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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Sequences of powers with second differences equal to two and hyperbolicity
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by Natalia Garcia-Fritz PDF
Trans. Amer. Math. Soc. 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