Model theory of fields with operators- a survey
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paper_title: A note on canonical bases and one-based types in supersimple theories

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This paper studies the CBP, a model-theoretic property first discovered by Pillay and Ziegler. We first show a general decomposition result of types of canonical bases, which one can think of as a sort of primary decomposition. This decomposition is then used to show that existentially closed difference fields of any characteristic have the CBP. We also derive consequences of the CBP, and use these results for applications to differential and difference varieties, and algebraic dynamics.
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paper_title: Effective bounds for the number of transcendental points on subvarieties of semi-abelian varieties

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Let A be a semi-abelian variety, and X a subvariety of A, both denned over a number field. Assume that X does not contain X\ + X2 for any positive-dimensional subvarieties X\, X2 of A. Let r be a subgroup of A(C) of finite rational rank. We give doubly exponential bounds for the size of (X n 0\X(Q). Among the ingredients is a uniform bound, doubly exponential in the data, on finite sets which are quantifier-free definable in differentially closed fields. We also give uniform bounds on X n V in the case where X contains no translate of any semi-abelian subvariety of A and r is a subgroup of A(C) of finite rational rank which has trivial intersection with A(Q). (Here A is assumed to be defined over a number field, but X need not be.) algebraic torus (C*)2, and both A and X are defined over a number field. The question (asked by Bombieri) was to find uniform bounds for the number of generic points of X fl T, namely those points which are not rational over the al gebraic closure of Q. Buium obtained uniform bounds (of an iterated exponential nature) in the data (deg(X)9 rank(T)). Buium used differential algebraic methods. The aim in the present paper is to improve the bounds to being doubly exponen tial in the relevant data, as well as to treat the more general case of semi-abelian varieties. We are also able to slightly weaken the hypothesis on X. The methods are also differential-algebraic, closely following Buium, but there are additional model-theoretic ingredients (coming from the theory of commutative groups of finite Morley rank), and there is also one more key observation regarding the differential-algebraic set-up which yields the doubly exponential bounds. Before stating the results, let us discuss the nature of the data which for us will be quite naive. We work for now over the complex numbers. Let A be a semi-abelian variety of dimension n9 and X a subvariety of dimension ra, both defined over Q. Denote Q by k. Assume A to be given as a locally closed
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paper_title: The Mordell-Lang conjecture for function fields

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We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin’s homomorphism and of a certain analog in characteristic p. Department of Mathematics, Massachusetts Institute of Technology, 2-277, Cambridge, Massachusetts 02139 Current address: Department of Mathematics, Hebrew University, Jerusalem, Israel E-mail address: ehud@math.mit.edu
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paper_title: Difference fields and descent in algebraic dynamics, II

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We draw a connection between the model-theoretic notions of modularity (or onebasedness), orthogonality and internality, as applied to difference fields, and questions of descent in in algebraic dynamics. In particular we prove in any dimension a strong dynamical version of Northcott’s theorem for function fields, answering a question of Szpiro and Tucker and generalizing a theorem of Baker’s for the projective line. The paper comes in three parts. This first part contains an exposition some of the main results of the model theory of difference fields, and their immediate connection to questions of descent in algebraic dynamics. We present the model-theoretic notion of internality in a context that does not require a universal domain with quantifier-elimination. We also note a version of canonical heights that applies well beyond polarized algebraic dynamics. Part II sharpens the structure theory to arbitrary base fields and constructible maps where in part I we emphasize finite base change and correspondences. Part III will include precise structure theorems related to the Galois theory considered here, and will enable a sharpening of the descent results for non-modular dynamics.
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paper_title: The abc theorem for commutative algebraic groups in characteristic p

