Let ℛ be an o-minimal expansion of a real closed field R. We continue here the investigation we began in [11] of differentiability with respect to the algebraically closed field [Formula: see text]. We develop the basic theory of such K-differentiability for definable functions of several variables, proving theorems on removable singularities as well as analogues of the Weierstrass preparation and division theorems for definable functions. We consider also definably meromorphic functions and prove that every definable function (...) which is meromorphic on Kn is necessarily a rational function. We finally discuss definable analogues of complex analytic manifolds, with possible connections to the model theoretic work on compact complex manifolds, and present two examples of "nonstandard manifolds" in our setting. (shrink)

Fix an algebraically closed field of characteristic zero and let G be its geometry of transcendence degree one extensions. Let X be a set of points of G. We show that X extends to a projective subgeometry of G exactly if the partial derivatives of the polynomials inducing dependence on its elements satisfy certain separability conditions. This analysis produces a concrete representation of the coordinatizing fields of maximal projective subgeometries of G.

LetKbe an algebraically closed field and letLbe itscanonical language; that is,Lconsists of all relations onKwhich are definable from addition, multiplication, and parameters fromK. Two sublanguagesL1andL2ofLaredefinably equivalentif each relation inL1can be defined by anL2-formula with parameters inK, and vice versa. The equivalence classes of sublanguages ofLform a quotient lattice of the power set ofLabout which very little is known. We will not distinguish between a sublanguage and its equivalence class.LetLmdenote the language of multiplication alone, and letLadenote the language of (...) addition alone. Letf∈K[X, Y] and consider thealgebraic functiondefined byf(x, y) = 0 forx, y∈K. LetLfdenote the language consisting of the relation defined byf. The possibilities forLm∨Lfare examined in §2, and the possibilities forLa∨Lfare examined in §3. In fact the only comprehensive results known are under the additional hypothesis thatfactually defines a rational function (i.e., whenfis linear in one of the variables), and in positive characteristic, only expansions of addition by polynomials (i.e., whenfis linear and monic in one of the variables) are understood. It is hoped that these hypotheses will turn out to be unnecessary, so that reasonable generalizations of the theorems described below to algebraic functions will be true. The conjecture is thatLcoversLmand that the only languages betweenLaandLare expansions ofLaby scalar multiplications. (shrink)

We study structures on the fields of characteristic zero obtained by introducing operations of raising to power. Using Hrushovski–Fraisse construction we single out among the structures exponentially-algebraically closed once and prove, under certain Diophantine conjecture, that the first order theory of such structures is model complete and every its completion is superstable.

This paper studies unbounded pseudo-algebraically closed fields and shows an amalgamation result for types over algebraically closed sets. It discusses various applications, for instance that omega-free PAC fields have the property NSOP3. It also contains a description of imaginaries in PAC fields.

We prove that there exists no sentence F of the language of rings with an extra binary predicat I2 satisfying the following property: for every definable set X ⊆ ℂ2, X is connected if and only if ⊧ F, where I2 is interpreted by X. We conjecture that the same result holds for closed subset of ℂ2. We prove some results motivated by this conjecture.

Let n ≥ 3. The following theorems are proved. Theorem. The theory of the class of strictly upper triangular n × n matrix rings over fields is finitely axiomatizable. Theorem. If R is a strictly upper triangular n × n matrix ring over a field K, then there is a recursive map σ from sentences in the language of rings with constants for K into sentences in the language of rings with constants for R such that $K \vDash \varphi$ (...) if and only if $R \vDash \sigma$. Theorem. The theory of a strictly upper triangular n × n matrix ring over an algebraically closed field is ℵ 1 -categorical. (shrink)

We construct and study structures imitating the field of complex numbers with exponentiation. We give a natural, albeit non first-order, axiomatisation for the corresponding class of structures and prove that the class has a unique model in every uncountable cardinality. This gives grounds to conjecture that the unique model of cardinality continuum is isomorphic to the field of complex numbers with exponentiation.

We consider the theory P of pairs Falgebraically closed fields of a given characteristic p. We exhibit a collection of additional sorts in which this theory has geometric elimination of imaginaries. The sorts are essentially of the form aBVa/Ga, where G,V,B are varieties over the prime field, G is a group scheme over B and V is a scheme over B , G acts algebraically on V over B, and for generic bB the action (...) of Gb on Vb is generically free. (shrink)

The generalization properties of algebraically closed fields $ACF_p$ of characteristic $p > 0$ and $ACF_0$ of characteristic 0 are investigated in the sequent calculus with blocks of quantifiers. It is shown that $ACF_p$ admits finite term bases, and $ACF_0$ admits term bases with primality constraints. From these results the analogs of Kreisel's Conjecture for these theories follow: If for some $k$ , $A(1 + \cdots + 1)$ ( $n$ 1's) is provable in $k$ steps, then $(\forall x)A(x)$ (...) is provable. (shrink)

The first two steps of the construction of motivic integration in the fundamental work of Hrushovski and Kazhdan [8] have been presented in Yin [12]. In this paper we present the final third step. As in Yin [12], we limit our attention to the theory of algebraically closed valued fields of pure characteristic 0 expanded by a -generated substructure S in the language . A canonical description of the kernel of the homomorphism is obtained.

