Strongly minimal expansions of (ℂ, +) definable in o-minimal fields

Type: Article
Publication Date: 2008-01-04
Citations: 6
DOI: https://doi.org/10.1112/plms/pdm052

Abstract

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.

Locations

  • Proceedings of the London Mathematical Society

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We consider expansions of o-minimal structures on the real field by collections of restrictions to the positive real line of the canonical Weierstrass products associated to sequences such as $(-n^s)_{n>0}$ … We consider expansions of o-minimal structures on the real field by collections of restrictions to the positive real line of the canonical Weierstrass products associated to sequences such as $(-n^s)_{n>0}$ (for $s>0$) and $(-s^n)_{n>0}$ (for $s>1$), and also expansions by associated functions such as logarithmic derivatives. There are only three possible outcomes known for the resulting structures: (i)~o-minimality; (ii)~d-minimality (but not o-minimality); (iii)~definability of $\mathbb{Z}$.
We prove Zilber’s Trichotomy Conjecture for strongly minimal expansions of 2-dimensional groups, definable in o-minimal structures: Theorem. Let \mathcal{M} be an o-minimal expansion of a real closed field, \langle G;+\rangle … We prove Zilber’s Trichotomy Conjecture for strongly minimal expansions of 2-dimensional groups, definable in o-minimal structures: Theorem. Let \mathcal{M} be an o-minimal expansion of a real closed field, \langle G;+\rangle a 2-dimensional group definable in \mathcal{M} , and \mathcal{D}=\langle G;+,\ldots\rangle a strongly minimal structure, all of whose atomic relations are definable in \mathcal{M} . If \mathcal{D} is not locally modular, then an algebraically closed field K is interpretable in \mathcal{D} , and the group G , with all its induced \mathcal{D} -structure, is definably isomorphic in \mathcal{D} to an algebraic K -group with all its induced K -structure.
We prove the higher dimensional case of the o-minimal variant of Zilber's Restricted Trichotomy Conjecture. More precisely, let $\mathcal R$ be an o-minimal expansion of a real closed field, let … We prove the higher dimensional case of the o-minimal variant of Zilber's Restricted Trichotomy Conjecture. More precisely, let $\mathcal R$ be an o-minimal expansion of a real closed field, let $M$ be an interpretable set in $\mathcal R$, and let $\mathcal M=(M,...)$ be a reduct of the induced structure on $M$. If $\mathcal M$ is strongly minimal and not locally modular, then $\dim_{\mathcal R}(M)=2$. As an application, we prove the Zilber trichotomy for all strongly minimal structures interpreted in the theory of compact complex manifolds.
We establish the choice property, a weak analogue of definable choice, for certain tame expansions of o-minimal structures. Most noteworthily, dense pairs of real closed fields have this property. We establish the choice property, a weak analogue of definable choice, for certain tame expansions of o-minimal structures. Most noteworthily, dense pairs of real closed fields have this property.
We establish the choice property, a weak analogue of definable choice, for certain tame expansions of o-minimal structures. Most noteworthily, dense pairs of real closed fields have this property. We establish the choice property, a weak analogue of definable choice, for certain tame expansions of o-minimal structures. Most noteworthily, dense pairs of real closed fields have this property.
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 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.
Every o-minimal expansion of the real field has an o-minimal expansion in which the solutions to Pfaffian equations with definable C1 coefficients are definable. Every o-minimal expansion of the real field has an o-minimal expansion in which the solutions to Pfaffian equations with definable C1 coefficients are definable.
Given an o-minimal expansion R of the real field, we show that the structure obtained from R by iterating the operation of adding all total Pfaffian functions over R defines … Given an o-minimal expansion R of the real field, we show that the structure obtained from R by iterating the operation of adding all total Pfaffian functions over R defines the same sets as the Pfaffian closure of R.
We give some examples of polynomially bounded o-minimal expansions of the ordered field of real numbers where the Weierstrass division theorem does not hold in the ring of germs, at … We give some examples of polynomially bounded o-minimal expansions of the ordered field of real numbers where the Weierstrass division theorem does not hold in the ring of germs, at the origin of $\mathbb{R}^n$, of definable $C^{\infty}$ functions.
In 1934, H. Whitney asked how one can determine whether a real-valued function on a closed subset of \mathbb{R}^n is the restriction of a C^m -function on \mathbb{R}^n . A … In 1934, H. Whitney asked how one can determine whether a real-valued function on a closed subset of \mathbb{R}^n is the restriction of a C^m -function on \mathbb{R}^n . A complete answer to this question was found much later by C. Fefferman in the early 2000s. Here, we work in an o-minimal expansion of a real closed field and solve the C^1 -case of Whitney's extension problem in this context. Our main tool is a definable version of Michael's selection theorem, and we include other another application of this theorem, to solving linear equations in the ring of definable continuous functions.

