Jonathan Kirby

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The theory of the exponential differential equations of semiabelian varieties

2009-07-16

Jonathan Kirby

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On quasiminimal excellent classes

2010-03-18

Jonathan Kirby

A careful exposition of Zilber's quasiminimal excellent classes and their categoricity is given, leading to two new results: the L_w1,w(Q)-definability assumption may be dropped, and each class is determined by … A careful exposition of Zilber's quasiminimal excellent classes and their categoricity is given, leading to two new results: the L_w1,w(Q)-definability assumption may be dropped, and each class is determined by its model of dimension aleph_0.

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Exponential algebraicity in exponential fields

2010-06-17

Jonathan Kirby

I give an algebraic proof that the exponential algebraic closure operator in an exponential field is always a pregeometry, and show that its dimension function satisfies a weak Schanuel property.A … I give an algebraic proof that the exponential algebraic closure operator in an exponential field is always a pregeometry, and show that its dimension function satisfies a weak Schanuel property.A corollary is that there are at most countably many essential counterexamples to Schanuel's conjecture.

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Quasiminimal structures and excellence

2013-10-24

Martin Bays , Bradd Hart , Tapani Hyttinen +2 more

We show that the excellence axiom in the definition of Zilber's quasiminimal excellent classes is redundant, in that it follows from the other axioms. This substantially simplifies a number of … We show that the excellence axiom in the definition of Zilber's quasiminimal excellent classes is redundant, in that it follows from the other axioms. This substantially simplifies a number of categoricity proofs.

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Pseudo-exponential maps, variants, and quasiminimality

2018-06-12

Martin Bays , Jonathan Kirby

We give a construction of quasiminimal fields equipped with pseudo-analytic maps, generalising Zilber's pseudo-exponential function. In particular we construct pseudo-exponential maps of simple abelian varieties, including pseudo-$\wp$-functions for elliptic curves. … We give a construction of quasiminimal fields equipped with pseudo-analytic maps, generalising Zilber's pseudo-exponential function. In particular we construct pseudo-exponential maps of simple abelian varieties, including pseudo-$\wp$-functions for elliptic curves. We show that the complex field with the corresponding analytic function is isomorphic to the pseudo-analytic version if and only the appropriate version of Schanuel's conjecture is true and the corresponding version of the strong exponential-algebraic closedness property holds. Moreover, we relativize the construction to build a model over a fairly arbitrary countable subfield and deduce that the complex exponential field is quasiminimal if it is exponentially-algebraically closed. This property asks only that the graph of exponentiation have non-trivial intersection with certain algebraic varieties but does not require genericity of these points. Furthermore Schanuel's conjecture is not required as a condition for quasiminimality.

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Exponentially closed fields and the conjecture on intersections with tori

2014-07-08

Jonathan Kirby , Boris Zilber

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A Schanuel property for exponentially transcendental powers

2010-08-04

Martin Bays , Jonathan Kirby , A. J. Wilkie

We prove the analogue of Schanuel's conjecture for raising to the power of an exponentially transcendental real number. All but countably many real numbers are exponentially transcendental. We also give … We prove the analogue of Schanuel's conjecture for raising to the power of an exponentially transcendental real number. All but countably many real numbers are exponentially transcendental. We also give a more general result for several powers in a context which encompasses the complex case.

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Finitely presented exponential fields

2013-08-29

Jonathan Kirby

The algebra of exponential fields and their extensions is developed. The focus is on ELA-fields, which are algebraically closed with a surjective exponential map. In this context, finitely presented extensions … The algebra of exponential fields and their extensions is developed. The focus is on ELA-fields, which are algebraically closed with a surjective exponential map. In this context, finitely presented extensions are defined, it is shown that finitely generated strong extensions are finitely presented, and these extensions are classified. An algebraic construction is given of Zilber's pseudo-exponential fields. As applications of the general results and methods of the paper, it is shown that Zilber's fields are not model-complete, answering a question of Macintyre, and a precise statement is given explaining how Schanuel's conjecture answers all transcendence questions about exponentials and logarithms. Connections with the Kontsevich-Zagier, Grothendieck, and Andr\'e transcendence conjectures on periods are discussed, and finally some open problems are suggested.

