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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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Asymptotic gcd and divisible sequences for entire functions
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by Ji Guo and Julie Tzu-Yueh Wang PDF
Trans. Amer. Math. Soc. 371 (2019), 6241-6256 Request permission


Let $f$ and $g$ be algebraically independent entire functions. We first give an estimate of the Nevanlinna counting function for the common zeros of $f^n-1$ and $g^n-1$ for sufficiently large $n$. We then apply this estimate to study divisible sequences in the sense that $f^n-1$ is divisible by $g^n-1$ for infinitely many $n$. For the first part of establishing our gcd estimate, we need to formulate a truncated second main theorem for effective divisors by modifying a theorem from a paper by Hussein and Ru and explicitly computing the constants involved for a blowup of $\mathbb {P}^1\times \mathbb {P}^1$ along a point with its canonical divisor and the pull-back of vertical and horizontal divisors of $\mathbb {P}^1\times \mathbb {P}^1$.
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Additional Information
  • Ji Guo
  • Affiliation: Department of Mathematics, National Tsing Hua University, No. 101, Section 2, Kuang-Fu Road, Hsinchu 30013, Taiwan
  • Email:
  • Julie Tzu-Yueh Wang
  • Affiliation: Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan
  • MR Author ID: 364623
  • ORCID: 0000-0003-2133-1178
  • Email:
  • Received by editor(s): July 24, 2017
  • Received by editor(s) in revised form: September 1, 2017
  • Published electronically: January 16, 2019
  • Additional Notes: The second author was supported in part by Taiwan’s MoST grant 106-2115-M-001-001-MY2.
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 371 (2019), 6241-6256
  • MSC (2010): Primary 30D30; Secondary 32H30, 11J97
  • DOI:
  • MathSciNet review: 3937323