## Sequences of powers with second differences equal to two and hyperbolicity

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- by Natalia Garcia-Fritz PDF
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**370**(2018), 3441-3466 Request permission

## Abstract:

By explicitly finding the complete set of curves of genus $0$ or $1$ in some surfaces of general type, we prove that under the Bombieri-Lang conjecture for surfaces, there exists an absolute bound $M>0$ such that there are only finitely many sequences of length $M$ formed by $k$-th rational powers with second differences equal to $2$. Moreover, we prove the unconditional analogue of this result for function fields, with $M$ depending only on the genus of the function field. We also find new examples of Brody-hyperbolic surfaces arising from the previous arithmetic problem. Finally, under the Bombieri-Lang conjecture and the ABC-conjecture for four terms, we prove analogous results for sequences of integer powers with possibly different exponents, in which case some exceptional sequences occur.## References

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## Additional Information

**Natalia Garcia-Fritz**- Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street BA6103, Toronto, Ontario, Canada, M5S 2E4
- Address at time of publication: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Campus San Joaquín, Avenida Vicuña Mackenna 4860, Santiago, Chile
- Email: natalia.garcia@mat.uc.cl
- Received by editor(s): March 7, 2016
- Received by editor(s) in revised form: August 8, 2016
- Published electronically: December 27, 2017
- Additional Notes: This work was part of the author’s thesis at Queen’s University and was partially supported by a Becas Chile Scholarship
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 3441-3466 - MSC (2010): Primary 11D41, 32Q45, 14G05
- DOI: https://doi.org/10.1090/tran/7040
- MathSciNet review: 3766854