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Thue Diophantine equations. (English) Zbl 07220059
Chakraborty, Kalyan (ed.) et al., Class groups of number fields and related topics. Collected papers presented at the first international conference, ICCGNFRT, Harish-Chandra Research Institute, Allahabad, India, September 4–7, 2017. Singapore: Springer (ISBN 978-981-15-1513-2/hbk; 978-981-15-1514-9/ebook). 25-41 (2020).
Summary: This text includes an extended abstract of a keynote talk under the title Families of Thue equations associated with a rank one subgroup of the unit group of a number field given on September 4, 2017 at the Harish-Chandra Research Institute (HRI), Allahabad (India), for the International Conference on Class Groups of Number Fields and Related Topics (ICCGNFRT-2017) based on notes by Kristýna Zemková. Some more information is added, including references, especially to joint works with Claude Levesque.
For the entire collection see [Zbl 1444.11004].
MSC:
11D59 Thue-Mahler equations
11D61 Exponential Diophantine equations
11D41 Higher degree equations; Fermat’s equation
11D25 Cubic and quartic Diophantine equations
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[1] F. Amoroso, D. Masser, U. Zannier, Bounded height in pencils of finitely generated subgroups. Duke Math. J. 166, 2599-2642 (2017), arXiv:1509.04963 [math.NT] · Zbl 1431.11084
[2] A. Baker, Transcendental Number Theory, Cambridge Mathematical Library (1975), 2nd edn (1990) · Zbl 0297.10013
[3] A. Baker, G. Wüstholz, Logarithmic Forms and Diophantine Geometry, vol. 9. New Mathematical Monographs (Cambridge University Press , Cambridge, 2007) · Zbl 1145.11004
[4] M.A. Bennett, Effective measures of irrationality for certain algebraic numbers. J. Austral. Math. Soc. Ser. A 62, 329-344 (1997) · Zbl 0880.11055
[5] M.A. Bennett, Explicit lower bounds for rational approximation to algebraic numbers. Proc. London Math. Soc. 75(3), 63-78 (1997) · Zbl 0879.11038
[6] Y. Bugeaud, Approximation by Algebraic Numbers, vol. 160. Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, 2004) · Zbl 1055.11002
[7] Y. Bugeaud, Linear Forms in Logarithms and Applications, IRMA Lectures in Mathematics and Theoretical Physics, vol. 28. European Mathematical Society (2018). http://www.ems-ph.org/books/book.php?proj_nr=228 · Zbl 1394.11001
[8] Y. Bugeaud, C. Levesque, M. Waldschmidt, Équations de Fermat-Pell-Mahler simultanées. Publ. Math. Debrecen. 79(3-4), 357-366 (2011) · Zbl 1249.11053
[9] P. Corvaja, Integral Points on Algebraic Varieties, vol. 3. Institute of Mathematical Sciences Lecture Notes (Hindustan Book Agency, New Delhi, 2016)
[10] J.-H. Evertse, K. Győry, Unit Equations in Diophantine Number Theory, vol. 146. Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 2015) · Zbl 1339.11001
[11] A. Faisant, L’équation diophantienne du second degré, vol. 1430 of Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], Hermann, Paris, 1991. Collection Formation des Enseignants et Formation Continue. [Collection on Teacher Education and Continuing Education]
[12] E. Fouvry, C. Levesque, M. Waldschmidt, Representation of integers by cyclotomic binary forms. Acta Arithmetica 184.1, 67-86 (2018). arXiv:1712.09019 [math.NT] · Zbl 1417.11028
[13] K. Győry, Représentation des nombres entiers par des formes binaires. Publ. Math. Debrecen 24(3-4), 363-375 (1977) · Zbl 0389.10018
[14] K. Győry, Solving Diophantine, equations by Baker’s theory, in A Panorama of Number Theory or the View from Baker’s Garden (Zürich (Cambridge University Press, Cambridge, 2002), pp. 38-72 · Zbl 1074.11021
[15] C. Heuberger, Parametrized Thue equations –a survey, in Proceedings of the RIMS Symposium “Analytic Number Theory and Surrounding Areas, RIMS Kôkyûroku 2004, vol. 1511 (Kyoto, 2006), pp. 82-91
[16] P.-C. Hu, C.-C. Yang, Distribution Theory of Algebraic Numbers, vol. 45. De Gruyter Expositions in Mathematics (2008)
[17] S. Lang, Fundamentals of Diophantine geometry (Springer, New York, 1983) · Zbl 0528.14013
