We show that a large class of divisible abelian-groups (lattice ordered groups) of continuous fun... more We show that a large class of divisible abelian-groups (lattice ordered groups) of continuous functions is interpretable (in a certain sense) in the lattice of the zero sets of these functions. This has various applications to the model theory of these-groups, including decidability results. Contents 1. Introduction 1 2. Constructible-groups 2 3. Translating into the lattice of cozero sets 6 4. Applications 13 References 17
We introduce the notion of differential largeness for fields equipped with several commuting deri... more We introduce the notion of differential largeness for fields equipped with several commuting derivations (as an analogue to largeness of fields). We lay out the foundations of this new class of "tame" differential fields. We state several characterizations and exhibit plenty of examples and applications. Our results strongly indicate that differentially large fields will play a key role in differential field arithmetic. For instance, we characterise differential largeness in terms of being existentially closed in their power series field (furnished with natural derivations), we give explicit constructions of differentially large fields in terms of iterated powers series, we prove that the class of differentially large fields is elementary, and we show that differential largeness is preserved under algebraic extensions, therefore showing that their algebraic closure is differentially closed. Contents 1. Introduction 1 2. Preliminaries 4 3. The Taylor Morphism 9 4. Differentially large fields and algebraic characterisations 13 5. Fundamental properties and constructions 18 6. Algebraic-geometric axioms 25 References 29
In this short note a differential version of the classical Weil descent is established in all cha... more In this short note a differential version of the classical Weil descent is established in all characteristics. This yields a ready-to-deploy tool of differential restriction of scalars for differential varieties over finite differential field extensions.
Quantifier elimination of matrix rings $M_n(K)$ for $K$ a formally real field is characterized in... more Quantifier elimination of matrix rings $M_n(K)$ for $K$ a formally real field is characterized in the language of rings extended by trace and transposition, in terms of invariant theory. This is used to prove quantifier elimination when $K$ is an intersection of real closed fields. For dimension-free matrices it is shown that no such result can hold by establishing various undecidability results.
A differential version of the classical Weil descent is established in all characteristics. It yi... more A differential version of the classical Weil descent is established in all characteristics. It yields a theory of differential restriction of scalars for differential varieties over finite differential field extensions. This theory is then used to prove that in characteristic 0, \textit{differential largeness} (a notion introduced here as an analogue to largeness of fields) is preserved under algebraic extensions. This provides many new differential fields with minimal differential closures. A further application is Kolchin-density of rational points in differential algebraic groups defined over differentially large fields.
We study the model theoretic strength of various lattices that occur naturally in topology, like ... more We study the model theoretic strength of various lattices that occur naturally in topology, like closed (semi-linear or semi-algebraic or convex) sets. The method is based on weak monadic second order logic and sharpens previous results by Grzegorczyk. We also answers a question of Grzegorczyk on the 'algebra of convex sets'.
Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valu... more Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: see text]-minimal, or an expansion of a local field, we prove that the ring of integers [Formula: see text] is interpretable in [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] is definably connected of pure dimension [Formula: see text], then [Formula: see text] defines the subring [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] has no isolated points, then there is a discrete ring [Formula: see text] contained ...
Annales de la Faculté des sciences de Toulouse : Mathématiques, 2012
L'accès aux articles de la revue « Annales de la faculté des sciences de Toulouse Mathématiques »... more L'accès aux articles de la revue « Annales de la faculté des sciences de Toulouse Mathématiques » (http://afst.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http://afst.cedram. org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/
A box type is an n-type of an o-minimal structure which is uniquely determined by the projections... more A box type is an n-type of an o-minimal structure which is uniquely determined by the projections to the coordinate axes. We characterize heirs of box types of a polynomially bounded o-minimal structure M. From this, we deduce various structure theorems for subsets of Mk , definable in the expansion of M by all convex subsets of the line. We show that after naming constants, is model complete provided M is model complete.
We explain how the field of logarithmic-exponential series constructed in [DMM1] and [DMM2] embed... more We explain how the field of logarithmic-exponential series constructed in [DMM1] and [DMM2] embeds as an exponential field in any field of exponential-logarithmic series constructed in [KK1], [K] and [KS]. On the other hand, we explain why no field of exponential-logarithmic series embeds in the field of logarithmic-exponential series. This clarifies why the two constructions are intrinsically different, in the sense that they produce non-isomorphic models of Th(Ran, exp); the elementary theory of the ordered field of real numbers, with the exponential function and restricted analytic functions. Contents 1. Introduction 1 2. Reminder on generalized power series. 3 3. Prelogarithmic sections and associated prelogarithms. 4 4. Main examples of prelogarithmic series fields 5 5. The exponential extension of a prelogarithmic field. 6 6. The exponential closure of a prelogarithmic field 7 7. Morphisms of prelogarithmic fields. 9 8. Groups satisfying the growth axiom. 12 9. LE-series constructions. 13 10. Finding the LE-series field in the exponential field generated by logarithmic words. 17 11. The L-operation. 18 References 19
We investigate connections between arithmetic properties of rings and topological properties of t... more We investigate connections between arithmetic properties of rings and topological properties of their prime spectrum. Any property that the prime spectrum of a ring may or may not have, defines the class of rings whose prime spectrum has the given property. We ask whether a class of rings defined in this way is axiomatizable in the model theoretic sense. Answers are provided for a variety of different properties of prime spectra, e.g., normality or complete normality, Hausdorffness of the space of maximal points, compactness of the space of minimal points.
