Mann pairs

Lou van den Dries, Ayhan Günaydin

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)2393-2414
Number of pages22
JournalTransactions of the American Mathematical Society
Volume362
Issue number5
DOIs
StatePublished - May 1 2010

Fingerprint

Roots of Unity
Field extension
Finite Set
Multiplicative
Arbitrary
Model

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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

In: Transactions of the American Mathematical Society, Vol. 362, No. 5, 01.05.2010, p. 2393-2414.

Research output: Contribution to journalArticle

van den Dries, Lou ; Günaydin, Ayhan. / Mann pairs. In: Transactions of the American Mathematical Society. 2010 ; Vol. 362, No. 5. pp. 2393-2414.
@article{14fab20d8913471cae9fc572e3295c09,
title = "Mann pairs",
abstract = "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.",
author = "{van den Dries}, Lou and Ayhan G{\"u}naydin",
year = "2010",
month = "5",
day = "1",
doi = "10.1090/S0002-9947-09-05020-X",
language = "English (US)",
volume = "362",
pages = "2393--2414",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "5",

}

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 -