Abstract
Assume Vojta’s Conjecture for blowups of \({\mathbb {P}}^1 \times {\mathbb {P}}^1\). Suppose \(a, b, \alpha ,\beta \in {\mathbb {Z}}\), and \(f(x),g(x)\in {\mathbb {Z}}[x]\) are polynomials of degree \(d\ge 2\). Assume that the sequence \((f^{\circ n}(a), g^{\circ n}(b))_n\) is generic and \(\alpha ,\beta \) are not exceptional for f, g respectively. We prove that for each given \(\varepsilon >0\), there exists a constant \(C = C(\varepsilon ,a,b,\alpha ,\beta ,f,g)>0 \), such that for all \(n\ge 1\), we have
We prove an estimate for rational functions and for a more general gcd and then obtain the above inequality as a consequence.
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Acknowledgements
I want to express my gratitude to my advisor Thomas Tucker for suggesting this project and for many valuable discussions. I also would like to thank Joseph Silverman for useful comments on an earlier draft of this paper. I would like to thank the referee for many useful comments and suggestions. In addition, I want to thank Shouman Das, Wayne Peng, and Uǧur Yiǧit for proofreading this paper.
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Communicated by Adrian Constantin.
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Huang, K. Generalized greatest common divisors for orbits under rational functions. Monatsh Math 191, 103–123 (2020). https://doi.org/10.1007/s00605-019-01350-1
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DOI: https://doi.org/10.1007/s00605-019-01350-1