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Transactions of the American Mathematical Society
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
Published electronically: November 4, 2013
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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.


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-06029-1
PII: S 0002-9947(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.