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 ...

Motivated by the problem of determining the structure of integral points on subvarieties of semiabelian varieties defined over finite fields, we prove a quantifier elimination result for certain modules over finite simple extensions of the integers given together with predicates for orbits of the distinguished generator of the ring.Remark 2.2. For fixed R = Z[F ], the following observations are immediate consequences of the definitions:• The class of F -spaces is closed under taking products and passing to quotients by F -subspaces.

The notion of a prolongation of an algebraic variety is developed in an abstract setting that generalizes the difference and (Hasse) differential contexts. An interpolating map that compares these prolongation spaces with algebraic jet spaces is introduced and studied.

Building on the abstract notion of prolongation developed by Moosa and Scanlon (‘Jet and prolongation spaces’, J. Inst. Math. Jussieu 9 (2010), 391–430), the theory of iterative Hasse–Schmidt rings and schemes is introduced, simultaneously generalizing difference and (Hasse–Schmidt) differential rings and schemes. This work provides a unified formalism for studying difference and differential algebraic geometry, as well as other related geometries. As an application, Hasse–Schmidt jet spaces are constructed generally, allowing the development of the theory for arbitrary systems of algebraic partial difference/differential equations, where constructions by earlier authors applied only to the finite‐dimensional case. In particular, it is shown that under appropriate separability assumptions a Hasse–Schmidt variety is determined by its jet spaces at a point.

L'accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http://jtnb.cedram. org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Journal de Théorie des Nombres de Bordeaux 31 (2019), 101-130 F-sets and finite automata par Jason BELL et Rahim MOOSA Résumé. On observe que le théorème de Skolem-Mahler-Lech de Derksen est un cas particulier du théorème de Mordell-Lang isotrivial en caractéristique positive dû au second auteur et Scanlon. Cela motive une extension de la notion classique d'un sous-ensemble k-automatique des nombres naturels à celle d'un ensemble F-automatique d'un groupe abélien de type fini Γ équipé d'un endomorphisme F. Dans le contexte de Mordell-Lang, où F est l'action de Frobenius sur un groupe algébrique commutatif G sur un corps fini, et Γ est un sous-groupe F-invariant de G, il est montré que les « Fsous-ensembles » de Γ introduits par le second auteur et Scanlon sont Fautomatiques. Il en découle que lorsque G est semi-abélien et X ⊆ G est une sous-variété fermée, X ∩ Γ est F-automatique. La notion d'un sous-ensemble k-normal des nombres naturels au sens de Derksen est également généralisée au contexte abstrait cité ci-dessus, et il est démontré que les F-sous-ensembles sont F-normaux. En particulier, les ensembles X ∩ Γ qui apparaissent dans le problème de Mordell-Lang sont F-normaux. Cela généralise le théorème de Skolem-Mahler-Lech de Derksen au contexte de Mordell-Lang.

Abstract. Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds for any complex affine Poisson algebra; that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier-Moeglin equivalence in finite GK dimension. A weaker version of the Poisson Dixmier-Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero.

A criterion is given for a strong type in a finite rank stable theory T to be (almost) internal to a given nonmodular minimal type. The motivation comes from results of Campana [5] which give criteria for a compact complex analytic space to be "algebraic" (namely Moishezon). The canonical base property for a stable theory states that the type of the canonical base of a stationary type over a realisation is almost internal to the minimal types of the theory. It is conjectured that every finite rank stable theory has the canonical base property. It is shown here, that in a theory with the canonical base property, if p is a stationary type for which there exists a family of types q b , each internal to a non-locally modular minimal type r, and such that any pair of independent realisations of p are "connected" by the q b 's, then p is almost internal to r.

A differential-algebraic geometric analogue of the Dixmier-Moeglin equivalence is articulated, and proven to hold for D-groups over the constants. The model theory of differentially closed fields of characteristic zero, in particular the notion of analysability in the constants, plays a central role. As an application it is shown that if R is a commutative affine Hopf algebra over a field of characteristic zero, and A is an Ore extension to which the Hopf algebra structure extends, then A satisfies the classical Dixmier-Moeglin equivalence. Along the way it is shown that all such A are Hopf Ore extensions in the sense of [Brown et al., "Connected Hopf algebras and iterated Ore extensions", Journal of Pure and Applied Algebra, 219 (6), 2015].