Model theory of fields with free operators in characteristic zero

Abstract

Generalising and unifying the known theorems for difference and differential fields, it is shown that for every finite free algebra scheme D over a field A of characteristic zero, the theory of D-fields has a model companion D -CF 0 which is simple and satisfies the Zilber dichotomy for finite-dimensional minimal types.Contents the multiplication of D(R) may be expressed by insisting that certain polynomial relations hold amongst ∂ 0 (x), . . . , ∂ n−1 (x); ∂ 0 (y), . . . , ∂ n−1 (y); ∂ 0 (xy), . . . , ∂ n−1 (xy). In this way, the class of D-fields is easily seen to be first order in the language of rings augmented by unary function symbols for the operators ∂ 0 , . . . , ∂ n−1 . On the other hand, the interpretation of the operators as components of a ring homomorphism permits us to apply ideas from commutative algebra and algebraic geometry to analyze these theories.Our first main theorem is that for any ring scheme D (meeting the requirements set out in Section 3), the theory of D-fields of characteristic zero has a model companion, which we denote by D -CF 0 and call the theory of D-closed fields. Our axiomatization of D -CF 0 follows the geometric style which first appeared in the Chatzidakis-Hrushovski axioms for ACFA [3] and was then extended to differential fields by Pierce and Pillay [14]. Moreover, the proofs will be familiar to anyone who has worked through the corresponding results for difference and differential fields. Following the known proofs for difference and differential fields, we establish a quantifier simplification theorem and show that D -CF 0 is always simple.As noted above, the theory DCF 0 is the quintessential ω-stable theory, but ACFA, the model companion of the theory of difference fields is not even stable. At a technical level, the instability of ACFA may be traced to the failure of quantifier elimination which, algebraically, is due to the non-uniqueness (up to isomorphism) of the extension of an automorphism of a field to an automorphism of its algebraic closure. We show that this phenomenon, namely that instability is tied to the nonuniqueness of extensions of automorphisms, pervades the theory of D-fields. That is, for each D there is a finite list of associated endomorphisms expressible as linear combinations of the basic operators. Since we require that ∂ 0 is the identity map, one of these associated endomorphisms is always the identity map. If there are any others, then the theory of D -CF 0 suffers from instability and the failure of quantifier elimination just as does ACFA. On the other hand, if there are no other associated endomorphisms, then D -CF 0 is stable.The deepest of the fine structural theorems for types in DCF 0 and in ACFA is the Zilber dichotomy for minimal types, first established by Sokolović and Hrushovski for DCF 0 using Zariski geometries [8], for ACFA 0 by Chatzidakis and Hrushovski through a study of ramification [3], and by Chatzidakis, Hrushovski and Peterzil for ACFA in all characteristics using the theory of limit types and a refined form of the ...

Model Theory of Fields With Operators – a Survey

“…The following is a particular case of a setup introduced in [68] by Moosa and Scanlon, and which permits to treat some of the above considered cases in a uniform fashion. Consider a Q-algebra D which is finite-dimensional over Q and has basis {ε 0 , .…”

“…The main result of [68] is that under certain conditions on the algebra D, the theory of D-rings which are integral domains has a model companion, D-CF 0 . The axiomatisation is geometric, once one has introduced the appropriate twisted tangent bundle.…”

“…Moosa and Scanlon show in [68] that the dichotomy theorem holds for finite-dimensional types of SU-rank 1, i.e., types over a set A of tuples a such that the transcendence degree over acl(A) of acl(Aa) is finite. This brings me to another remark.…”

“…By the above mentioned result of Pillay, their result gives in particular a direct proof of the dichotomy for DCF 0 and ACFA 0 . Their proof can be (and was) adapted to the so-called finite-dimensional types of DCFA and of DCF 0,n , D-CF 0 , DCFA n , see [14] and [68], [87]. In the case of DCF 0,n , and later of DCFA, the method of arcspaces introduced in [69] yields the full dichotomy.…”

The Model Companion of Differential Fields With Free Operators

“…A new rather general formalism for the model theory of fields with operators was introduced and developed by the second author and Thomas Scanlon in [7]. That theory treats differential fields and difference fields uniformly, and also allows for a wide variety of other additive operators, or systems of operators, whose multiplicative rules are induced by a ring homomorphism from the field to a finite algebra over the field.…”