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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Mann pairs


Authors: Lou van den Dries and Ayhan Günaydin
Journal: Trans. Amer. Math. Soc. 362 (2010), 2393-2414
MSC (2010): Primary 03C35, 03C60, 03C98, 11U09
Published electronically: December 8, 2009
Erratum: Tran. Amer. Math. Soc. 363 (2011), 5057-5057
MathSciNet review: 2584604
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Abstract: Mann proved in the 1960s that for any $ n\ge 1$ there is a finite set $ E$ of $ n$-tuples $ (\eta_1,\dots, \eta_n)$ of complex roots of unity with the following property: if $ a_1,\dots,a_n$ are any rational numbers and $ \zeta_1,\dots,\zeta_n$ are any complex roots of unity such that $ \sum_{i=1}^n a_i\zeta_i=1$ and $ \sum_{i\in I} a_i \zeta_i\ne 0$ for all nonempty $ I\subseteq \{1,\dots,n\}$, then $ (\zeta_1,\dots,\zeta_n)\in E$. Taking an arbitrary field $ \mathbf{k}$ instead of $ \mathbb{Q}$ and any multiplicative group in an extension field of $ \mathbf{k}$ instead of the group of roots of unity, this property defines what we call a Mann pair $ (\mathbf{k}, \Gamma)$. We show that Mann pairs are robust in certain ways, construct various kinds of Mann pairs, and characterize them model-theoretically.


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Additional Information

Lou van den Dries
Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, Illinois 61801
Email: vddries@math.uiuc.edu

Ayhan Günaydin
Affiliation: Fields Institute, 222 College Street, Second Floor, Toronto, Ontario, Canada M5T 3J1
Email: agunaydi@fields.utoronto.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-09-05020-X
PII: S 0002-9947(09)05020-X
Received by editor(s): November 1, 2007
Published electronically: December 8, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.