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Existentially closed exponential fields

2021-01-15

Levon Haykazyan , Jonathan Kirby

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A Note on the Axioms for Zilber’s Pseudo-Exponential Fields

2013-01-01

Jonathan Kirby

We show that Zilber's conjecture that complex exponentiation is isomorphic to his pseudo-exponentiation follows from the a priori simpler conjecture that they are elementarily equivalent. An analysis of the first-order … We show that Zilber's conjecture that complex exponentiation is isomorphic to his pseudo-exponentiation follows from the a priori simpler conjecture that they are elementarily equivalent. An analysis of the first-order types in pseudo-exponentiation leads to a description of the elementary embeddings, and the result that pseudo-exponential fields are precisely the models of their common first-order theory which are atomic over exponential transcendence bases. We also show that the class of all pseudo-exponential fields is an example of a nonfinitary abstract elementary class, answering a question of Kesälä and Baldwin.

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Differential existential closedness for the 𝑗-function

2020-11-30

Vahagn Aslanyan , Sebastian Eterović , Jonathan Kirby

We prove the Existential Closedness conjecture for the differential equation of the $j$-function and its derivatives. It states that in a differentially closed field certain equations involving the differential equation … We prove the Existential Closedness conjecture for the differential equation of the $j$-function and its derivatives. It states that in a differentially closed field certain equations involving the differential equation of the $j$-function have solutions. Its consequences include a complete axiomatisation of $j$-reducts of differentially closed fields, a dichotomy result for strongly minimal sets in those reducts, and a functional analogue of the Modular Zilber-Pink with Derivatives conjecture.

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A Schanuel condition for Weierstrass equations

2005-06-01

Jonathan Kirby

Abstract I prove a version of Schanuel's conjecture for Weierstrass equations in differential fields, answering a question of Zilber, and show that the linear independence condition in the statement cannot … Abstract I prove a version of Schanuel's conjecture for Weierstrass equations in differential fields, answering a question of Zilber, and show that the linear independence condition in the statement cannot be relaxed.

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Excellence and uncountable categoricity of Zilber's exponential fields

2013-01-01

Martin Bays , Jonathan Kirby

We prove that Zilber's class of exponential fields is quasiminimal excellent and hence uncountably categorical, filling two gaps in Zilber's original proof. We prove that Zilber's class of exponential fields is quasiminimal excellent and hence uncountably categorical, filling two gaps in Zilber's original proof.

The algebraic numbers definable in various exponential fields

2012-04-02

Jonathan Kirby , Angus Macintyre , Alf Onshuus

Abstract We prove the following theorems. Theorem 1: for any E-field with cyclic kernel, in particular ℂ or the Zilber fields, all real abelian algebraic numbers are pointwise definable. Theorem … Abstract We prove the following theorems. Theorem 1: for any E-field with cyclic kernel, in particular ℂ or the Zilber fields, all real abelian algebraic numbers are pointwise definable. Theorem 2: for the Zilber fields, the only pointwise definable algebraic numbers are the real abelian numbers.

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A Geometric Approach to Some Systems of Exponential Equations

2021-11-10

Vahagn Aslanyan , Jonathan Kirby , Vincenzo Mantova

Zilber's Exponential Algebraic Closedness conjecture (also known as Zilber's Nullstellensatz) gives conditions under which a complex algebraic variety should intersect the graph of the exponential map of a semiabelian variety. … Zilber's Exponential Algebraic Closedness conjecture (also known as Zilber's Nullstellensatz) gives conditions under which a complex algebraic variety should intersect the graph of the exponential map of a semiabelian variety. We prove the special case of the conjecture where the variety has dominant projection to the domain of the exponential map, for abelian varieties and for algebraic tori. Furthermore, in the situation where the intersection is 0-dimensional, we exhibit structure in the intersection by parametrizing the sufficiently large points as the images of the period lattice under a (multivalued) analytic map. Our approach is complex geometric, in contrast to a real analytic proof given by Brownawell and Masser just for the case of algebraic tori.

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LOCAL INTERDEFINABILITY OF WEIERSTRASS ELLIPTIC FUNCTIONS

2014-11-24

Gareth A. Jones , Jonathan Kirby , Tamara Servi

We explain which Weierstrass elliptic functions are locally definable from other elliptic functions and exponentiation in the context of o-minimal structures. The proofs make use of the predimension method from … We explain which Weierstrass elliptic functions are locally definable from other elliptic functions and exponentiation in the context of o-minimal structures. The proofs make use of the predimension method from model theory to exploit functional transcendence theorems in a systematic way.

