On the equation $x+y=1$ in finitely generated groups in 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 $n\ge 1$. 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 $n\ge 1$ 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.

arXiv commentFixed some typos

Cited by in corpus (1)

• Unit equations on quaternions