On the equation X1 + X2 = 1 infinitely generated multiplicative groups in positive characteristic
| Autor: | Peter Koymans, Carlo Pagano |
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| Rok vydání: | 2017 |
| Předmět: | |
| Zdroj: | The Quarterly Journal of Mathematics. 68:923-934 |
| ISSN: | 1464-3847 0033-5606 |
| DOI: | 10.1093/qmath/hax007 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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