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Buium proved what he called the abc theorem for abelian varieties over function fields in characteristic zero [3]. Using methods of algebraic model theory we prove an analog of his theorem for commutative algebraic groups in characteristic p. In what follows, k is an algebraically closed field of characteristic p, C is a smooth projective curve over k, and K = k(C) is the function field of C. Identify each point x ∈ C(k) with its corresponding valuation vx := ordx on K. The purpose of this paper is to demonstrate: Theorem 0.1. Let A be an abelian variety over K. Let f : A → P be a rational function. Let r be a positive integer, then there is a bound Br ∈ Z such that for any P ∈ A(K) either there are some a ∈ A(K) and Q ∈ A(K) such that f(Q) ∈ {0,∞} and P = Q+ [p]a or vx(f(P )) ≤ Br for any x ∈ C(k). Theorem 0.1 is the characteristic p analog of Buium’s abc theorem [3]. Our proof works for more general commutative algebraic groups and for distances computed to subvarieties of codimension greater than one. More general statements are in Section 1. Our proof follows the general form of Buium’s proof. We construct uniformly continuous homomorphisms from A(K) to some unipotent group with kernel [p]A(K) using differential algebra. The maps allow us to bound the distance from points in A(K)\[p]A(K) to 0. We then use a general lemma on approximations to reduce the theorem to the case of bounding the distance to a point. These estimates dovetail to give a proof of the theorem. Using Hrushovski’s Mordell-Lang theorem [8] we can give qualitative estimates on the growth of Br with r in many cases. The results of this paper formed a chapter of my Ph. D. thesis [14] written under the direction of E. Hrushovski whom I now thank for his advice. I thank B. Mazur for his advice and for insisting that a more geometric presentation of these arguments be given (though I admit that the argument is still not geometric). I thank D. Abramovich and J. F. Voloch for their comments on an earlier version. 1. A More General Formulation We give now a definition of the distance to a subvariety. Definition 1.1. Let Y ⊆ AK be a subvariety of affine space over K. Let P ∈ A(K). Let x ∈ C(k). Then the distance from P to Y is dvx(P, Y ) := min{vx(f(P )) : f ∈ IY ∩ OK,vx [X1, . . . , Xn]}
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paper_title: The Manin-Mumford conjecture and the model theory of difference fields

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Abstract Using methods of geometric stability (sometimes generalized to finite S1 rank), we determine the structure of Abelian groups definable in ACFA, the model companion of fields with an automorphism. We also give general bounds on sets definable in ACFA. We show that these tools can be used to study torsion points on Abelian varieties; among other results, we deduce a fairly general case of a conjecture of Tate and Voloch on p -adic distances of torsion points from subvarieties.
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paper_title: Differentially closed fields: a model-theoretic tour

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Publisher Summary This chapter discusses background model theoretic work in differential algebra and the notion of model completions. Model theory and its applications exhibit very specific and concrete results following in a natural way from logical and model theoretic properties of theories and very general principles. Abraham Robison, using Seidenberg's elimination theory, showed that the theory of differential fields possessed a model completion, and then making use of general properties of theories with model completions, he was able to give purely model theoretic proofs of certain results about differential fields. The theory of algebraically closed fields is the model completion of the theory of fields; the theory of real closed ordered fields is the model completion of the theory of ordered fields; the theory of Hensel fields is the model completion of the theory of valued fields; the theory of differentially closed fields is the model completion of the theory of differential fields. The chapter also highlights some properties of differential closures and some immediate consequences.
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paper_title: The model theory of differential fields with finitely many commuting derivations

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In this paper we set out the basic model theory of differential fields of characteristic 0, which have finitely many commuting derivations. We give axioms for the theory of differentially closed differential fields with mn derivations and show that this theory is co-stable, model complete, and quantifier-eliminable, and that it admits elimination of imaginaries. We give a characterization of forking and compute the rank of this theory to be co" + 1. ?
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paper_title: FIELDS WITH SEVERAL COMMUTING DERIVATIONS

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For every natural number $m$, the existentially closed models of the theory of fields with $m$ commuting derivations can be given a first-order geometric characterization in several ways. In particular, the theory of these differential fields has a model-companion. The axioms are that certain differential varieties determined by certain ordinary varieties are nonempty. There is no restriction on the characteristic of the underlying field.
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paper_title: Geometric Axioms for Differentially Closed Fields with Several Commuting Derivations

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In the proof of Lemma 2.6 (2) the iteration of the map {\tau} was not performed properly and in fact the lemma is wrong; a counterexample is given by f = \bar{x}_1and k = 2. This error does not, however, affect the geometric characterization given in Theorem 3.4 but only the attempt in Theorem 4.3 to express it as a first-order set of axioms. That attempt is incorrect; the main problem being that in general {\tau}V(f_1,..., f_s) 6= V(f_1..., f_s, {\tau}f_1,..., {\tau}f_s). But a different, indeed simpler, set of first-order axioms, which we will now describe, does express the geometric characterization.
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paper_title: Note sur les corps différentiellement clos valués

