Counterexamples to the uniformity conjecture

The Exact Geometric Computing approach requires a zero test for numbers which are built up using standard operations starting with the natural numbers. The uniformity conjecture, part of an attempt to solve this problem, postulates a simple linear relationship between the syntactic length of expressions built up from the natural numbers using field operations, radicals and exponentials and logarithms, and the smallness of non zero complex numbers defined by such expressions. It is shown in this article that this conjecture is incorrect, and a technique is given for generating counterexamples. The technique may be useful to check other conjectured constructive root bounds of this kind. A revised form of the uniformity conjecture is proposed which avoids all the known counterexamples.