[18] C. Levesque, M. Waldschmidt, Some remarks on diophantine equations and diophantine approximation. Vietnam J. Math. 393, 343-368 (2011). arXiv:1312.7200 [math.NT] · Zbl 1247.11041
[19] C. Levesque, M. Waldschmidt, Familles d’équations de Thue-Mahler n’ayant que des solutions triviales. Acta Arith. 155, 117-138 (2012). arXiv:1312.7202 [math.NT] · Zbl 1304.11016
[20] C. Levesque, M. Waldschmidt, Approximation of an algebraic number by products of rational numbers and units. J. Aust. Math. Soc. 93(1-2), 121-131 (2013). arXiv:1312.7203 [math.NT] · Zbl 1300.11026
[21] C. Levesque, M. Waldschmidt, Families of cubic Thue equations with effective bounds for the solutions, Number Theory and Related Fields, in Memory of Alf van der Poorten, Springer Proceedings in Mathematics and Statistics, vol. 43, ed by J.M. Borwein et al., pp. 229-243 (2013). arXiv:1312.7204 [math.NT] · Zbl 1358.11050
[22] C. Levesque, M. Waldschmidt, Solving effectively some families of Thue Diophantine equations. Moscow J. Comb. Number Theory 3(3-4), 118-144 (2013). arXiv:1312.7205 [math.NT] · Zbl 1352.11035
[23] C. Levesque, M. Waldschmidt, Familles d’équations de Thue associées à un sous-groupe de rang \(1\) d’unités totalement réelles d’un corps de nombres, in SCHOLAR—a Scientific Celebration Highlighting Open Lines of Arithmetic Research, 2013 (volume dedicated to Ram Murty), CRM collection Contemporary Mathematics”, AMS, vol. 655, 117-134 (2015). http://www.ams.org/books/conm/655/, arXiv: 1505.06656 [math.NT] · Zbl 1394.11030
[24] C. Levesque, M. Waldschmidt, A family of Thue equations involving powers of units of the simplest cubic fields. J. Théor. Nombres Bordx. 27(2), 537-563 (2015). arXiv:1505.06708 [math.NT] · Zbl 1395.11059
[25] C. Levesque, M. Waldschmidt, Solving simultaneously Thue Diophantine equations: almost totally imaginary case, in Ramanujan Mathematical Society, Lecture Notes Series, vol. 23, 137-156 (2016). arXiv: 1505.06653 [math.NT] · Zbl 1418.11053
[26] C. Levesque, M. Waldschmidt, Families of Thue equations associated with a rank one subgroup of the unit group of a number field. Mathematika 63(3), 1060-1080 (2017). arXiv: 1701.01230 [math.NT] · Zbl 1428.11060
[27] D. Masser, Auxiliary Polynomials in Number Theory, vol. 207. Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, 2016)
[28] L.J. Mordell, Diophantine Equations, Pure and Applied Mathematics, vol. 30 (Academic, New York, 1969) · Zbl 0188.34503
[29] N.J. Sloane, The On-line Encyclopedia of Integer Sequences. https://oeis.org/ · Zbl 1044.11108
[30] W.M. Schmidt, Diophantine Approximations and Diophantine Equations, vol. 1467. Lecture Notes in Mathematics (Springer, Berlin, 1991) · Zbl 0754.11020
[31] J.-P. Serre, Lectures on the Mordell-Weil theorem, 3rd edn. 1997. Aspects of Mathematics (Friedr. Vieweg and Sohn, Braunschweig, 1989)
[32] T.N. Shorey, R. Tijdeman, Exponential Diophantine Equations, vol. 87. Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, 1986)
[33] T.N. Shorey, A.J. Van der Poorten, R. Tijdeman, A. Schinzel, Applications of the Gel’fond-Baker method to Diophantine equations, in Transcendence Theory: Advances and Applications (Cambridge, 1977), pp. 59-77 · Zbl 0371.10015
[34] V.G. Sprindžuk, Classical Diophantine Equations, vol. 1559. Lecture Notes in Mathematics (Springer, Berlin, 1993). Translated from the 1982 Russian original
[35] M. Waldschmidt, Diophantine equations and transcendental methods (written by Noriko Hirata), in Transcendental Numbers and Related Topics, RIMS Kôkyûroku, vol. \(599. n^\circ 8\) (Kyoto, 1986), pp. 82-94. http://www.kurims.kyoto-u.ac.jp/ kyodo/kokyuroku/contents/599.html
[36] U. Zannier, Some Applications of Diophantine Approximation to Diophantine Equations with Special Emphasis on the Schmidt Subspace Theorem (Forum Editrice Universitaria Udine srl, Udine, 2003)
[37] U. Zannier, Lecture Notes on Diophantine Analysis, vol. 8. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) (Edizioni della Normale, Pisa, 2009) · Zbl 1186.11001
[38] U.
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