We characterize those functions f:ℂ → ℂ definable in o-minimal expansions of the reals for which the structure (ℂ,+, f) is strongly minimal: such functions must be complex constructible, possibly …
We characterize those functions f:ℂ → ℂ definable in o-minimal expansions of the reals for which the structure (ℂ,+, f) is strongly minimal: such functions must be complex constructible, possibly after conjugating by a real matrix. In particular we prove a special case of the Zilber Dichotomy: an algebraically closed field is definable in certain strongly minimal structures which are definable in an o-minimal field.
Abstract An exponential $\exp $ on an ordered field $(K,+,-,\cdot ,0,1,<)$ is an order-preserving isomorphism from the ordered additive group $(K,+,0,<)$ to the ordered multiplicative group of positive elements $(K^{>0},\cdot …
Abstract An exponential $\exp $ on an ordered field $(K,+,-,\cdot ,0,1,<)$ is an order-preserving isomorphism from the ordered additive group $(K,+,0,<)$ to the ordered multiplicative group of positive elements $(K^{>0},\cdot ,1,<)$ . The structure $(K,+,-,\cdot ,0,1,<,\exp )$ is then called an ordered exponential field (cf. [6]). A linearly ordered structure $(M,<,\ldots )$ is called o-minimal if every parametrically definable subset of M is a finite union of points and open intervals of M . The main subject of this thesis is the algebraic and model theoretic examination of o-minimal exponential fields $(K,+,-,\cdot ,0,1,<,\exp )$ whose exponential satisfies the differential equation $\exp ' = \exp $ with initial condition $\exp (0) = 1$ . This study is mainly motivated by the Transfer Conjecture, which states as follows: Any o-minimal exponential field $(K,+,-,\cdot ,0,1,<,\exp )$ whose exponential satisfies the differential equation $\exp ' = \exp $ with initial condition $\exp (0)=1$ is elementarily equivalent to $\mathbb {R}_{\exp }$ . Here, $\mathbb {R}_{\exp }$ denotes the real exponential field $(\mathbb {R},+,-,\cdot ,0,1,<,\exp )$ , where $\exp $ denotes the standard exponential $x \mapsto \mathrm {e}^x$ on $\mathbb {R}$ . Moreover, elementary equivalence means that any first-order sentence in the language $\mathcal {L}_{\exp } = \{+,-,\cdot ,0,1, <,\exp \}$ holds for $(K,+,-,\cdot ,0,1,<,\exp )$ if and only if it holds for $\mathbb {R}_{\exp }$ . The Transfer Conjecture, and thus the study of o-minimal exponential fields, is of particular interest in the light of the decidability of $\mathbb {R}_{\exp }$ . To the date, it is not known if $\mathbb {R}_{\exp }$ is decidable, i.e., whether there exists a procedure determining for a given first-order $\mathcal {L}_{\exp }$ -sentence whether it is true or false in $\mathbb {R}_{\exp }$ . However, under the assumption of Schanuel’s Conjecture—a famous open conjecture from Transcendental Number Theory—a decision procedure for $\mathbb {R}_{\exp }$ exists (cf. [7]). Also a positive answer to the Transfer Conjecture would result in the decidability of $\mathbb {R}_{\exp }$ (cf. [1]). Thus, we study o-minimal exponential fields with regard to the Transfer Conjecture, Schanuel’s Conjecture, and the decidability question of $\mathbb {R}_{\exp }$ . Overall, we shed light on the valuation theoretic invariants of o-minimal exponential fields—the residue field and the value group—with additional induced structure. Moreover, we explore elementary substructures and extensions of o-minimal exponential fields to the maximal ends—the smallest elementary substructures being prime models and the maximal elementary extensions being contained in the surreal numbers. Further, we draw connections to models of Peano Arithmetic, integer parts, density in real closure, definable Henselian valuations, and strongly NIP ordered fields. Parts of this thesis were published in [2–5]. Abstract prepared by Lothar Sebastian Krapp E-mail : [email protected] URL : https://d-nb.info/1202012558/34
Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in …
Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in M. We show that every M–definable ring without zero divisors is definably isomorphic to R, R(√(−l)) or the ring of quaternions over R. One corollary is that no model of Texp is interpretable in a model of Tan.