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In the paper by Guzy and Point, Differential topological fields, the model-completion (OVF)D∗ of the theory of ordered valued differential fields OVFD is established. Models of this theory are closed ordered differential fields (the theory CODF was studied by Singer) which have a non-trivial convex (for the order) subring as valuation ring. Here we prove the valued analogue of a result of Singer: if K is a model of (OVF)D∗ then K(i) (i2=−1) is a model of the theory of differentially closed valued fields which is the model-completion of the theory of non-trivially valued differential fields of characteristic zero. To cite this article: N. Guzy, C. R. Acad. Sci. Paris, Ser. I 341 (2005).
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paper_title: The theory of closed ordered differential fields with m commuting derivations

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We generalize the work of M. Singer (1978) on the theory of closed ordered differential fields to the case of m-ODF, the theory of ordered fields equipped with m commuting derivations. We give an algebraic axiomatization of the model completion (denoted by m-CODF) of this theory and we can immediately deduce that m-CODF has quantifier elimination in the natural language of ordered Δ-rings. To cite this article: C. Riviere, C. R. Acad. Sci. Paris, Ser. I 343 (2006).
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paper_title: Some elements of Lie-differential algebra and a uniform companion for large Lie-differential fields

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Abstract In this paper, we develop the beginning of Lie-differential algebra, in the sense of Kolchin (see [E.R. Kolchin, Differential algebra and algebraic groups, in: Pure and Applied Mathematics, vol. 54, Academic Press, 1973]) by using tools introduced by Hubert in [E. Hubert, Differential algebra for derivations with nontrivial commutation rules, J. Pure Appl. Algebra 200 (2005) 163–190]. In particular it allows us to adapt the results of Tressl (see [M. Tressl, A uniform companion for large differential fields of characteristic zero, Trans. Amer. Math. Soc. 357 (10) (2005) 3933–3951]) by showing the existence of a theory ( UC Lie , m ) of Lie-differential fields of characteristic zero. This theory will serve as a model companion for every theory of large and Lie-differential fields extending a model complete theory of pure fields. As an application, we introduce the Lie counterpart of classical theories of differential fields in several commuting derivations.
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paper_title: The model theory of differential fields revisited

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The intent of this article is to provide a general and elementary account of the model theory of differential fields, collecting together various results (many without proof) and offering a few algebraic details for the logician reader. The first model-theoretic look at differential fields was taken by Abraham Robinson in the context of model completeness, while later developments have served to illustrate concepts developed by Morley and Shelah.
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paper_title: 0-D-valued fields

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In [12]. T. Scanlon proved a quantifier elimination result for valued Z)-fields in a three-sorted language by using angular component functions. Here we prove an analogous theorem in a different language Jz?2 which was introduced by F. Delon in her thesis. This language allows us to lift the quantifier elimination result to a one-sorted language by a process described in the Appendix. As a byproduct, we state and prove a "positivstellensatz,, theorem for the differential analogue of the theory of real-series closed fields in the valued D-field setting. ?
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paper_title: Geometrical Axiomatization for Model Complete Theories of Differential Topological Fields

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In this paper we give a differential lifting principle which provides a general method to geometrically axiomatize the model companion (if it exists) of some theories of differential topological fields. The topological fields we consider here are in fact topological systems in the sense of [vdD1, 2.11], and the lifting principle we develop is a generalization of the geometric axiomatization of the theory DCF0 given by D. Pierce and A. Pillay in [PP]. Moreover it provides a geometric alternative to the axiomatizations obtained in [Tr] and [GP] where the authors also build general schemes of axioms for some model complete theories of differential fields. We first characterize the existentially closed models of a given theory of differential topological fields and then, under an additional hypothesis of largeness, we show how to modify this characterization to get a general scheme of first-order axioms for the model companion of any large theory of differential topological fields. We conclude with an application of this lifting principle proving that in existentially closed models of a large theory of differential topological fields, the jet-spaces are dense in their ambient topological space. 1 Basic algebraic geometry In what follows K is a field of characteristic zero, Ω is a sufficiently saturated elementary extension of K (in particular Ω is of infinite transcendence degree over K) and Ω denotes the algebraic closure of Ω. ∗The first author is supported by a FNRS grant. The second author is supported by a FRIA grant.
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paper_title: TOPOLOGICAL DIFFERENTIAL FIELDS

