Mann pairs

Van Den Dries L., GÜNAYDIN A.

Transactions of the American Mathematical Society, cilt.362, sa.5, ss.2393-2414, 2010 (SCI-Expanded, Scopus)

Özet

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. © 2009 American Mathematical Society.