Model theory begins with an audacious idea: to consider statements about mathematical structures as mathematical objects of study in their own right. While inherently important as a tool of mathematical …
Model theory begins with an audacious idea: to consider statements about mathematical structures as mathematical objects of study in their own right. While inherently important as a tool of mathematical logic, it also enjoys connections to and applications in diverse branches of mathematics, including algebra, number theory and analysis. Despite this, traditional introductions to model theory assume a graduate-level background of the reader. In this innovative textbook, Jonathan Kirby brings model theory to an undergraduate audience. The highlights of basic model theory are illustrated through examples from specific structures familiar from undergraduate mathematics, paying particular attention to definable sets throughout. With numerous exercises of varying difficulty, this is an accessible introduction to model theory and its place in mathematics.
The Pila-Wilkie theorem states that if a set $X\subseteq \mathbb R^n$ is definable in an o-minimal structure $\mathcal R$ and contains `many' rational points, then it contains an infinite semialgebraic …
The Pila-Wilkie theorem states that if a set $X\subseteq \mathbb R^n$ is definable in an o-minimal structure $\mathcal R$ and contains `many' rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion $\widetilde{\mathcal R}=\langle \mathcal R, P\rangle$ of $\mathcal R$ by a dense set $P$, which is either an elementary substructure of $\mathcal R$, or it is independent, as follows. If $X$ is definable in $\widetilde{\mathcal R}$ and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is $\emptyset$-definable in $\langle \overline{\mathbb R}, P\rangle$, where $\overline {\mathbb R}$ is the real field.
Abstract The Pila–Wilkie theorem states that if a set $X\subseteq \mathbb{R}^n$ is definable in an o-minimal structure $\mathcal{R}$ and contains ‘many’ rational points, then it contains an infinite semialgebraic set. …
Abstract The Pila–Wilkie theorem states that if a set $X\subseteq \mathbb{R}^n$ is definable in an o-minimal structure $\mathcal{R}$ and contains ‘many’ rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion $\widetilde{\mathcal{R}}=\langle {\mathcal{R}}, P\rangle$ of ${\mathcal{R}}$ by a dense set P, which is either an elementary substructure of ${\mathcal{R}}$, or it is $\mathrm{dcl}$-independent, as follows. If X is definable in $\widetilde{\mathcal{R}}$ and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is ${\emptyset}$-definable in $\langle \overline{\mathbb{R}}, P\rangle$, where $\overline{\mathbb{R}}$ is the real field. Along the way we introduce the notion of the ‘algebraic trace part’ $X^{{\, alg}}_t$ of any set $X\subseteq \mathbb{R}^n$, and we show that if X is definable in an o-minimal structure, then $X^{{\, alg}}_t$ coincides with the usual algebraic part of X.
The Pila-Wilkie theorem states that if a set $X\subseteq \mathbb R^n$ is definable in an o-minimal structure $\mathcal R$ and contains `many' rational points, then it contains an infinite semialgebraic …
The Pila-Wilkie theorem states that if a set $X\subseteq \mathbb R^n$ is definable in an o-minimal structure $\mathcal R$ and contains `many' rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion $\widetilde{\mathcal R}=\langle \mathcal R, P\rangle$ of $\mathcal R$ by a dense set $P$, which is either an elementary substructure of $\mathcal R$, or it is independent, as follows. If $X$ is definable in $\widetilde{\mathcal R}$ and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is $\emptyset$-definable in $\langle \overline{\mathbb R}, P\rangle$, where $\overline {\mathbb R}$ is the real field.
