On the $abc$ and the $abcd$ conjectures

TL;DR

This paper revisits and refines a subexponential bound related to the $abc$ conjecture, introducing a variation using linear forms in logarithms and applying it to the 4-terms $abc$ conjecture under certain hypotheses.

Contribution

It provides a new variation of a subexponential bound for the $abc$ conjecture using linear forms in logarithms and applies it to the 4-terms $abc$ conjecture unconditionally under specific assumptions.

Findings

Established a variation of the subexponential bound using linear forms in logarithms.

Proved an unconditional subexponential bound towards the 4-terms $abc$ conjecture under a size hypothesis.

Revisited a previous bound for the $abc$ conjecture with new techniques.

Abstract

We revisit a subexponential bound for the $abc$ conjecture due to the first author, and we establish a variation of it using linear forms in logarithms. As an application, we prove an unconditional subexponential bound towards the $4$-terms $abc$ conjecture under a suitable hypothesis on the size of the variables.