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Blurred complex exponentiation

2019-11-13

Jonathan Kirby

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A closure operator respecting the modular j-function

2022-10-05

Vahagn Aslanyan , Sebastian Eterović , Jonathan Kirby

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Blurrings Of The J-Function

2021-08-04

Vahagn Aslanyan , Jonathan Kirby

Inspired by the idea of blurring the exponential function, we define blurred variants of the $j$-function and its derivatives, where blurring is given by the action of a subgroup of … Inspired by the idea of blurring the exponential function, we define blurred variants of the $j$-function and its derivatives, where blurring is given by the action of a subgroup of $\rm{GL}_2(\mathbb{C})$. For a dense subgroup (in the complex topology) we prove an Existential Closedness theorem which states that all systems of equations in terms of the corresponding blurred $j$ with derivatives have complex solutions, except where there is a functional transcendence reason why they should not. For the $j$-function without derivatives we prove a stronger theorem, namely, Existential Closedness for $j$ blurred by the action of a subgroup which is dense in $\rm{GL}_2^+(\mathbb{R})$, but not necessarily in $\rm{GL}_2(\mathbb{C})$. We also show that for a suitably chosen countable algebraically closed subfield $C \subseteq \mathbb{C}$, the complex field augmented with a predicate for the blurring of the $j$-function by $\rm{GL}_2(C)$ is model theoretically tame, in particular, $\omega$-stable and quasiminimal.

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On Local definability of holomorphic functions

2019-05-01

Gareth A. Jones , Jonathan Kirby , Olivier Le Gal +1 more

Abstract Given a collection $\mathcal {A}$ of holomorphic functions, we consider how to describe all the holomorphic functions locally definable from $\mathcal {A}$. The notion of local definability of holomorphic … Abstract Given a collection $\mathcal {A}$ of holomorphic functions, we consider how to describe all the holomorphic functions locally definable from $\mathcal {A}$. The notion of local definability of holomorphic functions was introduced by Wilkie, who gave a complete description of all functions locally definable from $\mathcal {A}$ in the neighbourhood of a generic point. We prove that this description is no longer complete in the neighbourhood of non-generic points. More precisely, we produce three examples of holomorphic functions that suggest that at least three new operations need to be added to Wilkie’s description in order to capture local definability in its entirety. The constructions illustrate the interaction between resolution of singularities and definability in the o-minimal setting.

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THE UNIFORM SCHANUEL CONJECTURE OVER THE REAL NUMBERS

2006-07-24

Jonathan Kirby , Boris Zilber

We prove that Schanuel's conjecture for the reals is equivalent to a uniform version of itself. 2000 Mathematics Subject Classification 11J81, 11U09, 03C64. We prove that Schanuel's conjecture for the reals is equivalent to a uniform version of itself. 2000 Mathematics Subject Classification 11J81, 11U09, 03C64.

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The rational field is not universally definable in pseudo-exponentiation

2015-10-19

Jonathan Kirby

We show that the field of rational numbers is not definable by a universal formula in Zilber's pseudo-exponential field. We show that the field of rational numbers is not definable by a universal formula in Zilber's pseudo-exponential field.

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Quasiminimality of complex powers

2024-01-01

Francesco Gallinaro , Jonathan Kirby

Abstract The complex field, equipped with the multivalued functions of raising to each complex power, is quasiminimal, proving a conjecture of Zilber and providing evidence towards his stronger conjecture that … Abstract The complex field, equipped with the multivalued functions of raising to each complex power, is quasiminimal, proving a conjecture of Zilber and providing evidence towards his stronger conjecture that the complex exponential field is quasiminimal.

Corrigendum to “A Schanuel Condition for Weierstrass Equations”

2005-07-22

Jonathan Kirby

An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

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Some pseudo-analytic functions on commutative algebraic groups

2015-12-14

Martin Bays , Jonathan Kirby

We give a construction of quasiminimal fields equipped with pseudo-analytic functions, generalising Zilber's pseudo-exponential function and correcting some errors. In particular we construct pseudo-Weierstrass p-functions and generic power functions. For … We give a construction of quasiminimal fields equipped with pseudo-analytic functions, generalising Zilber's pseudo-exponential function and correcting some errors. In particular we construct pseudo-Weierstrass p-functions and generic power functions. For each structure we give uncountably categorical axioms. We show that the complex field with the corresponding analytic function is isomorphic to the pseudo-analytic version if and only the appropriate version of Schanuel's conjecture is true and a corresponding strong $\Gamma$-closedness property holds. As an application we show that the complex exponential field is quasiminimal under the much weaker hypothesis that it is generically strongly exponentially-algebraically closed, without the assumption of Schanuel's conjecture.