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Abstract We consider first-order theories of topological fields admitting a model-completion and their expansion to differential fields (requiring no interaction between the derivation and the other primitives of the language). We give a criterion under which the expansion still admits a model-completion which we axiomatize. It generalizes previous results due to M. Singer for ordered differential fields and of C. Michaux for valued differential fields. As a corollary, we show a transfer result for the N I P property. We also give a geometrical axiomatization of that model-completion. Then, for certain differential valued fields, we extend the positive answer of Hilbert’s seventeenth problem and we prove an Ax–Kochen–Ershov theorem. Similarly, we consider first-order theories of topological fields admitting a model-companion and their expansion to differential fields, and under a similar criterion as before, we show that the expansion still admits a model-companion. This last result can be compared with those of M. Tressl: on one hand we are only dealing with a single derivation whereas he is dealing with several, on the other hand we are not restricting ourselves to definable expansions of the ring language, taking advantage of our topological context. We apply our results to fields endowed with several valuations (respectively several orders).
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paper_title: The model theory of m‐ordered differential fields

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In his Ph.D. thesis [7], L. van den Dries studied the model theory of fields (more precisely domains) with finitely many orderings and valuations where all open sets according to the topology defined by an order or a valuation is globally dense according with all other orderings and valuations. Van den Dries proved that the theory of these fields is companionable and that the theory of the companion is decidable (see also [8]). ::: ::: ::: ::: In this paper we study the case where the fields are expanded with finitely many orderings and an independent derivation. We show that the theory of these fields still admits a model companion in the language L = {+, –, ·, D, <1, …, <m, 1, 0}. We denote this model companion by CODFm and give a geometric axiomatization of this theory which uses basic notions of algebraic geometry and some generalized open subsets which appear naturally in this context. This axiomatization allows to recover (just by putting m = 1) the one given in [4] for the theory CODF of closed ordered differential fields. Most of the technics we use here are already present in [2] and [4]. ::: ::: ::: ::: Finally, we prove that it is possible to describe the completions of CODFm and to obtain quantifier elimination in a slightly enriched (infinite) language. This generalizes van den Dries' results in the “derivation free” case. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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paper_title: Further notes on cell decomposition in closed ordered differential fields

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In [T. Brihaye, C. Michaux, C. Riviere, Cell decomposition and dimension function in the theory of closed ordered differential fields, Ann. Pure Appl. Logic (in press).] the authors proved a cell decomposition theorem for the theory of closed ordered differential fields (CODF) which generalizes the usual Cell Decomposition Theorem for o-minimal structures. As a consequence of this result, a well-behaving dimension function on definable sets in CODF was introduced. Here we continue the study of this cell decomposition in CODF by proving three additional results. We first discuss the relation between the δ-cells introduced in the above-mentioned reference and the notion of Kolchin polynomial (or dimensional polynomial) in differential algebra. We then prove two generalizations of classical decomposition theorems in o-minimal structures. More exactly we give a theorem of decomposition into definably d-connected components (d-connectedness is a weak differential generalization of usual connectedness w.r.t. the order topology) and a differential cell decomposition theorem for a particular class of definable functions in CODF.
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paper_title: Cell decomposition and dimension function in the theory of closed ordered differential fields

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Abstract In this paper we develop a differential analogue of o-minimal cell decomposition for the theory C O D F of closed ordered differential fields. Thanks to this differential cell decomposition we define a well-behaving dimension function on the class of definable sets in C O D F . We conclude this paper by proving that this dimension (called δ -dimension) is closely related to both the usual differential transcendence degree and the topological dimension associated, in this case, with a natural differential topology on ordered differential fields.
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paper_title: Model theory of difference fields, II: Periodic ideals and the trichotomy in all characteristics

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We classify all possible combinatorial geometries associated with one-dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the automorphism by its nth power. We show that these reducts admit a successively smoother theory as n becomes large; and we succeed in defining a limit structure to these reducts, or rather to the structure they induce on one-dimensional sets. This limit structure is shown to be a Zariski geometry in (roughly) the sense of Hrushovski and Zil'ber. The trichotomy is thus obtained for the limit structure as a consequence of a general theorem, and then shown to be inherited by the original theory. 2000 Mathematical Subject Classification: 03C60; (primary) 03C45, 03C98, 08A35, 12H10 (secondary)
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paper_title: Generic automorphisms of fields

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Abstract It is shown that the theory of fields with an automorphism has a decidable model companion. Quantifier-elimination is established in a natural language. The theory is intimately connected (via fixed fields) to Ax's theory of pseudofinite fields, and analogues are obtained for most of Ax's classical results. Some indication is given of the connection to nonstandard Frobenius maps.
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paper_title: On subgroups of semi-abelian varieties defined by difference equations