Abstract. We explain how the field of logarithmic-exponential series con-structed in [DMM1] and [... more Abstract. We explain how the field of logarithmic-exponential series con-structed in [DMM1] and [DMM2] embeds as an exponential field in any field of exponential-logarithmic series (constructed in [KK1], [K] and [KS]). On the other hand, we explain why no field of ...
We explain how the field of logarithmic-exponential series constructed in [DMM1] and [DMM2] embed... more We explain how the field of logarithmic-exponential series constructed in [DMM1] and [DMM2] embeds as an exponential field in any field of exponential-logarithmic series constructed in [KK1], [K] and [KS]. On the other hand, we explain why no field of exponential-logarithmic series embeds in the field of logarithmic-exponential series. This clarifies why the two constructions are intrinsically different, in the sense that they produce non-isomorphic models of Th(Ran, exp); the elementary theory of the ordered field of real numbers, with the exponential function and restricted analytic functions. of prelogarithmic fields. We would like to thank Mickaël Matusinski for reading a preliminary version of this paper and providing helpful comments.
This is an extended abstract of work in progress on a new class of rings called p-adically closed... more This is an extended abstract of work in progress on a new class of rings called p-adically closed rings. These generalise the notion of a p-adically closed field to commutative rings and serve as rings of sections of what one might call abstract padic functions associated to an arbitrary commutative ring. What we have in mind is an approach to the topology of p-adic sets parallel to the real case where Niels Schwartz in [Schw] has developed abstract semi-algebraic spaces and functions; these are certain ringed spaces whose affine models have so-called real closed rings as rings of sections. A direct parallel approach in the p-adic case seems difficult and we still do not have a good algebraic description of p-adically closed rings. We hope to be able to generalise Luc Bélair's work , where local p-adically closed rings are studied, to obtain such an explicit description.
We show that a large class of divisible abelian-groups (lattice ordered groups) of continuous fun... more We show that a large class of divisible abelian-groups (lattice ordered groups) of continuous functions is interpretable (in a certain sense) in the lattice of the zero sets of these functions. This has various applications to the model theory of these-groups, including decidability results. Contents 1. Introduction 1 2. Constructible-groups 2 3. Translating into the lattice of cozero sets 6 4. Applications 13 References 17
We introduce the notion of differential largeness for fields equipped with several commuting deri... more We introduce the notion of differential largeness for fields equipped with several commuting derivations (as an analogue to largeness of fields). We lay out the foundations of this new class of "tame" differential fields. We state several characterizations and exhibit plenty of examples and applications. Our results strongly indicate that differentially large fields will play a key role in differential field arithmetic. For instance, we characterise differential largeness in terms of being existentially closed in their power series field (furnished with natural derivations), we give explicit constructions of differentially large fields in terms of iterated powers series, we prove that the class of differentially large fields is elementary, and we show that differential largeness is preserved under algebraic extensions, therefore showing that their algebraic closure is differentially closed. Contents 1. Introduction 1 2. Preliminaries 4 3. The Taylor Morphism 9 4. Differentially large fields and algebraic characterisations 13 5. Fundamental properties and constructions 18 6. Algebraic-geometric axioms 25 References 29
In this short note a differential version of the classical Weil descent is established in all cha... more In this short note a differential version of the classical Weil descent is established in all characteristics. This yields a ready-to-deploy tool of differential restriction of scalars for differential varieties over finite differential field extensions.
Quantifier elimination of matrix rings $M_n(K)$ for $K$ a formally real field is characterized in... more Quantifier elimination of matrix rings $M_n(K)$ for $K$ a formally real field is characterized in the language of rings extended by trace and transposition, in terms of invariant theory. This is used to prove quantifier elimination when $K$ is an intersection of real closed fields. For dimension-free matrices it is shown that no such result can hold by establishing various undecidability results.
A differential version of the classical Weil descent is established in all characteristics. It yi... more A differential version of the classical Weil descent is established in all characteristics. It yields a theory of differential restriction of scalars for differential varieties over finite differential field extensions. This theory is then used to prove that in characteristic 0, \textit{differential largeness} (a notion introduced here as an analogue to largeness of fields) is preserved under algebraic extensions. This provides many new differential fields with minimal differential closures. A further application is Kolchin-density of rational points in differential algebraic groups defined over differentially large fields.