Variants of Schanuel's conjecture

2018-01-01

Jonathan Kirby

This is a collection of variants of Schanuel's conjecture and the known dependencies between them. It was originally written in 2007, and made available for a time on my webpage. … This is a collection of variants of Schanuel's conjecture and the known dependencies between them. It was originally written in 2007, and made available for a time on my webpage. I have been asked by a few people to make it available again and have taken the opportunity to make some minor revisions now in January 2018. The treatment is far from exhaustive of the literature, and for the most part consists of statements which were under discussion in the Oxford logic group between around 2002 and 2007.

Higher amalgamation properties in stable theories

2018-01-01

David M. Evans , Jonathan Kirby , Tim Zander

For a complete, stable theory $T$ we construct, in a reasonably canonical way, a related stable theory $T^*$ which has higher independent amalgamation properties over the algebraic closure of the … For a complete, stable theory $T$ we construct, in a reasonably canonical way, a related stable theory $T^*$ which has higher independent amalgamation properties over the algebraic closure of the empty-set. The theory $T^*$ is an algebraic cover of $T$ and we give an explicit description of the finite covers involved in the construction of $T^*$ from $T$. This follows an approach of E. Hrushovski. If $T$ is almost strongly minimal with a $0$-definable strongly minimal set, then we show that $T^*$ has higher amalgamation over any algebraically closed subset.

Power Sets and Boolean Algebras

2019-04-04

Jonathan Kirby

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.

Categoricity and Completeness

2019-04-04

Jonathan Kirby

The Successor Structure

2019-04-04

Jonathan Kirby

Substructures and Quantifiers

2019-04-04

Jonathan Kirby

Characterising Definable Sets

2019-04-04

Jonathan Kirby

Semi-algebraic Sets

2019-04-04

Jonathan Kirby

Hilbert’s Nullstellensatz

2019-04-04

Jonathan Kirby

A closure operator respecting the modular j-function

2020-09-30

Vahagn Aslanyan , Sebastian Eterović , Jonathan Kirby

A geometric approach to some systems of exponential equations

2021-11-09

Vahagn Aslanyan , Jonathan Kirby , Vincenzo Mantova

Linear Orders

2019-04-04

Jonathan Kirby

The Complex and Real Fields

2019-04-04

Jonathan Kirby

Theories and Axioms

2019-04-04

Jonathan Kirby

Vector Spaces and Categoricity

2019-04-04

Jonathan Kirby

Theories and Compactness

2019-04-04

Jonathan Kirby

Constructing Models from Syntax

2019-04-04

Jonathan Kirby

Real Vector Spaces and Parameters

2019-04-04

Jonathan Kirby

Axiomatisable Classes

2019-04-04

Jonathan Kirby

Definable Sets

2019-04-04

Jonathan Kirby

Compactness and New Constants

2019-04-04

Jonathan Kirby

Algebraic Closures of Fields

2019-04-04

Jonathan Kirby

Substructure Completeness

2019-04-04

Jonathan Kirby

Structures

2019-04-04

Jonathan Kirby

Fields and Their Extensions

2019-04-04

Jonathan Kirby

Algebraic types in Zilber’s exponential field

2025-01-17

Vahagn Aslanyan , Jonathan Kirby

We characterise the model-theoretic algebraic closure in Zilber's exponential field.A key step involves showing that certain algebraic varieties have finite intersections with certain finite-rank subgroups of the graph of exponentiation.Mordell-Lang … We characterise the model-theoretic algebraic closure in Zilber's exponential field.A key step involves showing that certain algebraic varieties have finite intersections with certain finite-rank subgroups of the graph of exponentiation.Mordell-Lang for algebraic tori (a theorem of Laurent) plays a central role in our proof.

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Algebraic types in Zilber's exponential field

2024-05-02

Vahagn Aslanyan , Jonathan Kirby

We characterise the model-theoretic algebraic closure in Zilber's exponential field. A key step involves showing that certain algebraic varieties have finite intersections with certain finite-rank subgroups of the graph of … We characterise the model-theoretic algebraic closure in Zilber's exponential field. A key step involves showing that certain algebraic varieties have finite intersections with certain finite-rank subgroups of the graph of exponentiation. Mordell-Lang for algebraic tori (a theorem of Laurent) plays a central role in our proof.

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Up with Categories, Down with Sets; Out with Categories, In with Sets!

