Zero estimates for polynomials in 3 and 4 variables using orbits and stabilisers
To solve many Diophantine equations it often requires good lower bounds for linear forms in the logarithms of a small number of algebraic numbers. This in turn depends on good zero estimates: A non-zero polynomial cannot have a "grid" of zeros of size N unless it has large degree (in terms of N). Building on ideas of Wustholz (but using orbits and stabilisers) we obtain smaller bounds for the zero estimate for polynomials in 3 or 4 variables. We give an application to Catalan's conjecture.
C. Bennett, L. Elderbrock, & A.M.W. Glass, Zero-Estimates for Polynomials in 3 and 4 Variables using Orbits and Stabilizers, in Hilbert’s Tenth Problem: Relations with Arithmetic and Algebraic Geometry, AMS, 2000.