On the equation $x + y = 1$ in finitely generated groups in positive characteristic

TL;DR

This paper proves a conjecture by Voloch that bounds the solutions to the equation $ax + by = 1$ in finitely generated groups over fields of positive characteristic, using diophantine approximation methods adapted to positive characteristic.

Contribution

The paper confirms Voloch's conjecture by establishing a uniform upper bound on solutions depending only on the rank $r$, extending diophantine approximation techniques to positive characteristic.

Findings

Bound on solutions is at most $31 imes 19^{r+1}$ for given conditions.

Generalization of Beukers and Schlickewei's work to positive characteristic.

Successful adaptation of diophantine approximation methods to positive characteristic.

Abstract

Let $K$ be a field of characteristic $p > 0$ and let $G$ be a subgroup of $K^\ast \times K^\ast$ with $\text{dim}_\mathbb{Q}(G \otimes_\mathbb{Z} \mathbb{Q}) = r$ finite. Then Voloch proved that the equation $ax + by = 1 \text{ in } (x, y) \in G$ for given $a, b \in K^\ast$ has at most $p^r(p^r + p - 2)/(p - 1)$ solutions $(x, y) \in G$, unless $(a, b)^n \in G$ for some $n \geq 1$. Voloch also conjectured that this upper bound can be replaced by one depending only on $r$. Our main theorem answers this conjecture positively. We prove that there are at most $31 \cdot 19^{r + 1}$ solutions $(x, y)$ unless $(a, b)^n \in G$ for some $n \geq 1$ with $(n, p) = 1$. During the proof of our main theorem we generalize the work of Beukers and Schlickewei to positive characteristic, which heavily relies on diophantine approximation methods. This is a surprising feat on its own, since usually these methods can not be transferred to positive characteristic.