The unit equation has no solutions in number fields of degree prime to $3$ where $3$ splits completely

TL;DR

This paper proves that the unit equation has no solutions in certain number fields where 3 splits completely and the degree is not divisible by 3, using an elementary p-adic approach inspired by the Skolem-Chabauty-Coleman method.

Contribution

It establishes a new non-existence result for the unit equation in number fields with specific splitting and degree conditions, with applications to arithmetic dynamics.

Findings

No solutions to the unit equation in the specified fields.

Finite cyclic orbits in these fields have length only 1, 2, or 4.

Elementary p-adic proof technique inspired by Chabauty-Coleman method.

Abstract

Let $K$ be a number field with ring of integers $\mathcal O_{K}$. We prove that if $3$ does not divide $ [K:\mathbb Q]$ and $3$ splits completely in $K$, then the unit equation has no solutions in $K$. In other words, there are no $x, y \in \mathcal O_{K}^{\times}$ with $x + y = 1$. Our elementary $p$-adic proof is inspired by the Skolem-Chabauty-Coleman method applied to the restriction of scalars of the projective line minus three points. Applying this result to a problem in arithmetic dynamics, we show that if $f \in \mathcal O_{K}[x]$ has a finite cyclic orbit in $\mathcal O_{K}$ of length $n$ then $n \in \{1, 2, 4\}$.