +y = if the unknowns are algebraic units in the union of all number elds which have unit group of rank 1.