Heights on groups and small multiplicative dependencies
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- by Jeffrey D. Vaaler PDF
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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.References
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Additional Information
- Jeffrey D. Vaaler
- Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
- MR Author ID: 176405
- Email: vaaler@math.utexas.edu
- 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.
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3180748