Using a modification of Wilkie''s recent proof of o-minimality for Pfaffian functions, we gave an invariant characterization of o-minimal expansions of IR. We apply this to construct the Pfaffian closure …
Using a modification of Wilkie''s recent proof of o-minimality for Pfaffian functions, we gave an invariant characterization of o-minimal expansions of IR. We apply this to construct the Pfaffian closure of an arbitrary o-minimal expansion of IR.
We consider the Riemann mapping theorem in the case of a bounded simply connected and semianalytic domain. We show that the germ at 0 of the Riemann map (that is, …
We consider the Riemann mapping theorem in the case of a bounded simply connected and semianalytic domain. We show that the germ at 0 of the Riemann map (that is, biholomorphic map) from the upper half plane to such a domain can be realized in a certain quasianalytic class if the angle of the boundary at the point to which 0 is mapped is greater than 0. This quasianalytic class was introduced and used by Ilyashenko in his work on Hilbert's 16th problem. With this result, we can prove that the Riemann map from a bounded simply connected semianalytic domain onto the unit ball is definable in an o-minimal structure, provided that at singular boundary points the angles of the boundary are irrational multiples of π.
Abstract We use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions …
Abstract We use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.
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 …
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 K n 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.
The notion of an analytic-geometric category was introduced by v. d. Dries and Miller in [Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. …
The notion of an analytic-geometric category was introduced by v. d. Dries and Miller in [Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540.]. It is a category of subsets of real analytic manifolds which extends the category of subanalytic sets. This paper discusses connections between the subanalytic category, or more generally analytic-geometric categories, and complex analytic geometry. The questions are of the following nature: We start with a subset A of a complex analytic manifold M and assume that A is an object of an analytic-geometric category (by viewing M as a real analytic manifold of double dimension). We then formulate conditions under which A, its closure or its image under a holomorphic map is a complex analytic set.
We prove the characteristic zero case of Zilber’s Restricted Trichotomy Conjecture. That is, we show that if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal M</mml:annotation></mml:semantics></mml:math></inline-formula>is any non-locally modular …
We prove the characteristic zero case of Zilber’s Restricted Trichotomy Conjecture. That is, we show that if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal M</mml:annotation></mml:semantics></mml:math></inline-formula>is any non-locally modular strongly minimal structure interpreted in an algebraically closed field<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding="application/x-tex">K</mml:annotation></mml:semantics></mml:math></inline-formula>of characteristic zero, then<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal M</mml:annotation></mml:semantics></mml:math></inline-formula>itself interprets<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding="application/x-tex">K</mml:annotation></mml:semantics></mml:math></inline-formula>; in particular, any non-1-based structure interpreted in<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding="application/x-tex">K</mml:annotation></mml:semantics></mml:math></inline-formula>is mutually interpretable with<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"><mml:semantics><mml:mi>K</mml:mi><mml:annotation encoding="application/x-tex">K</mml:annotation></mml:semantics></mml:math></inline-formula>. Notably, we treat both the ‘one-dimensional’ and ‘higher-dimensional’ cases of the conjecture, introducing new tools to resolve the higher-dimensional case and then using the same tools to recover the previously known one-dimensional case.
We prove Zilber's Trichotomy Conjecture for strongly minimal expansions of two-dimensional groups, definable in o-minimal structures: Theorem. Let M be an o-minimal expansion of a real closed field, (G;+) a …
We prove Zilber's Trichotomy Conjecture for strongly minimal expansions of two-dimensional groups, definable in o-minimal structures:
Theorem. Let M be an o-minimal expansion of a real closed field, (G;+) a 2-dimensional group definable in M, and D = (G;+,...) a strongly minimal structure, all of whose atomic relations are definable in M. If D is not locally modular, then an algebraically closed field K is interpretable in D, and the group G, with all its induced D-structure, is definably isomorphic in D to an algebraic K-group with all its induced K-structure.