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Consider the algebraic dynamics on a torus T=G_m^n given by a matrix M in GL_n(Z). Assume that the characteristic polynomial of M is prime to all polynomials X^m-1. We show that any finite equivariant map from another algebraic dynamics onto (T,M) arises from a finite isogeny T \to T. A similar and more general statement is shown for Abelian and semi-abelian varieties. In model-theoretic terms, our result says: Working in an existentially closed difference field, we consider a definable subgroup B of a semi-abelian variety A; assume B does not have a subgroup isogenous to A'(F) for some twisted fixed field F, and some semi-Abelian variety A'. Then B with the induced structure is stable and stably embedded. This implies in particular that for any n>0, any definable subset of B^n is a Boolean combination of cosets of definable subgroups of B^n. This result was already known in characteristic 0 where indeed it holds for all commutative algebraic groups ([CH]). In positive characteristic, the restriction to semi-abelian varieties is necessary.
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paper_title: Geometric Axioms for Differentially Closed Fields with Several Commuting Derivations

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In the proof of Lemma 2.6 (2) the iteration of the map {\tau} was not performed properly and in fact the lemma is wrong; a counterexample is given by f = \bar{x}_1and k = 2. This error does not, however, affect the geometric characterization given in Theorem 3.4 but only the attempt in Theorem 4.3 to express it as a first-order set of axioms. That attempt is incorrect; the main problem being that in general {\tau}V(f_1,..., f_s) 6= V(f_1..., f_s, {\tau}f_1,..., {\tau}f_s). But a different, indeed simpler, set of first-order axioms, which we will now describe, does express the geometric characterization.
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paper_title: Model Theory of Differentially Closed Fields with a Generic Automorphism

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E. Hrushovski proved tha the theory of difference-differential fields has a model companion. We prove this result and other maind properties of this theory that we call DCFA. We describe the SU rank a its relation with transcendence degree. We prove a version of Zilber's dichotomy and we give a description of Abelian definable groups.
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paper_title: Model theory of fields with free operators in characteristic zero

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Generalizing 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-CF0 which is simple and satisfies the Zilber dichotomy for finite-dimensional minimal types.
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paper_title: A new strongly minimal set

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Abstract We construct a new class of ℵ 1 categorical structures, disproving Zilber's conjecture, and study some of their properties.
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paper_title: Strongly minimal expansions of algebraically closed fields

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(1) We construct a strongly minimal expansion of an algebraically closed field of a given characteristic. Actually we show a much more general result, implying for example the existence of a strongly minimal set with two different field structures of distinct characteristics.
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paper_title: The Mordell-Lang conjecture for function fields

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We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin’s homomorphism and of a certain analog in characteristic p. Department of Mathematics, Massachusetts Institute of Technology, 2-277, Cambridge, Massachusetts 02139 Current address: Department of Mathematics, Hebrew University, Jerusalem, Israel E-mail address: ehud@math.mit.edu
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paper_title: Weakly Normal Groups

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Publisher Summary This chapter describes that a group G is weakly normal if and only if every definable X ⊂ G n is a Boolean combination of cosets of acl (ϕ)-definable subgroups ∀n if and only if every definable X ⊂ G n is a Boolean combination of cosets of definable subgroups (of G n ) ∀n. The chapter describes some equivalent conditions to that of weak normality and some basic results on stable groups. A locally connected definable subgroup of a weakly normal group is acl (ϕ)-definable. The chapter shows that if G is a weakly normal group, any p ∊ S(G) is determined by the definable cosets in p. The above result is enough to prove equivalence in the theorem, “an Abelian structure is an Abelian group A, together with distinguished subgroups of A n for various n's. Any Abelian structure is interpretable in a module.”
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paper_title: Differential arcs and regular types in differential fields

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We introduce differential arc spaces in analogy to the algebraic arc spaces and show that a differential variety is determined by its arcs at a point. Using differential arcs, we show that if (K,+,×, δ1, . . . , δn) is a differentially closed field of characteristic zero with n commuting derivations and p ∈ S(K) is a regular type over K, then either p is locally modular or there is a definable subgroup G ≤ (K,+) of the additive group having a regular generic type that is nonorthogonal to p.
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paper_title: Model theory of fields with free operators in characteristic zero

