Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Asymptotic gcd and divisible sequences for entire functions


Authors: Ji Guo and Julie Tzu-Yueh Wang
Journal: Trans. Amer. Math. Soc. 371 (2019), 6241-6256
MSC (2010): Primary 30D30; Secondary 32H30, 11J97
DOI: https://doi.org/10.1090/tran/7435
Published electronically: January 16, 2019
MathSciNet review: 3937323
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 30D30, 32H30, 11J97

Retrieve articles in all journals with MSC (2010): 30D30, 32H30, 11J97


Additional Information

Ji Guo
Affiliation: Department of Mathematics, National Tsing Hua University, No. 101, Section 2, Kuang-Fu Road, Hsinchu 30013, Taiwan
Email: s104021881@m104.nthu.edu.tw

Julie Tzu-Yueh Wang
Affiliation: Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan
Email: jwang@math.sinica.edu.tw

DOI: https://doi.org/10.1090/tran/7435
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.
Article copyright: © Copyright 2019 American Mathematical Society