## 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

## 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

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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: 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
- MR Author ID: 364623
- ORCID: 0000-0003-2133-1178
- Email: jwang@math.sinica.edu.tw
- 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: https://doi.org/10.1090/tran/7435
- MathSciNet review: 3937323