In, Marker and Steinhorn characterized models of an o‐minimal theory such that all types over M realized in N are definable. In this article we characterize pairs of algebraically closed valued fields satisfying the same property. In o‐minimal theories, a pair of models for which all 1‐types over M realized in N are definable has already the desired property. Although it is true that if M is an algebraically closed valued field such that all 1‐types (...) over M are definable then all types over M are definable, we build a counterexample for the relative statement, i.e., we show for any that there is a pair of algebraically closed valued fields such that all n‐types over M realized in N are definable but there is an ‐type over M realized in N which is not definable. (shrink)

We give some general criteria for the stable embeddedness of a definable set. We use these criteria to establish the stable embeddedness in algebraically closed valued fields of two definable sets: The set of balls of a given radius r < 1 contained in the valuation ring and the set of balls of a given multiplicative radius r < 1. We also show that in an algebraically closed valued field a 0-definable set is stably embedded (...) if and only if its algebraic closure is stably embedded. (shrink)

We present two of the three major steps in the construction of motivic integration, that is, a homomorphism between Grothendieck semigroups that are associated with a first-order theory of algebraically closed valued fields, in the fundamental work of Hrushovski and Kazhdan [8]. We limit our attention to a simple major subclass of V-minimal theories of the form ACV FS, that is, the theory of algebraically closed valued fields of pure characteristic 0 expanded by a (...) -generated substructure S in the language . The main advantage of this subclass is the presence of syntax. It enables us to simplify the arguments with many different technical details while following the major steps of the Hrushovski–Kazhdan theory. (shrink)

In this paper we extend of the notion of algebraically closed given in the case of groups and skew fields to an arbitrary h-inductive theory. The main subject of this paper is the study of the notion of positive algebraic closedness and its relationship with the notion of positive closedness and the amalgamation property.

We construct Hrushovski–Kazhdan style motivic integration in certain expansions of ACVF. Such an expansion is typically obtained by adding a full section or a cross-section from the RV-sort into the VF-sort and some extra structure in the RV-sort. The construction of integration, that is, the inverse of the lifting map , is rather straightforward. What is a bit surprising is that the kernel of is still generated by one element, exactly as in the case of integration in ACVF. The overall (...) construction is more or less parallel to the main construction of Hrushovski and Kazhdan , as presented in Yin and Yin . As an application, we show uniform rationality of Igusa zeta functions for non-archimedean local fields with unbounded ramification degrees. (shrink)

The theory of algebraically closed non-Archimedean valued fields is proved to eliminate quantifiers in an analytic language similar to the one used by Cluckers, Lipshitz, and Robinson. The proof makes use of a uniform parameterized normalization theorem which is also proved in this paper. This theorem also has other consequences in the geometry of definable sets. The method of proving quantifier elimination in this paper for an analytic language does not require the algebraic quantifier elimination theorem of (...) Weispfenning, unlike the customary method of proof used in similar earlier analytic quantifier elimination theorems. (shrink)

We present a canonical form for definable subsets of algebraically closed valued fields by means of decompositions into sets of a simple form, and do the same for definable subsets of real closed valued fields. Both cases involve discs, forming "Swiss cheeses" in the algebraically closed case, and cuts in the real closed case. As a step in the development, we give a proof for the fact that in "most" valued fields (...) F, if f(x),g(x) ∈ F[ x] and v is the valuation map, then the set {x : v(f(x)) ≤ v(g(x))} is a Boolean combination of discs; in fact, it is a finite union of Swiss cheeses. The development also depends on the introduction of "valued trees", which we define formally. (shrink)

Elimination of imaginaries for 1-variable definable equivalence relations is proved for a theory of algebraically closed valued fields with new sorts for the disc spaces. The proof is constructive, and is based upon a new framework for proving elimination of imaginaries, in terms of prototypes which form a canonical family of formulas for defining each set that is definable with parameters. The proof also depends upon the formal development of the tree-like structure of valued fields, in (...) terms of valued trees, and a decomposition of valued trees which is used in the coding of certain sets of discs. (shrink)

Recall that a group G has finitely satisfiable generics (fsg) or definable f-generics (dfg) if there is a global type p on G and a small model $M_0$ such that every left translate of p is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a p-adically closed field is an extension of a definably compact fsg definable group by a dfg definable group. We discuss an approach which might prove a (...) similar statement for interpretable abelian groups. In the case where G is an abelian group definable in the standard model $\mathbb {Q}_p$, we show that $G^0 = G^{00}$, and that G is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb {Q}_p$. (shrink)

We point out that the theory of difference fields with algebraically closed fixed field has no model companion.

