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Recursive spectra of strongly minimal theories satisfying the Zilber trichotomy. (English) Zbl 1349.03033

Summary: We conjecture that for a strongly minimal theory \(T\) in a finite signature satisfying the Zilber Trichotomy, there are only three possibilities for the recursive spectrum of \(T\): all countable models of \(T\) are recursively presentable; none of them are recursively presentable; or only the zero-dimensional model of \(T\) is recursively presentable. We prove this conjecture for disintegrated (formerly, trivial) theories and for modular groups. The conjecture also holds via known results for fields. The conjecture remains open for finite covers of groups and fields.

MSC:

03C57 Computable structure theory, computable model theory
03C15 Model theory of denumerable and separable structures
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References:

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