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Aaron Levin
Variations on a theme of Runge: effective determination of integral points on certain varieties
Journal de théorie des nombres de Bordeaux, 20 no. 2 (2008), p. 385-417, doi: 10.5802/jtnb.634
Article PDF | Reviews MR 2477511 | Zbl pre05543169

Résumé - Abstract

We consider some variations on the classical method of Runge for effectively determining integral points on certain curves. We first prove a version of Runge’s theorem valid for higher-dimensional varieties, generalizing a uniform version of Runge’s theorem due to Bombieri. We then take up the study of how Runge’s method may be expanded by taking advantage of certain coverings. We prove both a result for arbitrary curves and a more explicit result for superelliptic curves. As an application of our method, we completely solve certain equations involving squares in products of terms in an arithmetic progression.

Bibliography

[1] A. Baker, Transcendental number theory, second ed. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990.  MR 1074572 |  Zbl 0715.11032
[2] M. A. Bennett, N. Bruin, K. Győry, L. Hajdu, Powers from products of consecutive terms in arithmetic progression. Proc. London Math. Soc. (3) 92 (2006), no. 2, 273–306.  MR 2205718 |  Zbl pre05014379
[3] E. Bombieri, On Weil’s “théorème de décomposition”. Amer. J. Math. 105 (1983), no. 2, 295–308.  Zbl 0516.12009
[4] E. Bombieri, W. Gubler, Heights in Diophantine geometry. New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006.  MR 2216774 |  Zbl 1115.11034
[5] R. F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), no. 3, 765–770. Article |  MR 808103 |  Zbl 0588.14015
[6] P. Erdős, J. L. Selfridge, The product of consecutive integers is never a power. Illinois J. Math. 19 (1975), 292–301. Article |  MR 376517 |  Zbl 0295.10017
[7] D. L. Hilliker, E. G. Straus, Determination of bounds for the solutions to those binary Diophantine equations that satisfy the hypotheses of Runge’s theorem. Trans. Amer. Math. Soc. 280 (1983), no. 2, 637–657.  Zbl 0528.10011
[8] M. Hindry, J. H. Silverman, Diophantine geometry. Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000.  MR 1745599 |  Zbl 0948.11023
[9] N. Hirata-Kohno, S. Laishram, T. N. Shorey, R. Tijdeman, An extension of a theorem of Euler. Acta Arith. 129 (2007), no. 1, 71–102.  MR 2326488 |  Zbl 1137.11022
[10] S. Laishram, T. N. Shorey, Squares in products in arithmetic progression with at most two terms omitted and common difference a prime power. Acta Arith. (to appear).  Zbl pre05376844
[11] S. Laishram, T. N. Shorey, S. Tengely, Squares in products in arithmetic progression with at most one term omitted and common difference a prime power. (to appear).  MR 2453529 |  Zbl pre05376844
[12] A. Levin, Ideal class groups and torsion in Picard groups of varieties. (submitted). arXiv
[13] A. Levin, Ideal class groups, Hilbert’s irreducibility theorem, and integral points of bounded degree on curves. J. Théor. Nombres Bordeaux 19 (2007), no. 2, 485–499. Cedram |  Zbl pre05302786
[14] D. W. Masser, G. Wüstholz, Fields of large transcendence degree generated by values of elliptic functions. Invent. Math. 72 (1983), no. 3, 407–464.  MR 704399 |  Zbl 0516.10027
[15] A. Mukhopadhyay, T. N. Shorey, Almost squares in arithmetic progression. II Acta Arith. 110 (2003), no. 1, 1–14.  MR 2007540 |  Zbl 1030.11010
[16] A. Mukhopadhyay, T. N. Shorey, Almost squares in arithmetic progression. III. Indag. Math. (N.S.) 15 (2004), no. 4, 523–533.  MR 2114935 |  Zbl pre02192843
[17] A. Mukhopadhyay, T. N. Shorey, Square free part of products of consecutive integers. Publ. Math. Debrecen 64 (2004), no. 1-2, 79–99.  MR 2035890 |  Zbl 1049.11037
[18] R. Obláth, Über das Produkt fünf aufeinander folgender Zahlen in einer arithmetischen Reihe. Publ. Math. Debrecen 1 (1950), 222–226.  MR 39745 |  Zbl 0038.17901
[19] C. Runge, Über ganzzahlige Lösungen von Gleichungen zwischen zwei Veränderlichen. J. Reine Angew. Math. 100 (1887), 425–435.  JFM 19.0076.03
[20] N. Saradha, T. N. Shorey, Almost squares and factorisations in consecutive integers. Compositio Math. 138 (2003), no. 1, 113–124.  MR 2002956 |  Zbl 1038.11020
[21] N. Saradha, T. N. Shorey, Almost squares in arithmetic progression. Compositio Math. 138 (2003), no. 1, 73–111.  MR 2002955 |  Zbl 1036.11007
[22] A. Schinzel, W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers. Acta Arith. 4 (1958), 185–208; erratum 5 (1958), 259. Article |  MR 106202 |  Zbl 0082.25802
[23] T. N. Shorey, Exponential Diophantine equations involving products of consecutive integers and related equations. Number theory, Trends Math., Birkhäuser, Basel, 2000, pp. 463–495.  MR 1764814 |  Zbl 0958.11026
[24] T. N. Shorey, Powers in arithmetic progressions. III. The Riemann zeta function and related themes: papers in honour of Professor K. Ramachandra, Ramanujan Math. Soc. Lect. Notes Ser., vol. 2, Ramanujan Math. Soc., Mysore, 2006, pp. 131–140.  MR 2335192 |  Zbl 1127.11027
[25] T. N. Shorey, R. Tijdeman, Some methods of Erdős applied to finite arithmetic progressions. The mathematics of Paul Erdős, I, Algorithms Combin., vol. 13, Springer, Berlin, 1997, pp. 251–267.  MR 1425190 |  Zbl 0874.11035
[26] V. G. Sprindžuk, Reducibility of polynomials and rational points on algebraic curves. Dokl. Akad. Nauk SSSR 250 (1980), no. 6, 1327–1330.  MR 564338 |  Zbl 0447.12010
[27] W. Stein, Sage: Open Source Mathematical Software (Version 2.10.2). The Sage Group, 2008, http://www.sagemath.org.
[28] S. Tengely, Note on a paper “An extension of a theorem of Euler” by Hirata-Kohno et al. arXiv:0707.0596v1 [math.NT]. arXiv |  Zbl pre05354500
[29] The PARI Group, Bordeaux, PARI/GP, version 2.3.3, 2005, available from http://pari.math.u-bordeaux.fr/.
[30] P. G. Walsh, A quantitative version of Runge’s theorem on Diophantine equations. Acta Arith. 62 (1992), no. 2, 157–172. Article |  Zbl 0769.11017