On the $abc$ Conjecture in Algebraic Number Fields

TL;DR

This paper extends bounds related to the abc conjecture in algebraic number fields, improving previous results and applying them to the Skolem-Mahler-Lech problem, with implications for number theory conjectures.

Contribution

It generalizes Stewart and Yu's method to improve Gy"ory's bounds for algebraic integers and provides a sub-exponential bound under certain conditions.

Findings

Improved bounds on the projective height in algebraic number fields.

Application to the effective Skolem-Mahler-Lech problem.

Sub-exponential bounds for specific algebraic number conditions.

Abstract

While currently the $abc$ conjecture and work towards it remains open or is disputed, at the same time much work has been done on weaker versions, as well as on its generalisation to number fields. Given integers satisfying $a+b=c$, Stewart and Yu were able to give an exponential bound for $\max(a,\,b,\,c)$ in terms of the radical over the integers, while Gy\"{o}ry was able to give an exponential bound in the algebraic number field case for the projective height $H_{K}(a,\,b,\,c)$ in terms of the radical for algebraic numbers. We generalise Stewart and Yu's method to give an improvement on Gy\"{o}ry's bound for algebraic integers. Finally, we will give an application to the effective Skolem-Mahler-Lech problem. Of importance is to note that, given some conditions, we obtain a sub-exponential bound for $\log H_{L}(a,\,b,\,c)$. We use these results to give an improvement on a result by Lagarias and Soundararajan. At the final stages of preparation, we were made aware that a similar result to our main theorem has been obtained independently by Gy\"ory, using different methods.