Cited by (6)

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.
The idea of applying methods and results from stability theory to unstable theories has been an important theme over the past 25 years, with o-minimality, smoothly approximable structures, and simple … The idea of applying methods and results from stability theory to unstable theories has been an important theme over the past 25 years, with o-minimality, smoothly approximable structures, and simple theories being key examples. But there have been some key recent developments which bring new ideas and techniques to the table. One of these is the investigation of abstract notions of independence, leading for example to the notions of thorn forking and rosiness. Another is the discovery that forking, weight, and related notions from stability are meaningful in dependent theories. Another is the formulation of notions of stable, compact, or more general domination, coming from the analysis of theories such as algebraically closed valued fields and o-minimal theories. The level of different approaches and techniques which end up overlapping was the reason we decided it would be a perfect time for a research meeting where the most prominent researchers would come together and discuss the ideas, results and goals that were showing up in different contexts. The dominant subjects of the meeting were the following.
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.

References (21)

We work in an o-minimal expansion of a real closed field. Using piecewise smoothness of definable functions we define the topological degree for definable continuous functions. Using this notion of … We work in an o-minimal expansion of a real closed field. Using piecewise smoothness of definable functions we define the topological degree for definable continuous functions. Using this notion of the degree we obtain a new proof for the existence of torsion points in a definably compact group, and also a new proof of an o-minimal analogue of the Brouwer fixed point theorem.
The article surveys some topics related to o-minimality, and is based on three lectures. The emphasis is on o-minimality as an analogue of strong minimality, rather than as a setting … The article surveys some topics related to o-minimality, and is based on three lectures. The emphasis is on o-minimality as an analogue of strong minimality, rather than as a setting for the model theory of expansions of the reals. Section 2 gives some basics (the Monotonicity and Cell Decomposition Theorems) together with a discussion of dimension. Section 3 concerns the Peterzil–Starchenko Trichotomy Theorem (an o-minimal analogue of Zil’ber Trichotomy). There follows some material on definable groups, with powerful applications of the Trichotomy Theorem in work by Peterzil, Pillay and Starchenko. The final section introduces weak o-minimality, P -minimality, and C-minimality. These are analogues of o-minimality intended as settings for certain henselian valued fields with extra structure.
1. Naming of parts 2. Classifying structures 3. Structures that look alike 4. Interpretations 5. The first order case: compactness 6. The countable case 7. The existential case 8. Saturation … 1. Naming of parts 2. Classifying structures 3. Structures that look alike 4. Interpretations 5. The first order case: compactness 6. The countable case 7. The existential case 8. Saturation 9. Structure and categoricity.
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}=\langle G, \cdot \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a group definable in an o-minimal structure <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>. A subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\langle G,\cdot \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (while <italic>definable</italic> means definable in the structure <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>). Assume <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable proper subgroup of finite index. In this paper we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no nontrivial abelian normal subgroup, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the direct product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable subgroups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1 comma ellipsis comma upper H Subscript k Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">H_1,\ldots ,H_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript i"> <mml:semantics> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">H_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin’s conjecture.
