Multiplicative independence of algebraic numbers and expressions

AbstractA list of complex numbers is multiplicatively independent if no integral–exponent power product of them is equal to 1, unless all exponents are zero. A method of deciding multiplicative independence is given, for complex numbers in a finitely generated field, with given proper set of generators. This is based on computing an upper bound on absolute value for possible minimal 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 given as closed-form expressions using exp,log, radicals, and field operations. It is argued, however, that an efficient solution of this problem is likely to use numerical approximation, together with an upper bound, depending on the syntax of the expressions for the numbers, for the amount of precision needed to distinguish the numbers if they are not the same. A conjecture is stated (the uniformity conjecture) which attempts to provide such an upper bound