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Transactions of the American Mathematical Society

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The $ ABC$-Conjecture implies uniform bounds on dynamical Zsigmondy sets


Author: Nicole R. Looper
Journal: Trans. Amer. Math. Soc. 373 (2020), 4627-4647
MSC (2010): Primary 11G50, 11R32, 37P15; Secondary 14G05, 37P45
DOI: https://doi.org/10.1090/tran/8082
Published electronically: April 16, 2020
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Abstract: We prove that the $ abc$-Conjecture implies upper bounds on Zsigmondy sets that are uniform over families of unicritical polynomials over number fields. As an application, we use the $ abc$-Conjecture to prove that there exist uniform bounds on the index of the associated arboreal Galois representations.


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Additional Information

Nicole R. Looper
Affiliation: Department of Mathematics, Brown University, 151 Thayer Street, Providence, Rhode Island 02912

DOI: https://doi.org/10.1090/tran/8082
Received by editor(s): November 7, 2017
Received by editor(s) in revised form: August 24, 2019
Published electronically: April 16, 2020
Additional Notes: This research was supported in part by an NSF Graduate Research Fellowship
Article copyright: © Copyright 2020 American Mathematical Society