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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Mann pairs
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by Lou van den Dries and Ayhan Günaydın PDF
Trans. Amer. Math. Soc. 362 (2010), 2393-2414 Request permission

Erratum: Trans. Amer. Math. Soc. 363 (2011), 5057-5057.


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:
  • Ayhan Günaydın
  • Affiliation: Fields Institute, 222 College Street, Second Floor, Toronto, Ontario, Canada M5T 3J1
  • Email:
  • Received by editor(s): November 1, 2007
  • Published electronically: December 8, 2009
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 362 (2010), 2393-2414
  • MSC (2010): Primary 03C35, 03C60, 03C98, 11U09
  • DOI:
  • MathSciNet review: 2584604