We study the model theoretic strength of various lattices that occur naturally in topology, like ... more We study the model theoretic strength of various lattices that occur naturally in topology, like closed (semi-linear or semi-algebraic or convex) sets. The method is based on weak monadic second order logic and sharpens previous results by Grzegorczyk. We also answers a question of Grzegorczyk on the 'algebra of convex sets'.
Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valu... more Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: see text]-minimal, or an expansion of a local field, we prove that the ring of integers [Formula: see text] is interpretable in [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] is definably connected of pure dimension [Formula: see text], then [Formula: see text] defines the subring [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] has no isolated points, then there is a discrete ring [Formula: see text] contained ...
Annales de la Faculté des sciences de Toulouse : Mathématiques, 2012
L'accès aux articles de la revue « Annales de la faculté des sciences de Toulouse Mathématiques »... more L'accès aux articles de la revue « Annales de la faculté des sciences de Toulouse Mathématiques » (http://afst.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http://afst.cedram. org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/
A box type is an n-type of an o-minimal structure which is uniquely determined by the projections... more A box type is an n-type of an o-minimal structure which is uniquely determined by the projections to the coordinate axes. We characterize heirs of box types of a polynomially bounded o-minimal structure M. From this, we deduce various structure theorems for subsets of Mk , definable in the expansion of M by all convex subsets of the line. We show that after naming constants, is model complete provided M is model complete.
We explain how the field of logarithmic-exponential series constructed in [DMM1] and [DMM2] embed... more We explain how the field of logarithmic-exponential series constructed in [DMM1] and [DMM2] embeds as an exponential field in any field of exponential-logarithmic series constructed in [KK1], [K] and [KS]. On the other hand, we explain why no field of exponential-logarithmic series embeds in the field of logarithmic-exponential series. This clarifies why the two constructions are intrinsically different, in the sense that they produce non-isomorphic models of Th(Ran, exp); the elementary theory of the ordered field of real numbers, with the exponential function and restricted analytic functions. Contents 1. Introduction 1 2. Reminder on generalized power series. 3 3. Prelogarithmic sections and associated prelogarithms. 4 4. Main examples of prelogarithmic series fields 5 5. The exponential extension of a prelogarithmic field. 6 6. The exponential closure of a prelogarithmic field 7 7. Morphisms of prelogarithmic fields. 9 8. Groups satisfying the growth axiom. 12 9. LE-series constructions. 13 10. Finding the LE-series field in the exponential field generated by logarithmic words. 17 11. The L-operation. 18 References 19
We investigate connections between arithmetic properties of rings and topological properties of t... more We investigate connections between arithmetic properties of rings and topological properties of their prime spectrum. Any property that the prime spectrum of a ring may or may not have, defines the class of rings whose prime spectrum has the given property. We ask whether a class of rings defined in this way is axiomatizable in the model theoretic sense. Answers are provided for a variety of different properties of prime spectra, e.g., normality or complete normality, Hausdorffness of the space of maximal points, compactness of the space of minimal points.
Abstract. We explain how the field of logarithmic-exponential series con-structed in [DMM1] and [... more Abstract. We explain how the field of logarithmic-exponential series con-structed in [DMM1] and [DMM2] embeds as an exponential field in any field of exponential-logarithmic series (constructed in [KK1], [K] and [KS]). On the other hand, we explain why no field of ...
We explain how the field of logarithmic-exponential series constructed in [DMM1] and [DMM2] embed... more We explain how the field of logarithmic-exponential series constructed in [DMM1] and [DMM2] embeds as an exponential field in any field of exponential-logarithmic series constructed in [KK1], [K] and [KS]. On the other hand, we explain why no field of exponential-logarithmic series embeds in the field of logarithmic-exponential series. This clarifies why the two constructions are intrinsically different, in the sense that they produce non-isomorphic models of Th(Ran, exp); the elementary theory of the ordered field of real numbers, with the exponential function and restricted analytic functions. of prelogarithmic fields. We would like to thank Mickaël Matusinski for reading a preliminary version of this paper and providing helpful comments.
This is an extended abstract of work in progress on a new class of rings called p-adically closed... more This is an extended abstract of work in progress on a new class of rings called p-adically closed rings. These generalise the notion of a p-adically closed field to commutative rings and serve as rings of sections of what one might call abstract padic functions associated to an arbitrary commutative ring. What we have in mind is an approach to the topology of p-adic sets parallel to the real case where Niels Schwartz in [Schw] has developed abstract semi-algebraic spaces and functions; these are certain ringed spaces whose affine models have so-called real closed rings as rings of sections. A direct parallel approach in the p-adic case seems difficult and we still do not have a good algebraic description of p-adically closed rings. We hope to be able to generalise Luc Bélair's work , where local p-adically closed rings are studied, to obtain such an explicit description.
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