Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Heights on groups and small multiplicative dependencies


Author: Jeffrey D. Vaaler
Journal: Trans. Amer. Math. Soc. 366 (2014), 3295-3323
MSC (2010): Primary 11J25, 11R04, 46B04
DOI: https://doi.org/10.1090/S0002-9947-2013-06029-1
Published electronically: November 4, 2013
MathSciNet review: 3180748
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We generalize the absolute logarithmic Weil height from elements of the multiplicative group $ \overline {\mathbb{Q}}^{\times }/\mathrm {Tor}\bigl (\overline {\mathbb{Q}}^{\times }\bigr )$ to finitely generated subgoups of $ \overline {\mathbb{Q}}^{\times }/\mathrm {Tor}\bigl (\overline {\mathbb{Q}}^{\times }\bigr )$. The height of a finitely generated subgroup is shown to equal the volume of a certain naturally occurring, convex, symmetric subset of Euclidean space. This connection leads to a bound on the norm of integer vectors that give multiplicative dependencies among finite sets of algebraic numbers.


References [Enhancements On Off] (What's this?)

  • [1] Daniel Allcock and Jeffrey D. Vaaler, A Banach space determined by the Weil height, Acta Arith. 136 (2009), no. 3, 279-298. MR 2475695 (2009j:11115), https://doi.org/10.4064/aa136-3-6
  • [2] Francesco Amoroso and Sinnou David, Le problème de Lehmer en dimension supérieure, J. Reine Angew. Math. 513 (1999), 145-179 (French, with English summary). MR 1713323 (2001a:11116), https://doi.org/10.1515/crll.1999.058
  • [3] Francesco Amoroso and Evelina Viada, Small points on rational subvarieties of tori, Comment. Math. Helv. 87 (2012), no. 2, 355-383. MR 2914852, https://doi.org/10.4171/CMH/256
  • [4] A. Baker, Linear forms in the logarithms of algebraic numbers. IV, Mathematika 15 (1968), 204-216. MR 0258756 (41 #3402)
  • [5] Daniel Bertrand, Duality on tori and multiplicative dependence relations, J. Austral. Math. Soc. Ser. A 62 (1997), no. 2, 198-216. MR 1433209 (98f:11070)
  • [6] Ethan D. Bolker, A class of convex bodies, Trans. Amer. Math. Soc. 145 (1969), 323-345. MR 0256265 (41 #921)
  • [7] Enrico Bombieri and Walter Gubler, Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006. MR 2216774 (2007a:11092)
  • [8] E. Bombieri and J. Vaaler, On Siegel's lemma, Invent. Math. 73 (1983), no. 1, 11-32. MR 707346 (85g:11049a), https://doi.org/10.1007/BF01393823
  • [9] B. Brindza, On the generators of $ S$-unit groups in algebraic number fields, Bull. Austral. Math. Soc. 43 (1991), no. 2, 325-329. MR 1102085 (92m:11124), https://doi.org/10.1017/S0004972700029129
  • [10] Yann Bugeaud and Kálmán Győry, Bounds for the solutions of unit equations, Acta Arith. 74 (1996), no. 1, 67-80. MR 1367579 (97b:11045)
  • [11] N. L. Carothers, A short course on Banach space theory, London Mathematical Society Student Texts, vol. 64, Cambridge University Press, Cambridge, 2005. MR 2124948 (2005k:46001)
  • [12] J. W. S. Cassels, An introduction to the geometry of numbers, Springer-Verlag, Berlin, 1971. Second printing, corrected; Die Grundlehren der mathematischen Wissenschaften, Band 99. MR 0306130 (46 #5257)
  • [13] L. Fuchs, Abelian groups, International Series of Monographs on Pure and Applied Mathematics, Pergamon Press, New York, 1960. MR 0111783 (22 #2644)
  • [14] L. Hajdu, A quantitative version of Dirichlet's $ S$-unit theorem in algebraic number fields, Publ. Math. Debrecen 42 (1993), no. 3-4, 239-246. MR 1229671 (94e:11118)
  • [15] G. H. Hardy, J. E. Littlewood and G. Pólya,
    Inequalities,
    Cambridge U. Press, 1934.
  • [16] L. Kronecker
    Zwei Sätz über Gleichungen mit ganzzahligen Coefficienten,
    J. Reine Angew. Math. 53 (1857), 173-175.
  • [17] D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), no. 3, 461-479. MR 1503118, https://doi.org/10.2307/1968172
  • [18] Thomas Loher and David Masser, Uniformly counting points of bounded height, Acta Arith. 111 (2004), no. 3, 277-297. MR 2039627 (2005a:11162), https://doi.org/10.4064/aa111-3-5
