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Strongly minimal expansions of algebraically closed fields. (English) Zbl 0773.12005
The 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.
Reviewer: G.Pestov (Tomsk)

12L12 Model theory of fields
03C60 Model-theoretic algebra
Full Text: DOI
[1] John Baldwin,\(\alpha\) T is finite for 1-categorical T, Trans. Amer. Math. Soc.181 (1973), 37–51.
[2] Lau Van den Dries,Model theory of fields: decidability, and bounds for polynomial ideals, Doctoral Thesis, Univ. Utrecht, 1978.
[3] E. Hrushovski and J. Loveys,Structure of strongly minimal modules, in preparation. · Zbl 1197.03040
[4] E. Hrushovski,A strongly minimal set, to appear. · Zbl 0804.03020
[5] E. Hrushovski and A. Pillay,Weakly normal groups, Logic Colloquium 85, North-Holland, Amsterdam, 1986. · Zbl 0636.03028
[6] Serge Lang,Introduction to Algebraic Geometry, Interscience, New York, 1964. · Zbl 0211.38501
[7] Dave Marker,Semi-algebraic expansions of C, Trans. Amer. Math. Soc.320 (1989), 581–592. · Zbl 0778.03009 · doi:10.2307/2001690
[8] Bruno Poizat,Missionary mathematics, J. Symb. Logic53 (1988), 137–145.
[9] Bruno Poizat,Une thĂ©orie de Galois imaginaire, J. Symb. Logic48 (1983), 1151–1170. · Zbl 0537.03023 · doi:10.2307/2273680
[10] A. Pillay,An Introduction to Stability Theory, Oxford University Press, 1983. · Zbl 0526.03014
[11] A. Seidenberg,Constructions in algebra, Trans. Amer. Math. Soc.197 (1974), 273–313. · Zbl 0356.13007 · doi:10.1090/S0002-9947-1974-0349648-2
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