Abstract

Siegel's classical proof of the Lindemann-Weierstrass Theorem introduced two new features into transcendence proofs:  

1) An application of Thue's use of the Box Principle to show the existence of an auxiliary function with high order of zero at a point.

2) An argument showing that, not only was the auxiliary function non-zero, but its order of zero was not much larger than that imposed.

Those two features occur in essentially all modern transcendence proofs dealing with differential equations or group varieties.  For the best known transcendence applications it usually suffices to show that the actual order of such zero is not more than a fixed constant, depending upon the setting, times the order imposed.  Useful bounds have required major contributions of many authors culminating in the work of Nesterenko and, in a different direction, Wuestholz.  Very little seems to be known about the best constant, except for Siegel's bound for the simplest auxiliary functions P_1(z)exp(a_1z) + ... + P_n(z)exp(a_nz), with P_i(x,y) non-zero polynomials and a_i distinct.

In joint work with David Masser, we begin to explore some aspects of this question, with only modest results, which show however that the strongest imaginable bounds are sometimes false.  Incidentally we provide an new instance consistent with a well-known conjecture of Zilber.