We consider a subanalytic subset A of a complex analytic manifold M (when M is viewed as a real manifold) and formulate conditions under which A is a complex analytic …
We consider a subanalytic subset A of a complex analytic manifold M (when M is viewed as a real manifold) and formulate conditions under which A is a complex analytic subset of M.
Abstract We give a classification of connected abelian locally (real) Nash groups of dimension two. We first consider Painlevé’s description of meromorphic maps admitting an algebraic addition theorem and analyse …
Abstract We give a classification of connected abelian locally (real) Nash groups of dimension two. We first consider Painlevé’s description of meromorphic maps admitting an algebraic addition theorem and analyse the algebraic dependence of such maps. We then give a classification of connected abelian locally complex Nash groups of dimension two, from which we deduce the corresponding real classification. As a consequence, we obtain a classification of two-dimensional abelian irreducible algebraic groups defined over $\mathbb{R}$.
Working in an o-minimal expansion of the real field, we investigate when a germ (around zero, say) of a complex analytic function has a definable analytic continuation to its Mittag–Leffler …
Working in an o-minimal expansion of the real field, we investigate when a germ (around zero, say) of a complex analytic function has a definable analytic continuation to its Mittag–Leffler star. As an application we show that any algebro-logarithmic function that is complex analytic in a neighborhood of the origin in C has an analytic continuation to all but finitely many points in C.
We make several observations about the category C of compact complex manifolds, considered as a many-sorted structure of finite Morley rank. We also point out that the Mordell-Lang conjecture holds …
We make several observations about the category C of compact complex manifolds, considered as a many-sorted structure of finite Morley rank. We also point out that the Mordell-Lang conjecture holds for complex tori: if A is a complex torus, Γ a finitely generated subgroup of A, and X an analytic subvariety of A, then X ∩ Γ is a finite union of translates of subgroups of A. This is implicit in the literature but we give an elementary reduction to the abelian variety case. We discuss analogies between C and the category of finite dimensional differential algebraic sets. (A brief survey of the Mordell-Lang conjecture and model-theoretic contributions is also included.)
Abstract Model theorists have been studying analytic functions since the late 1970s. Highlights include the seminal work of Denef and van den Dries on the theory of the p -adics …
Abstract Model theorists have been studying analytic functions since the late 1970s. Highlights include the seminal work of Denef and van den Dries on the theory of the p -adics with restricted analytic functions, Wilkie's proof of o-minimality of the theory of the reals with the exponential function, and the formulation of Zilber's conjecture for the complex exponential. My goal in this talk is to survey these main developments and to reflect on today's open problems, in particular for theories of valued fields.
It is shown that in an elementary extension of a compact complex manifold $M$, the $K$-analytic sets (where $K$ is the algebraic closure of the underlying real closed field) agree …
It is shown that in an elementary extension of a compact complex manifold $M$, the $K$-analytic sets (where $K$ is the algebraic closure of the underlying real closed field) agree with the ccm-analytic sets if and only if $M$ is essentially saturated. In
Given an algebraically closed field $K$ of characteristic zero, we prove the Abhyankar–Jung theorem for any excellent henselian ring whose completion is a formal power series ring $K[[z]]$. In particular, …
Given an algebraically closed field $K$ of characteristic zero, we prove the Abhyankar–Jung theorem for any excellent henselian ring whose completion is a formal power series ring $K[[z]]$. In particular, examples include the local rings which form
We consider the ordered field which is the completion of the Puiseux series field over ℝ equipped with a ring of analytic functions on [−1, 1]n which contains the standard …
We consider the ordered field which is the completion of the Puiseux series field over ℝ equipped with a ring of analytic functions on [−1, 1]n which contains the standard subanalytic functions as well as functions given by t-adically convergent power series, thus combining the analytic structures of Denef and van den Dries [Ann. of Math. 128 (1988) 79–138] and Lipshitz and Robinson [Bull. London Math. Soc. 38 (2006) 897–906]. We prove quantifier elimination and o-minimality in the corresponding language. We extend these constructions and results to rank n ordered fields ℝn (the maximal completions of iterated Puiseux series fields). We generalize the example of Hrushovski and Peterzil [J. Symbolic Logic 72 (2007) 119–122] of a sentence which is not true in any o-minimal expansion of ℝ (shown in [Bull. London Math. Soc. 38 (2006) 897–906] to be true in an o-minimal expansion of the Puiseux series field) to a tower of examples of sentences σn, true in ℝn, but not true in any o-minimal expansion of any of the fields ℝ, ℝ1, …, ℝn−1.