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Generalizing 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-CF0 which is simple and satisfies the Zilber dichotomy for finite-dimensional minimal types.
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paper_title: The Mordell-Lang conjecture for function fields

paper_content:

We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin’s homomorphism and of a certain analog in characteristic p. Department of Mathematics, Massachusetts Institute of Technology, 2-277, Cambridge, Massachusetts 02139 Current address: Department of Mathematics, Hebrew University, Jerusalem, Israel E-mail address: ehud@math.mit.edu
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paper_title: Model Theory of Differentially Closed Fields with a Generic Automorphism

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E. Hrushovski proved tha the theory of difference-differential fields has a model companion. We prove this result and other maind properties of this theory that we call DCFA. We describe the SU rank a its relation with transcendence degree. We prove a version of Zilber's dichotomy and we give a description of Abelian definable groups.
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paper_title: Differential arcs and regular types in differential fields

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We introduce differential arc spaces in analogy to the algebraic arc spaces and show that a differential variety is determined by its arcs at a point. Using differential arcs, we show that if (K,+,×, δ1, . . . , δn) is a differentially closed field of characteristic zero with n commuting derivations and p ∈ S(K) is a regular type over K, then either p is locally modular or there is a definable subgroup G ≤ (K,+) of the additive group having a regular generic type that is nonorthogonal to p.
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paper_title: A note on canonical bases and one-based types in supersimple theories

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This paper studies the CBP, a model-theoretic property first discovered by Pillay and Ziegler. We first show a general decomposition result of types of canonical bases, which one can think of as a sort of primary decomposition. This decomposition is then used to show that existentially closed difference fields of any characteristic have the CBP. We also derive consequences of the CBP, and use these results for applications to differential and difference varieties, and algebraic dynamics.
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paper_title: Jet spaces of varieties over differential and difference fields

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We prove some new structural results on finite-dimensional differential algebraic varieties and difference algebraic varieties in characteristic zero, using elementary methods involving jet spaces. Some partial results and problems are given in the positive characteristic cases. The impact of these methods and results on proofs of the Mordell-Lang conjecture for function fields will also be discussed.
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paper_title: On canonical bases and internality criteria

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A criterion is given for a type in a finite rank stable theory to be (almost) internal to a given nonmodular minimal type. The motivation comes from results of Campana which give criteria for compact complex analytic spaces to be algebraic (namely Moishezon), in terms of the existence of "generating" families of algebraic subvarieties. A model-theoretic anologue/generalisation of Campana's results is given under the hypothesis that the theory has the "canonical base property" (CBP), a property that is conjectured to hold in all stable finite rank theories and which states that the type of the canonical base over a realisation is almost internal to the minimal types of the theory.
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paper_title: A note on groups definable in difference fields

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We prove that a group definable in a model of ACFA is virtually definably embeddable in an algebraic group. We give an improved proof of the same result for groups definable in differentially closed fields. We also extend to the difference field context results on the unipotence of definable groups on affine spaces.
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paper_title: Indecomposability for differential algebraic groups

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Abstract We study a notion of indecomposability in differential algebraic groups which is inspired by both model theory and differential algebra. After establishing some basic definitions and results, we prove an indecomposability theorem for differential algebraic groups. The theorem establishes a sufficient criterion for the subgroup of a differential algebraic group generated by an infinite family of subvarieties to be a differential algebraic subgroup. This theorem is used for various definability results. For instance, we show that every noncommutative almost simple differential algebraic group is perfect, solving a problem of Cassidy and Singer. We also establish numerous bounds on Kolchin polynomials, some of which seem to be of a nature not previously considered in differential algebraic geometry; in particular, we establish bounds on the Kolchin polynomial of the generators of the differential field of definition of a differential algebraic variety.
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paper_title: Groups and fields interpretable in separably closed fields

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We prove that any infinite group interpretable in a separably closed field F of finite Ersov-invariant is definably isomorphic to an F-algebraic group. Using this result we show that any infinite field K interpretable in a separably closed field F is itself separably closed; in particular, in the finite invariant case K is definably isomorphic to a finite extension of F . This paper answers a question raised by D. Marker, whose help and guidance made this work possible. I would like to thank A. Pillay for helpful discussions, and E. Hrushovski for pointing out mistakes in the first version of this paper.
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paper_title: Quantifier elimination for algebraic D-groups