A difference field is a field with a distinguished automorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"><mml:semantics><mml:mi>σ</mml:mi><mml:annotation encoding="application/x-tex">\sigma</mml:annotation></mml:semantics></mml:math></inline-formula>. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory … A difference field is a field with a distinguished automorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"><mml:semantics><mml:mi>σ</mml:mi><mml:annotation encoding="application/x-tex">\sigma</mml:annotation></mml:semantics></mml:math></inline-formula>. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory on formulas, and in particular on difference equations. We show that an arbitrary formula may be reduced into one-dimensional ones, and analyze the possible internal structures on the one-dimensional formulas when the characteristic is<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"><mml:semantics><mml:mn>0</mml:mn><mml:annotation encoding="application/x-tex">0</mml:annotation></mml:semantics></mml:math></inline-formula>.
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.
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 … 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)
Let K be an algebraically closed field and let L be its canonical language ; that is, L consists of all relations on K which are definable from addition, multiplication, … Let K be an algebraically closed field and let L be its canonical language ; that is, L consists of all relations on K which are definable from addition, multiplication, and parameters from K . Two sublanguages L 1 and L 2 of L are definably equivalent if each relation in L 1 can be defined by an L 2 -formula with parameters in K , and vice versa. The equivalence classes of sublanguages of L form a quotient lattice of the power set of L about which very little is known. We will not distinguish between a sublanguage and its equivalence class. Let L m denote the language of multiplication alone, and let L a denote the language of addition alone. Let f ∈ K [ X, Y ] and consider the algebraic function defined by f ( x, y ) = 0 for x, y ∈ K . Let L f denote the language consisting of the relation defined by f . The possibilities for L m ∨ L f are examined in §2, and the possibilities for L a ∨ L f are examined in §3. In fact the only comprehensive results known are under the additional hypothesis that f actually defines a rational function (i.e., when f is linear in one of the variables), and in positive characteristic, only expansions of addition by polynomials (i.e., when f is 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 that L covers L m and that the only languages between L a and L are expansions of L a by scalar multiplications.
Abstract We show that the structure (C, +, ·) has no proper non locally modular reducts which contain +. In other words, if X ⊂ C n is constructible and … Abstract We show that the structure (C, +, ·) has no proper non locally modular reducts which contain +. In other words, if X ⊂ C n is constructible and not definable in the module structure (C, +, λ a ) a Є C (where λ a denotes multiplication by a ) then multiplication is definable in ( C , +, X ).
We fix an arbitrary o -minimal structure ( R , ω, …), where ( R , &lt;) is a dense linearly ordered set without end points. In this paper “definable” … We fix an arbitrary o -minimal structure ( R , ω, …), where ( R , &lt;) is a dense linearly ordered set without end points. In this paper “definable” means “definable with parameters from R ”, We equip R with the interval topology and R n with the induced product topology. The main result of this paper is the following. Theorem. Let V ⊆ R n be a definable open set and suppose that f : V → R n is a continuous injective definable map. Then f is open, that is, f(U) is open whenever U is an open subset of V . Woerheide [6] proved the above theorem for o -minimal expansions of a real closed field using ideas of homology. The case of an arbitrary o -minimal structure remained an open problem, see [4] and [1]. In this paper we will give an elementary proof of the general case. Basic definitions and notation . A box B ⊆ R n is a Cartesian product of n definable open intervals: B = ( a 1 , b 1 ) × … × ( a n , b n ) for some a i , b i , ∈ R ∪ {−∞, +∞}, with a i &lt; b i , Given A ⊆ R n , cl( A ) denotes the closure of A , int( A ) denotes the interior of A , bd( A ) ≔ cl( A ) − int( A) denotes the boundary of A , and ∂ A ≔ cl( A ) − A denotes the frontier of A , Finally, we let π: R n → R n − denote the projection map onto the first n − 1 coordinates. Background material . Without mention we will use notions and facts discussed in [5] and [3]. We will also make use of the following result, which appears in [2].
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 … 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998). By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998).
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.
Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine … Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine structure of models of stable theories, using the geometry of forking; this often achieves global results relevant to classification theory. Topics range from Zilber-Cherlin classification of infinite locally finite homogenous geometries, to regular types, their geometries, and their role in superstable theories. The structure and existence of definable groups is featured prominently, as is work by Hrushovski. The book is unique in the range and depth of material covered and will be invaluable to anyone interested in modern model theory.