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Transactions of the American Mathematical Society

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

 
 

 

Mann pairs


Authors: Lou van den Dries and Ayhan Günaydın
Journal: Trans. Amer. Math. Soc. 362 (2010), 2393-2414
MSC (2010): Primary 03C35, 03C60, 03C98, 11U09
DOI: https://doi.org/10.1090/S0002-9947-09-05020-X
Published electronically: December 8, 2009
Erratum: Trans. 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
MR Author ID: 59845
Email: vddries@math.uiuc.edu

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

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.