Let $K,R$ be an algebraically closed field (of characteristic zero) and a real closed field respectively with $K=R(\sqrt{-1}).$ We show that every $K$-analytic set definable in an o-minimal expansion of …
Let $K,R$ be an algebraically closed field (of characteristic zero) and a real closed field respectively with $K=R(\sqrt{-1}).$ We show that every $K$-analytic set definable in an o-minimal expansion of $R$ can be locally approximated by a sequence of $K$
Abstract We characterise strongly minimal groups interpretable in elementary extensions of compact complex analytic spaces.
Abstract We characterise strongly minimal groups interpretable in elementary extensions of compact complex analytic spaces.
Abstract We prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying …
Abstract We prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.
We characterize those functions f:ℂ → ℂ definable in o-minimal expansions of the reals for which the structure (ℂ,+, f) is strongly minimal: such functions must be complex constructible, possibly …
We characterize those functions f:ℂ → ℂ definable in o-minimal expansions of the reals for which the structure (ℂ,+, f) is strongly minimal: such functions must be complex constructible, possibly after conjugating by a real matrix. In particular we prove a special case of the Zilber Dichotomy: an algebraically closed field is definable in certain strongly minimal structures which are definable in an o-minimal field.
This thesis is a model-theoretic study of exponential differential equations in the context of differential algebra. I define the theory of a set of differential equations and give an axiomatization …
This thesis is a model-theoretic study of exponential differential equations in the context of differential algebra. I define the theory of a set of differential equations and give an axiomatization for the theory of the exponential differential equations of split semiabelian varieties. In particular, this includes the theory of the equations satisfied by the usual complex exponential function and the Weierstrass p-functions.
The theory consists of a description of the algebraic structure on the solution sets together with necessary and sufficient conditions for a system of equations to have solutions. These conditions are stated in terms of a dimension theory; their necessity generalizes Ax’s differential field version of Schanuel’s conjecture and their sufficiency generalizes recent work of Crampin. They are shown to apply to the solving of systems of equations in holomorphic functions away from singularities, as well as in the abstract setting.
The theory can also be obtained by means of a Hrushovski-style amalgamation construction, and I give a category-theoretic account of the method.
Restricting to the usual exponential differential equation, I show that a “blurring” of Zilber’s pseudo-exponentiation satisfies the same theory. I conjecture that this theory also holds for a suitable blurring of the complex exponential maps and partially resolve the question, proving the necessity but not the sufficiency of the aforementioned conditions.
As an algebraic application, I prove a weak form of Zilber’s conjecture on intersections with subgroups (known as CIT) for semiabelian varieties. This in turn is used to show that the necessary and sufficient conditions are expressible in the appropriate first order language.
The usual model-theoretic approach to complex algebraic geometry is to view complex algebraic varieties as living definably in the structure (C,+,×). Variousmodel-theoretic properties of algebraically closed fields (such as quantifier …
The usual model-theoretic approach to complex algebraic geometry is to view complex algebraic varieties as living definably in the structure (C,+,×). Variousmodel-theoretic properties of algebraically closed fields (such as quantifier elimination and strong minimality) are then used to obtain geometric information about the varieties. This approach extends to other geometric contexts by considering expansions of algebraically closed fields to which the methods of stability or simplicity apply. For example, differential algebraic varieties live in differentially closed fields, and difference algebraic varieties in algebraically closed fields equipped with a generic automorphism. Another approach would be to consider the variety as a structure in its own right, equipped with the algebraic (respectively differential or difference algebraic) subsets of its cartesian powers. This point of view is compatible with the theory of Zariski-type structures, developed by Hrushovski and Zilber (see [21] and [41]). While the two approaches are equivalent (i.e., bi-interpretable) in the case of complex algebraic varieties, the latter point of view extends to certain fragments of complex analytic geometry in a manner that does not seem accessible by the former.
