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Definable sets in Mann pairs. (English) Zbl 1243.03054

Suppose that \(\zeta_1, \dots, \zeta_n\) are roots of unity and \(a_1, \dots, a_n\) are rational numbers such that \(\sum^n_{i=1} a_i \zeta_i = 1\) and \(\sum_{i \in I} a_i \zeta_i \neq 0\) for any non-empty subset \(I \subseteq \{1, \dots, n\}\). H. B. Mann [Mathematika, Lond. 12, 107–117 (1965; Zbl 0138.03102)] proved that, for fixed \(n\), there are only finitely many tuples \(\zeta_1, \dots, \zeta_n\) for which this is possible.
In several articles, the present authors considered generalizations of this question where \(\zeta_1, \dots, \zeta_n\) are elements of a subgroup \(\Gamma\) of the multiplicative group of a big (algebraically closed) field \(\Omega\) and \(a_1, \dots, a_n\) are taken from a sub-field \(\mathbf{k} \subseteq \Omega\). They call \((\mathbf{k}, \Gamma)\) a Mann pair if the result of Mann holds in that setting.
In most of the present article, the authors assume that \((\mathbf{k}, \Gamma)\) is a Mann pair and they describe its model-theoretic properties. More precisely, given such a Mann pair, they consider the structure consisting of the algebraically closed field \(\Omega\) (with the ring language) with two predicates for \(\mathbf{k}\) and \(\Gamma\).
The first main result (Theorems 1.1 and 3.7) describes, under some conditions, when two such structures are elementarily equivalent. The second main result (Theorem 1.2) describes definable sets in such a structure (under a condition on \(\Gamma\)): any formula is equivalent to an existential formula with quantifiers ranging only over \(\mathbf{k}\) and \(\Gamma\).
The article also contains a section with some new examples of Mann pairs.

MSC:

03C60 Model-theoretic algebra
03C98 Applications of model theory
11U09 Model theory (number-theoretic aspects)
03C10 Quantifier elimination, model completeness, and related topics

Citations:

Zbl 0138.03102
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References:

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