  • [19] M. A. Lopez and S. Reisner, A special case of Mahler's conjecture, Discrete Comput. Geom. 20 (1998), no. 2, 163-177. MR 1637864 (99h:52009), https://doi.org/10.1007/PL00000076
  • [20] J. H. Loxton and A. J. van der Poorten, Multiplicative dependence in number fields, Acta Arith. 42 (1983), no. 3, 291-302. MR 729738 (86b:11052)
  • [21] K. Mahler,
    Ein Minimalproblem für konvexe Polygone,
    Mathematica (Zutphen) B7 (1939), 118-127.
  • [22] Kurt Mahler, Ein Übertragungsprinzip für konvexe Körper, Časopis Pěst. Mat. Fys. 68 (1939), 93-102 (German). MR 0001242 (1,202c)
  • [23] E. M. Matveev, Linear and multiplicative relations, Mat. Sb. 184 (1993), no. 4, 23-40 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 78 (1994), no. 2, 411-425. MR 1225976 (94j:11060), https://doi.org/10.1070/SM1994v078n02ABEH003477
  • [24] E. M. Matveev, On the index of multiplicative groups of algebraic numbers, Mat. Sb. 196 (2005), no. 9, 59-70 (Russian, with Russian summary); English transl., Sb. Math. 196 (2005), no. 9-10, 1307-1318. MR 2195707 (2007f:11118), https://doi.org/10.1070/SM2005v196n09ABEH003643
  • [25] Peter McMullen, Volumes of projections of unit cubes, Bull. London Math. Soc. 16 (1984), no. 3, 278-280. MR 738519 (85j:52019), https://doi.org/10.1112/blms/16.3.278
  • [26] Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, PWN--Polish Scientific Publishers, Warsaw, 1974. Monografie Matematyczne, Tom 57. MR 0347767 (50 #268)
  • [27] D. G. Northcott, An inequality in the theory of arithmetic on algebraic varieties, Proc. Cambridge Philos. Soc. 45 (1949), 502-509. MR 0033094 (11,390a)
  • [28] Shlomo Reisner, Random polytopes and the volume-product of symmetric convex bodies, Math. Scand. 57 (1985), no. 2, 386-392. MR 832364 (87g:52011)
  • [29] Shlomo Reisner, Zonoids with minimal volume-product, Math. Z. 192 (1986), no. 3, 339-346. MR 845207 (87g:52022), https://doi.org/10.1007/BF01164009
  • [30] Y. Gordon, M. Meyer, and S. Reisner, Zonoids with minimal volume-product--a new proof, Proc. Amer. Math. Soc. 104 (1988), no. 1, 273-276. MR 958082 (89i:52015), https://doi.org/10.2307/2047501
  • [31] Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157 (88k:00002)
  • [32] Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Inc., New York, 1991. MR 1157815 (92k:46001)
  • [33] L. A. Santaló, An affine invariant for convex bodies of $ n$-dimensional space, Portugaliae Math. 8 (1949), 155-161 (Spanish). MR 0039293 (12,526f)
  • [34] Rolf Schneider and Wolfgang Weil, Zonoids and related topics, Convexity and its applications, Birkhäuser, Basel, 1983, pp. 296-317. MR 731116 (85c:52010)
  • [35] H. M. Stark, Further advances in the theory of linear forms in logarithms, Diophantine approximation and its applications (Proc. Conf., Washington, D.C., 1972), Academic Press, New York, 1973, pp. 255-293. MR 0352016 (50 #4504)
  • [36] A. C. Thompson, Minkowski geometry, Encyclopedia of Mathematics and its Applications, vol. 63, Cambridge University Press, Cambridge, 1996. MR 1406315 (97f:52001)
  • [37] A. J. Van der Poorten and J. H. Loxton, Computing the effectively computable bound in Baker's inequality for linear forms in logarithms, Bull. Austral. Math. Soc. 15 (1976), no. 1, 33-57. MR 0417068 (54 #5129)
  • [38] A. J. Van der Poorten and J. H. Loxton, Multiplicative relations in number fields, Bull. Austral. Math. Soc. 16 (1977), no. 1, 83-98. MR 0491537 (58 #10776a)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11J25, 11R04, 46B04

Retrieve articles in all journals with MSC (2010): 11J25, 11R04, 46B04


Additional Information

Jeffrey D. Vaaler
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
Email: vaaler@math.utexas.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-06029-1
Keywords: Weil height
Received by editor(s): March 31, 2012
Received by editor(s) in revised form: November 20, 2012
Published electronically: November 4, 2013
Additional Notes: This research was supported by the National Science Foundation, DMS-06-03282.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society