Given an o-minimal structure expanding the field of reals, we show a piecewise Weierstrass preparation theorem and a piecewise Weierstrass division theorem for definable holomorphic functions. In the semialgebraic setting …
Given an o-minimal structure expanding the field of reals, we show a piecewise Weierstrass preparation theorem and a piecewise Weierstrass division theorem for definable holomorphic functions. In the semialgebraic setting and for the structure of globally subanalytic sets and functions we obtain the corresponding results for definable real analytic functions. As an application we show a definable global Nullstellensatz for principal ideals.
We present a unifying theory of fields with certain classes of analytic functions, called fields with analytic structure. Both real closed fields and Henselian valued fields are considered. For real …
We present a unifying theory of fields with certain classes of analytic functions, called fields with analytic structure. Both real closed fields and Henselian valued fields are considered. For real closed fields with analytic structure, o -minimality is shown. For Henselian valued fields, both the model theory and the analytic theory are developed. We give a list of examples that comprises, to our knowledge, all principal, previously studied, analytic structures on Henselian valued fields, as well as new ones. The b -minimality is shown, as well as other properties useful for motivic integration on valued fields. The paper is reminiscent of [Denef, van den Dries, p-adic and real subanalytic sets . Ann. of Math. (2) 128 (1988) 79–138], of [Cohen, Paul J. Decision procedures for real and p-adic fields . Comm. Pure Appl. Math. 22 (1969)131–151], and of [Fresnel, van der Put, Rigid analytic geometry and its applications . Progress in Mathematics, 218 Birkhäuser (2004)], and unifies work by van den Dries, Haskell, Macintyre, Macpherson, Marker, Robinson, and the authors.
In difference algebra, basic definable sets correspond to prime ideals that are invariant under a structural endomorphism.The main idea of [5] was that periodic prime ideals enjoy better geometric properties …
In difference algebra, basic definable sets correspond to prime ideals that are invariant under a structural endomorphism.The main idea of [5] was that periodic prime ideals enjoy better geometric properties than invariant ideals; and to understand a definable set, it is helpful to enlarge it by relaxing invariance to periodicity, obtaining better geometric properties at the limit.The limit in question was an intriguing but somewhat ephemeral setting called virtual ideals.However a serious technical error was discovered by Tom Scanlon's UCB seminar.In this text, we correct the problem via two different routes.We replace the faulty lemma by a weaker one, that still allows recovering all results of [5] for all virtual ideals.In addition, we introduce a family of difference equations ("cumulative" equations) that we expect to be useful more generally.Results in [4] imply that cumulative equations suffice to coordinatize all difference equation.For cumulative equations, we show that virtual ideals reduce to globally periodic ideals, thus providing a proof of Zilber's trichotomy for difference equations using periodic ideals alone.