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We prove that if G is an algebraic D-group (in the sense of Buium over a differentially closed field (K,∂) of characteristic 0, then the first order structure consisting of G together with the algebraic D-subvarieties of G, G x G,..., has quantifier-elimination. In other words, the projection on G of a D-constructible subset of G n+1 is D-constructible. Among the consequences is that any finite-dimensional differential algebraic group is interpretable in an algebraically closed field.
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paper_title: Algebraic D-groups and differential Galois theory

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We discuss various relationships between the algebraic D-groups of Buium, 1992, and differential Galois theory. In the first part we give another exposition of our general differential Galois theory, which is somewhat more explicit than Pillay, 1998, and where generalized logarithmic derivatives on algebraic groups play a central role. In the second part we prove some results with a constrained Galois cohomological flavor. For example, if G and H are connected algebraic D-groups over an algebraically closed differential field F, and G and H are isomorphic over some differential field extension of F, then they are isomorphic over some Picard-Vessiot extension of F. Suitable generalizations to isomorphisms of algebraic D-varieties are also given.
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paper_title: GROUPS DEFINABLE IN SEPARABLY CLOSED FIELDS

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We consider the groups which are infinitely definable in separably closed fields of finite degree of imperfection. We prove in particular that no new definable groups arise in this way: we show that any group definable in such a field L is definably isomorphic to the group of L-rational points of an algebraic group defined over L.
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paper_title: Model Theory of Differentially Closed Fields with a Generic Automorphism

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E. Hrushovski proved tha the theory of difference-differential fields has a model companion. We prove this result and other maind properties of this theory that we call DCFA. We describe the SU rank a its relation with transcendence degree. We prove a version of Zilber's dichotomy and we give a description of Abelian definable groups.
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paper_title: On algebraic σ-groups

paper_content:

We introduce the categories of algebraic σ-varieties and σ-groups over a difference field (K, σ). Under a "linearly σ-closed" assumption on (K, a) we prove an isotriviality theorem for σ-groups. This theorem immediately yields the key lemma in a proof of the Manin-Mumford conjecture. The present paper crucially uses ideas of Pilay and Ziegler (2003) but in a model theory free manner. The applications to Manin-Mumford are inspired by Hrushovski's work (2001) and are also closely related to papers of Pink and Roessler (2002 and 2004).
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Title: Model theory of fields with operators - a survey

Section 1: Introduction
Description 1: This section introduces the model theory of fields with operators, its applications, and sets the scope of the survey.

Section 2: Basic results
Description 2: Discusses the fundamental theorems and properties concerning differential fields of characteristic 0, including their model completions and stability properties.

Section 3: Several derivations
Description 3: Examines the extension of basic results to fields with multiple commuting or non-commuting derivations.

Section 4: Geometric axiomatisation
Description 4: Presents the geometric axiomatisation of differential fields, highlighting the work of Blum, Pierce, and Pillay.

Section 5: Other theories of differential fields of characteristic 0
Description 5: Covers various other differential fields by imposing different conditions on the pure field.

Section 6: Hasse-Schmidt derivations
Description 6: Details the theory of fields with Hasse-Schmidt derivations in characteristic p > 0.

Section 7: Derivations of the Frobenius
Description 7: Discusses the model theory of fields endowed with a derivation of the q-Frobenius.

Section 8: Difference fields
Description 8: Explores the theory of difference fields, their model companions, and the work of Macintyre and others in this area.

Section 9: Mixing operators: the commutative case
Description 9: Looks at fields with multiple types of operators, especially commuting automorphisms and their model companions.

Section 10: A general framework for fields with operators
Description 10: Introduces a generalized setup for fields with operators, as developed by Moosa and Scanlon.

Section 11: Stability theoretic notions
Description 11: Explains the stability properties of the fields under study, categorizing them based on the types of operators involved.

Section 12: Algebraic closure, independence
Description 12: Describes the model-theoretic concepts of algebraic closure and independence in the context of fields with operators.

Section 13: Zilber's conjecture - the trichotomy
Description 13: Discusses the trichotomy conjecture, its implications for fields with operators, and the results obtained for various types.

Section 14: Stability theoretic results for enriched fields
Description 14: Details results related to the dichotomy theorem and the CBP (Cannone-Büchi Property) in the context of enriched fields.

Section 15: Definable groups
Description 15: Investigates the structure of definable groups within the model theory of fields with operators and the generalisation to various theories.