Level curves of rational functions and unimodular points on rational curves
HTML articles powered by AMS MathViewer
- by Fedor Pakovich and Igor E. Shparlinski PDF
- Proc. Amer. Math. Soc. 148 (2020), 1829-1833 Request permission
Abstract:
We obtain an improvement and broad generalisation of a result of N. Ailon and Z. Rudnick (2004) on common zeros of shifted powers of polynomials. Our approach is based on reducing this question to a more general question of counting intersections of level curves of complex functions. We treat this question via classical tools of complex analysis and algebraic geometry.References
- Nir Ailon and Zéev Rudnick, Torsion points on curves and common divisors of $a^k-1$ and $b^k-1$, Acta Arith. 113 (2004), no. 1, 31–38. MR 2046966, DOI 10.4064/aa113-1-3
- F. Beukers and C. J. Smyth, Cyclotomic points on curves, Number theory for the millennium, I (Urbana, IL, 2000) A K Peters, Natick, MA, 2002, pp. 67–85. MR 1956219
- Pietro Corvaja, David Masser, and Umberto Zannier, Sharpening ‘Manin-Mumford’ for certain algebraic groups of dimension 2, Enseign. Math. 59 (2013), no. 3-4, 225–269. MR 3189035, DOI 10.4171/LEM/59-3-2
- Gerd Fischer, Plane algebraic curves, Student Mathematical Library, vol. 15, American Mathematical Society, Providence, RI, 2001. Translated from the 1994 German original by Leslie Kay. MR 1836037, DOI 10.1090/stml/015
- Dragos Ghioca, Liang-Chung Hsia, and Thomas J. Tucker, On a variant of the Ailon-Rudnick theorem in finite characteristic, New York J. Math. 23 (2017), 213–225. MR 3625452
- Liang-Chung Hsia and Thomas J. Tucker, Greatest common divisors of iterates of polynomials, Algebra Number Theory 11 (2017), no. 6, 1437–1459. MR 3687102, DOI 10.2140/ant.2017.11.1437
- Serge Lang, Division points on curves, Ann. Mat. Pura Appl. (4) 70 (1965), 229–234. MR 190146, DOI 10.1007/BF02410091
- Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605, DOI 10.1007/978-1-4757-1810-2
- Alina Ostafe, On some extensions of the Ailon-Rudnick theorem, Monatsh. Math. 181 (2016), no. 2, 451–471. MR 3539944, DOI 10.1007/s00605-016-0911-3
- Hector Pasten and Julie Tzu-Yueh Wang, GCD bounds for analytic functions, Int. Math. Res. Not. IMRN 1 (2017), 47–95. MR 3632098, DOI 10.1093/imrn/rnw028
- J. Rafael Sendra, Franz Winkler, and Sonia Pérez-Díaz, Rational algebraic curves, Algorithms and Computation in Mathematics, vol. 22, Springer, Berlin, 2008. A computer algebra approach. MR 2361646, DOI 10.1007/978-3-540-73725-4
- Umberto Zannier, Some problems of unlikely intersections in arithmetic and geometry, Annals of Mathematics Studies, vol. 181, Princeton University Press, Princeton, NJ, 2012. With appendixes by David Masser. MR 2918151
Additional Information
- Fedor Pakovich
- Affiliation: Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Beer Sheva, 8410501, Israel
- MR Author ID: 602219
- Email: pakovich@math.bgu.ac.il
- Igor E. Shparlinski
- Affiliation: Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
- MR Author ID: 192194
- Email: igor.shparlinski@unsw.edu.au
- Received by editor(s): May 16, 2018
- Received by editor(s) in revised form: July 29, 2018
- Published electronically: January 21, 2020
- Additional Notes: The work of the second author was supported in part by the Australian Research Council Grants DP170100786 and DP180100201.
- Communicated by: Alexander Iosevich
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 1829-1833
- MSC (2010): Primary 11D61, 12D10, 30C15, 30J10
- DOI: https://doi.org/10.1090/proc/14928
- MathSciNet review: 4078070