This thesis considers theories of expansions of the natural algebraic structure on the multiplicative group and on an elliptic curve by a predicate for a subgroup that are constructed by …
This thesis considers theories of expansions of the natural algebraic structure on the multiplicative group and on an elliptic curve by a predicate for a subgroup that are constructed by Hrushovski’s predimension method. In the case of the multiplicative group, these are the theories of fields with green points constructed by Poizat. The convention of calling the elements of the distinguished subgroup green points is maintained throughout this work, also in the elliptic curve case, and we speak of theories of green points. In the first part of the thesis, we give a detailed account of the construction of the theories of green points. The work of Poizat is extended to the case of elliptic curves and an open question is answered in order to complete the construction in the cases where the distinguished subgroup is allowed to have torsion. Proofs of the main model-theoretic properties of the theories, ω-stability and near model-completeness, are included, as well as rank calculations. In the second part, following ideas of Zilber, we find natural models of the constructed theories on the complex points of the corresponding algebraic group. In the case of elliptic curves, this is done under the assumption that the curve has no complex multiplication and is defined over the reals. In general, we also need to assume a consequence of the Schanuel Conjecture, in the multiplicative group case, and an analogous statement in the elliptic curve case. For the multiplicative group, the assumption is known to hold in generic cases by a theorem of Bays, Kirby and Wilkie; our result is therefore unconditional in these cases. Motivated by Zilber’s work on connections between model theory and noncommutative geometry, we prove similar results for variations of the above theories in which the distinguished subgroup is elementarily equivalent to the additive group of the integers, which we call theories of emerald points.
The main idea of [4] was that structures built from periodic prime ideals have better properties from the usual ones built from invariant ideals; but unable to work with periodic …
The main idea of [4] was that structures built from periodic prime ideals have better properties from the usual ones built from invariant ideals; but unable to work with periodic ideals alone, we had to generalise further to a somewhat ephemeral setting called virtual ideals. This text has two purposes. It corrects an error in [4] discovered by Tom Scanlon's UCB seminar, recovering all results for all virtual ideals. In addition, based on results in [3], we describe a wide family of difference equations where virtual ideals reduce to periodic ideals.
A bstract The Distance Conjecture states that an infinite tower of modes becomes exponentially light when approaching an infinite distance point in field space. We argue that the inherent path-dependence …
A bstract The Distance Conjecture states that an infinite tower of modes becomes exponentially light when approaching an infinite distance point in field space. We argue that the inherent path-dependence of this statement can be addressed when combining the Distance Conjecture with the recent Tameness Conjecture. The latter asserts that effective theories are described by tame geometry and implements strong finiteness constraints on coupling functions and field spaces. By exploiting these tameness constraints we argue that the region near the infinite distance point admits a decomposition into finitely many sectors in which path-independent statements for the associated towers of states can be established. We then introduce a more constrained class of tame functions with at most polynomial asymptotic growth and argue that they suffice to describe the known string theory effective actions. Remarkably, the multi-field dependence of such functions can be reconstructed by one-dimensional linear test paths in each sector near the boundary. In four-dimensional effective theories, these test paths are traced out as a discrete set of cosmic string solutions. This indicates that such cosmic string solutions can serve as powerful tool to study the near-boundary field space region of any four-dimensional effective field theory. To illustrate these general observations we discuss the central role of tameness and cosmic string solutions in Calabi-Yau compactifications of Type IIB string theory.
We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic …
We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.
numbers with exponentiation is model complete. When we combine this with Hovanskii's finiteness theorem [9], it follows that the real exponential field is o-minimal. In o-minimal expansions of the real …
numbers with exponentiation is model complete. When we combine this with Hovanskii's finiteness theorem [9], it follows that the real exponential field is o-minimal. In o-minimal expansions of the real field the definable subsets of R' share many of the nice structural properties of semialgebraic sets. For example, definable subsets have only finitely many connected components, definable sets can be stratified and triangulated, and continuous definable maps are piecewise trivial (see [5]). In this paper we will prove a quantifier elimination result for the real field augmented by exponentiation and all restricted analytic functions, and use this result to obtain o-minimality. We were led to this while studying work of Ressayre [13] and several of his ideas emerge here in simplified form. However, our treatment is formally independent of the results of [16], [17], [9], and [13].
We prove no nontrivial expansion of the field of complex numbers can be obtained from a reduct of the field of real numbers.
We prove no nontrivial expansion of the field of complex numbers can be obtained from a reduct of the field of real numbers.
We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets.
We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets.