Multiplicative independence of algebraic numbers and expressions

A list of complex numbers is multiplicatively independent if no integral-ex ponent power product of them is equal to 1, unless all exponents are zero. A method of deciding multiplicative independence is given, for complex numb ers in a finitely generated field, with given proper set of generators. Thi s is based on computing an upper bound on absolute value for possible minim al non-zero integral exponents. As a consequence of this, a solution which does not use numerical approximation, depending on the Schanuel conjecture, can be given for the problem of deciding equality between two numbers give n as closed-form. expressions using exp, log, radicals, and field operation s. It is argued, however, that an efficient solution of this problem is lik ely to use numerical approximation, together with an upper bound, depending on the syntax of the expressions for the numbers, for the amount of precis ion needed to distinguish the numbers if they are not the same. A conjectur e is stated (the uniformity conjecture) which attempts to provide such an u pper bound. (C) 2001 Elsevier Science B.V. All rights reserved.