### Abstract

Mann proved in the 1960s that for any n ≥ 1 there is a finite set E of n-tuples (n_{1},⋯,n_{n}) of complex roots of unity with the following property: if a_{1},⋯,a_{n} are any rational numbers and ζ_{1},⋯,ζ_{n} are any complex roots of unity such that ∑_{i=1}^{n} a_{i}ζ _{i= 1} and ∑_{i∈I} a_{i}ζ_{i},=0 for all nonempty I ⊆{1,⋯,n},then (Ζ_{1},⋯, Ζ_{n}) ∈ E. Taking an arbitrary field k instead of ℚ and any multiplicative group in an extension field of k instead of the group of roots of unity, this property defines what we call a Mann pair (k, Γ). We show that Mann pairs are robust in certain ways, construct various kinds of Mann pairs, and characterize them model-theoretically.

Original language | English (US) |
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Pages (from-to) | 2393-2414 |

Number of pages | 22 |

Journal | Transactions of the American Mathematical Society |

Volume | 362 |

Issue number | 5 |

DOIs | |

State | Published - May 1 2010 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*362*(5), 2393-2414. https://doi.org/10.1090/S0002-9947-09-05020-X

**Mann pairs.** / van den Dries, Lou; Günaydin, Ayhan.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 362, no. 5, pp. 2393-2414. https://doi.org/10.1090/S0002-9947-09-05020-X

}

TY - JOUR

T1 - Mann pairs

AU - van den Dries, Lou

AU - Günaydin, Ayhan

PY - 2010/5/1

Y1 - 2010/5/1

N2 - Mann proved in the 1960s that for any n ≥ 1 there is a finite set E of n-tuples (n1,⋯,nn) of complex roots of unity with the following property: if a1,⋯,an are any rational numbers and ζ1,⋯,ζn are any complex roots of unity such that ∑i=1n aiζ i= 1 and ∑i∈I aiζi,=0 for all nonempty I ⊆{1,⋯,n},then (Ζ1,⋯, Ζn) ∈ E. Taking an arbitrary field k instead of ℚ and any multiplicative group in an extension field of k instead of the group of roots of unity, this property defines what we call a Mann pair (k, Γ). We show that Mann pairs are robust in certain ways, construct various kinds of Mann pairs, and characterize them model-theoretically.

AB - Mann proved in the 1960s that for any n ≥ 1 there is a finite set E of n-tuples (n1,⋯,nn) of complex roots of unity with the following property: if a1,⋯,an are any rational numbers and ζ1,⋯,ζn are any complex roots of unity such that ∑i=1n aiζ i= 1 and ∑i∈I aiζi,=0 for all nonempty I ⊆{1,⋯,n},then (Ζ1,⋯, Ζn) ∈ E. Taking an arbitrary field k instead of ℚ and any multiplicative group in an extension field of k instead of the group of roots of unity, this property defines what we call a Mann pair (k, Γ). We show that Mann pairs are robust in certain ways, construct various kinds of Mann pairs, and characterize them model-theoretically.

UR - http://www.scopus.com/inward/record.url?scp=77950874735&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950874735&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-09-05020-X

DO - 10.1090/S0002-9947-09-05020-X

M3 - Article

AN - SCOPUS:77950874735

VL - 362

SP - 2393

EP - 2414

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 5

ER -