Kikyo, Hirotaka; Pillay, Anand The definable multiplicity property and generic automorphisms. (English) Zbl 0967.03030 Ann. Pure Appl. Logic 106, No. 1-3, 263-273 (2000). Let \(T\) be a complete theory with quantifier elimination in a language \(L\) and let \(T_{\sigma}\) denote the (possibly incomplete) expansion \(T \cup \{ \sigma\) is an automorphism} in the suitable language extending \(L\) by a new 1-ary function symbol. A well-known example in this setting concerns \(T = \text{ACF}\), the theory of algebraically closed fields (with a fixed characteristic to ensure completeness); \(\text{ACF}_{\sigma}\) has a model companion, called ACFA and admitting a nice axiomatization. For an arbitrary \(T\), one may ask under which conditions \(T_{\sigma}\) has a model companion. Here the authors deal with this problem when \(T\) is strongly minimal (so ACF lies in this framework). The leading conjecture is that, under the strongly minimal assumption, a model companion exists if and only if \(T\) satisfies the definable multiplicity property DMP (a condition essentially requiring the definability of the Morley degree). The implication from right to left is known, and the authors show here the converse implication when \(T\) is a finite cover of a strongly minimal theory which does have the DMP, in particular when \(T\) is locally modular. Notably, nicer results are obtained when working with \(\geq 2\) automorphisms \(\sigma_i\) \((i \in I)\). It turns out that, in this case, for every strongly minimal \(T\), \(T \cup \{\sigma_i\) is an automorphism\(: i \in I \}\) has a model companion exactly when \(T\) has the DMP. In particular, for a given strongly minimal \(T\), the existence of a model companion for the expansion of \(T\) by 2 automorphisms implies the existence of a model companion when infinitely many automorphisms occur. Reviewer: C.Toffalori (Camerino) Cited in 1 ReviewCited in 10 Documents MSC: 03C45 Classification theory, stability, and related concepts in model theory Keywords:generic automorphisms; strongly minimal theory; model companion; definable multiplicity property; Morley degree PDFBibTeX XMLCite \textit{H. Kikyo} and \textit{A. Pillay}, Ann. Pure Appl. Logic 106, No. 1--3, 263--273 (2000; Zbl 0967.03030) Full Text: DOI References: [1] Chatzidakis, Z.; Hrushovski, E., The model theory of difference fields, Trans. AMS, 351, 2997-3071 (1999) · Zbl 0922.03054 [2] Chatzidakis, Z.; Pillay, A., Generic structures and simple theories, Ann. Pure Appl. Logic, 95, 71-92 (1998) · Zbl 0929.03043 [3] Hrushovski, E., Strongly minimal expansions of algebraically closed fields, Israel J. Math., 79, 129-151 (1992) · Zbl 0773.12005 [4] Hrushovski, E., Locally modular regular types, (Baldwin, J. T., Classification Theory, Proceedings of Chicago 1985. Classification Theory, Proceedings of Chicago 1985, Lecture Notes in Mathematics, vol. 1292 (1987), Springer: Springer Berlin) · Zbl 0643.03024 [5] H. Kikyo, Model companions of theories with an automorphism, J. Symbolic Logic (2000), in preparation.; H. Kikyo, Model companions of theories with an automorphism, J. Symbolic Logic (2000), in preparation. · Zbl 0967.03031 [6] Lascar, D., Les beaux automorphismes, Arch. Math. Logic, 31, 55-68 (1991) · Zbl 0766.03022 [7] Pillay, A., Geometric Stability Theory (1996), Oxford University Press: Oxford University Press Oxford · Zbl 0871.03023 [8] Poizat, B., Paires de structures stables, J. Symbolic Logic, 48, 239-249 (1983) · Zbl 0525.03023 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.