Optimal systems of fundamental S-units for LLL-reduction

TL;DR

This paper introduces the concept of optimal systems of fundamental S-units to improve bounds in solving S-unit equations, demonstrating their effective construction and impact on reduction methods.

Contribution

It defines and proves the existence of optimal S-unit systems, enhancing the effectiveness of LLL-reduction in solving S-unit equations.

Findings

Better bounds for solutions of S-unit equations after reduction

Effective construction of optimal S-unit systems

Implications for Wildanger and Smart's resolution methods

Abstract

We show that a particular parameter plays a vital role in the resolution of S-unit equations, at the stage where LLL-reduction is applied. We define the notion of optimal system of fundamental S-units (with respect to this parameter), and prove that such a system exists and can be effectively constructed. Applying our results and methods, one can obtain much better bounds for the solutions of S-unit equations after the reduction step, than earlier. We briefly also discuss some effects of our results on the method of Wildanger and Smart for the resolution of S-unit equations.