Strongly minimal expansions of algebraically closed fields.

*(English)*Zbl 0773.12005The author constructs a strongly minimal expansion of an algebraically closed field of a given characteristic. This result is a part of the attempt to understand the nature of \(\aleph_ 0\)-categorical theories. As a matter of fact the author has proved a much more general result which implies for example the existence of a strongly minimal set with two field structures of distinct characteristics. It is shown that a strongly minimal expansion of an algebraically closed field that preserves the algebraic closure relation is an expansion by algebraic constants. This result was obtained as a lemma in the classification of the geometries of strongly minimal sets of Zil’ber type.

The author also studies the definable multiplicity property (DMP). For fields, in dimension 1, the DMP signifies that curves remain strongly minimal under specialisation of the parameters. It is shown that any two strongly minimal sets with the definable multiplicity property can be amalgamated to a single strongly minimal set. This provides many examples of strongly minimal sets with interesting geometries. It is proved that every algebraically closed field has the definable multiplicity property. No strongly minimal theory in a countable language with the DMP is maximally strongly minimal.

The author also studies the definable multiplicity property (DMP). For fields, in dimension 1, the DMP signifies that curves remain strongly minimal under specialisation of the parameters. It is shown that any two strongly minimal sets with the definable multiplicity property can be amalgamated to a single strongly minimal set. This provides many examples of strongly minimal sets with interesting geometries. It is proved that every algebraically closed field has the definable multiplicity property. No strongly minimal theory in a countable language with the DMP is maximally strongly minimal.

Reviewer: G.Pestov (Tomsk)

##### Keywords:

field language; strongly minimal expansion; algebraically closed field; definable multiplicity property
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\textit{E. Hrushovski}, Isr. J. Math. 79, No. 2--3, 129--151 (1992; Zbl 0773.12005)

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