## Greatest common divisors with moving targets and consequences for linear recurrence sequences

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- by Nathan Grieve and Julie Tzu-Yueh Wang PDF
- Trans. Amer. Math. Soc.
**373**(2020), 8095-8126 Request permission

## Abstract:

We establish consequences of the moving form of Schmidt’s Subspace Theorem. Indeed, we obtain inequalities that bound the logarithmic greatest common divisor of moving multivariable polynomials evaluated at moving $S$-unit arguments. In doing so, we complement recent work of Levin. As an additional application, we obtain results that pertain to the greatest common divisor problem for algebraic linear recurrence sequences. These observations are motivated by previous related works of Corvaja-Zannier, Levin, and others.## References

- Yann Bugeaud, Pietro Corvaja, and Umberto Zannier,
*An upper bound for the G.C.D. of $a^n-1$ and $b^n-1$*, Math. Z.**243**(2003), no. 1, 79–84. MR**1953049**, DOI 10.1007/s00209-002-0449-z - Pietro Corvaja and Umberto Zannier,
*Finiteness of integral values for the ratio of two linear recurrences*, Invent. Math.**149**(2002), no. 2, 431–451. MR**1918678**, DOI 10.1007/s002220200221 - P. Corvaja and U. Zannier,
*On the greatest prime factor of $(ab+1)(ac+1)$*, Proc. Amer. Math. Soc.**131**(2003), no. 6, 1705–1709. MR**1955256**, DOI 10.1090/S0002-9939-02-06771-0 - Pietro Corvaja and Umberto Zannier,
*A lower bound for the height of a rational function at $S$-unit points*, Monatsh. Math.**144**(2005), no. 3, 203–224. MR**2130274**, DOI 10.1007/s00605-004-0273-0 - Zhihua Chen, Min Ru, and Qiming Yan,
*Schmidt’s subspace theorem with moving hypersurfaces*, Int. Math. Res. Not. IMRN**15**(2015), 6305–6329. MR**3384479**, DOI 10.1093/imrn/rnu126 - Nathan Grieve,
*Generalized GCD for toric Fano varieties*, Acta Arith.**195**(2020), no. 4, 415–428. MR**4121877**, DOI 10.4064/aa190430-5-12 - Ji Guo,
*The quotient problem for entire functions*, Canad. Math. Bull.**62**(2019), no. 3, 479–489. MR**3998734**, DOI 10.4153/s0008439518000097 - Brendan Hassett,
*Introduction to algebraic geometry*, Cambridge University Press, Cambridge, 2007. MR**2324354**, DOI 10.1017/CBO9780511755224 - Santos Hernández and Florian Luca,
*On the largest prime factor of $(ab+1)(ac+1)(bc+1)$*, Bol. Soc. Mat. Mexicana (3)**9**(2003), no. 2, 235–244. MR**2029272** - Serge Lang,
*Algebra*, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR**1878556**, DOI 10.1007/978-1-4613-0041-0 - Michel Laurent,
*Équations diophantiennes exponentielles*, Invent. Math.**78**(1984), no. 2, 299–327 (French). MR**767195**, DOI 10.1007/BF01388597 - Aaron Levin,
*Greatest common divisors and Vojta’s conjecture for blowups of algebraic tori*, Invent. Math.**215**(2019), no. 2, 493–533. MR**3910069**, DOI 10.1007/s00222-018-0831-z - A. Levin and J. T.-Y. Wang,
*Greatest common divisors and Nevanlinna theory on algebraic tori*, J. Reine Angew. Math., (to appear, published online on 2019-11-09). - Florian Luca,
*On the greatest common divisor of $u-1$ and $v-1$ with $u$ and $v$ near $\scr S$-units*, Monatsh. Math.**146**(2005), no. 3, 239–256. MR**2184226**, DOI 10.1007/s00605-005-0303-6 - Min Ru,
*Nevanlinna theory and its relation to Diophantine approximation*, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR**1850002**, DOI 10.1142/9789812810519 - Min Ru and Paul Vojta,
*Schmidt’s subspace theorem with moving targets*, Invent. Math.**127**(1997), no. 1, 51–65. MR**1423025**, DOI 10.1007/s002220050114 - Hans Peter Schlickewei and Wolfgang M. Schmidt,
*The intersection of recurrence sequences*, Acta Arith.**72**(1995), no. 1, 1–44. MR**1346803**, DOI 10.4064/aa-72-1-1-44 - H. P. Schlickewei and W. M. Schmidt,
*The number of solutions of polynomial-exponential equations*, Compositio Math.**120**(2000), no. 2, 193–225. MR**1739179**, DOI 10.1023/A:1001719425893 - Joseph H. Silverman,
*Generalized greatest common divisors, divisibility sequences, and Vojta’s conjecture for blowups*, Monatsh. Math.**145**(2005), no. 4, 333–350. MR**2162351**, DOI 10.1007/s00605-005-0299-y - Alfred J. van der Poorten,
*Solution de la conjecture de Pisot sur le quotient de Hadamard de deux fractions rationnelles*, C. R. Acad. Sci. Paris Sér. I Math.**306**(1988), no. 3, 97–102 (French, with English summary). MR**929097** - Paul Vojta,
*Diophantine approximation and Nevanlinna theory*, Arithmetic geometry, Lecture Notes in Math., vol. 2009, Springer, Berlin, 2011, pp. 111–224. MR**2757629**, DOI 10.1007/978-3-642-15945-9_{3}

## Additional Information

**Nathan Grieve**- Affiliation: The Tutte Institute for Mathematics and Computation, P.O. Box 9703, Terminal, Ottawa, Ontario, K1G 3Z4, Canada
- MR Author ID: 1020249
- ORCID: 0000-0003-3166-0039
- Email: nathan.m.grieve@gmail.com
**Julie Tzu-Yueh Wang**- Affiliation: Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan
- MR Author ID: 364623
- ORCID: 0000-0003-2133-1178
- Email: jwang@math.sinica.edu.tw
- Received by editor(s): April 29, 2019
- Received by editor(s) in revised form: August 27, 2019, and March 26, 2020
- Published electronically: August 28, 2020
- Additional Notes: The second author was supported in part by Taiwan’s MoST grant 108-2115-M-001-001-MY2.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**373**(2020), 8095-8126 - MSC (2010): Primary 11J87; Secondary 11B37, 11J25
- DOI: https://doi.org/10.1090/tran/8220
- MathSciNet review: 4169683