Grouplike minimal sets in ACFA and in T_A

TL;DR

This paper generalizes the characterization of definable groups from ACFA to T_A, a broader model-theoretic setting, using algebraic geometry and model theory to analyze minimal formulae involving algebraic curves and rational functions.

Contribution

It provides a model-theoretic generalization of group characterizations from ACFA to T_A, extending the scope of definability results in algebraically closed fields with endomorphisms.

Findings

Characterization of minimal formulae in ACFA involving algebraic curves.

Extension of group definability results from ACFA to T_A.

Use of algebraic geometry to support model-theoretic results.

Abstract

This paper began as a generalization of a part of the author's PhD thesis about ACFA and ended up with a characterization of groups definable in T_A. The thesis concerns minimal formulae in ACFA of the form "p lies on an algebraic curve A and s(x)=f(x)" for some dominant rational function f from A to s(A), where s is the automorphism. These are shown to be uniform in the Zilber trichotomy, and the pairs (A,f) that fall into each of the three cases are characterized. These characterizations are definable in families. This paper covers approximately half of the thesis, namely those parts of it which can be made purely model-theoretic by moving from ACFA, the model companion of the class of algebraically closed fields with an endomorphism, to T_A, the model companion of the class of models of an arbitrary totally-transcendental theory T with an injective endomorphism, if this model-companion exists. A T_A analog of the characterization of groups definable in ACFA is obtained in the process. The full characterization is obtained from these intermediate results with heavy use of algebraic geometry: see the thesis or the forthcoming paper "Around Lattes functions".