We give a simplified proof of elimination of imaginaries in ACVF, based on ideas of Hrushovski. This proof manages to avoid many of the technical issues which arose in the original proof by Haskell, Hrushovski, and Macpherson.

We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general context of the model theory of fields with a group scheme action.

We give an intuitionistic axiomatisation of real closed fields which has the constructive reals as a model. The main result is that this axiomatisation together with just the decidability of the order relation gives the classical theory of real closed fields. To establish this we rely on the quantifier elimination theorem for real closed fields due to Tarski, and a conservation theorem of classical logic over intuitionistic logic for geometric theories.

In this paper, we consider reducts of p-adically closed fields. We introduce a notion of shadows: sets Mf={∈K2∣|y|=|f|}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_f = \{ \in K^2 \mid |y| = |f|\}}$$\end{document}, where f is a semi-algebraic function. Adding symbols for such sets to a reduct of the ring language, we obtain expansions of the semi-affine language where multiplication is nowhere definable, thus giving a negative answer to a question posed by Marker, Peterzil and Pillay. (...) The second main result of this paper is the fact that in p-adic fields, full multiplication becomes definable if we add a rational function to the semi-affine language. (shrink)

We introduce a new construction of real closed fields by using an elementary extension of an ordered field with an integer part satisfying. This method can be extend to a finite extension of an ordered field with an integer part satisfying. In general, a field obtained from our construction is either real closed or algebraically closed, so an analogy of Ostrowski's dichotomy holds. Moreover we investigate recursive saturation of an o‐minimal extension of a real (...) class='Hi'>closed field by finitely many function symbols. (shrink)

The main scope of this short article is to provide a modification of the axioms given by Messmer and Wood for the theory of separably closed fields of positive characteristic and finite imperfectness degree. As their original axioms failed to meet natural expectations, a new axiomatization was given (i.e., Ziegler’s one), but the new axioms do not follow Messmer and Wood’s initial idea. Therefore, we aim to give a correct axiomatization that is more similar to the original one (...) and that, as with the original axioms, involves only one Hasse–Schmidt derivation, this time based on the iterativity conditions corresponding to the Witt group. (shrink)

By RCA 0 , we denote a subsystem of second order arithmetic based on Δ0 1 comprehension and Δ0 1 induction. We show within this system that the real number system R satisfies all the theorems (possibly with non-standard length) of the theory of real closed fields under an appropriate truth definition. This enables us to develop linear algebra and polynomial ring theory over real and complex numbers, so that we particularly obtain Hilbert’s Nullstellensatz in RCA 0.

The following conjecture is due to Shelah–Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non‐trivial definable henselian valuation, in the language of rings. We specialise this conjecture to ordered fields in the language of ordered rings, which leads towards a systematic study of the class of strongly NIP almost real closed fields. As a result, we obtain a complete characterisation of this class.

It is shown that the physics and semantics of quantum measurement provide a natural interpretation of the weak neighborhoods of the states on observable algebras without invoking any idea of “a reading error” or “a measured range.” Then the state preparation process in quantum measurement theory is shown to give the normal (or locally normal) states on the observable algebra. Some remarks are made concerning the physical implications of normal states for systems with an infinite number of degrees of freedom, (...) including questions on open and closed algebraic theories. (shrink)

The stability of each of the theories of separably closed fields is proved, in the manner of Shelah's proof of the corresponding result for differentially closed fields. These are at present the only known stable but not superstable theories of fields. We indicate in § 3 how each of the theories of separably closed fields can be associated with a model complete theory in the language of differential algebra. We assume familiarity with some (...) basic facts about model completeness [4], stability [7], separably closed fields [2] or [3], and (for § 3 only) differential fields [8]. (shrink)

In this paper we deal with the model theory of differentially closed fields of characteristic zero with finitely many commuting derivations. First we observe that the only known lower bound for the Lascar rank of types in differentially closed fields, announced in a paper of McGrail, is false. This gives us a new class of regular types which are orthogonal to fields. Then we classify the subgroups of the additive group of Lascar rank omega with (...) differential-type 1 which are nonorthogonal to fields. The last parts consist of an analaysis of the quotients of the heat variety. We show that the generic type of such a quotient is locally modular. Finally, we answer a question of Phylliss Cassidy about the existence of certain Jordan-Holder type series in the negative. (shrink)