2024-04-12

Jonathan Kirby

Abstract Practical approaches to the notions of subsets and extension sets are compared, coming from broadly set-theoretic and category-theoretic traditions of mathematics. I argue that the set-theoretic approach is the … Abstract Practical approaches to the notions of subsets and extension sets are compared, coming from broadly set-theoretic and category-theoretic traditions of mathematics. I argue that the set-theoretic approach is the most practical for ‘looking down’ or ‘in’ at subsets and the category-theoretic approach is the most practical for ‘looking up’ or ‘out’ at extensions, and suggest some guiding principles for using these approaches without recourse to either category theory or axiomatic set theory.

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Quasiminimality of complex powers

2024-01-01

Francesco Gallinaro , Jonathan Kirby

Independence relations for exponential fields

2023-05-15

Vahagn Aslanyan , Robert Henderson , Mark Kamsma +1 more

We give four different independence relations on any exponential field. Each is a canonical independence relation on a suitable Abstract Elementary Class of exponential fields, showing that two of these … We give four different independence relations on any exponential field. Each is a canonical independence relation on a suitable Abstract Elementary Class of exponential fields, showing that two of these are NSOP$_1$-like and non-simple, a third is stable, and the fourth is the quasiminimal pregeometry of Zilber's exponential fields, previously known to be stable (and uncountably categorical). We also characterise the fourth independence relation in terms of the third, strong independence.

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Quasiminimality of complex powers

2023-01-01

Francesco Gallinaro , Jonathan Kirby

The complex field, equipped with the multivalued functions of raising to each complex power, is quasiminimal, proving a conjecture of Zilber and providing evidence towards his stronger conjecture that the … The complex field, equipped with the multivalued functions of raising to each complex power, is quasiminimal, proving a conjecture of Zilber and providing evidence towards his stronger conjecture that the complex exponential field is quasiminimal.

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Around Zilber's quasiminimality conjecture

2023-01-01

Jonathan Kirby

This is an extended abstract for a survey talk given in Oberwolfach on 1st December 2022, slightly updated in June 2023. I survey some work around the notion of quasiminimality … This is an extended abstract for a survey talk given in Oberwolfach on 1st December 2022, slightly updated in June 2023. I survey some work around the notion of quasiminimality and some of the progress towards Zilber's conjecture from the last 25 years.

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A closure operator respecting the modular j-function

2022-10-05

Vahagn Aslanyan , Sebastian Eterović , Jonathan Kirby

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Independence relations for exponential fields

2022-01-01

Vahagn Aslanyan , Robert Henderson , Mark Kamsma +1 more

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A Geometric Approach to Some Systems of Exponential Equations

2021-11-10

Vahagn Aslanyan , Jonathan Kirby , Vincenzo Mantova

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A geometric approach to some systems of exponential equations

2021-11-09

Vahagn Aslanyan , Jonathan Kirby , Vincenzo Mantova

Blurrings Of The J-Function

2021-08-04

Vahagn Aslanyan , Jonathan Kirby

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Existentially closed exponential fields

2021-01-15

Levon Haykazyan , Jonathan Kirby

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Differential existential closedness for the 𝑗-function

2020-11-30

Vahagn Aslanyan , Sebastian Eterović , Jonathan Kirby

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A closure operator respecting the modular j-function

2020-09-30

Vahagn Aslanyan , Sebastian Eterović , Jonathan Kirby

A closure operator respecting the modular $j$-function

2020-01-01

Vahagn Aslanyan , Sebastian Eterović , Jonathan Kirby

We prove some unconditional cases of the Existential Closedness problem for the modular $j$-function. For this, we show that for any finitely generated field we can find a "convenient" set … We prove some unconditional cases of the Existential Closedness problem for the modular $j$-function. For this, we show that for any finitely generated field we can find a "convenient" set of generators. This is done by showing that in any field equipped with functions replicating the algebraic behaviour of the modular $j$-function and its derivatives, one can define a natural closure operator in three equivalent different ways.

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Blurred complex exponentiation

2019-11-13

Jonathan Kirby

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On Local definability of holomorphic functions

2019-05-01

Gareth A. Jones , Jonathan Kirby , Olivier Le Gal +1 more

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Power Sets and Boolean Algebras

2019-04-04

Jonathan Kirby

Categoricity and Completeness

2019-04-04

Jonathan Kirby

The Successor Structure

2019-04-04

Jonathan Kirby

Substructures and Quantifiers

2019-04-04

Jonathan Kirby

Characterising Definable Sets

2019-04-04

Jonathan Kirby

Semi-algebraic Sets

2019-04-04

Jonathan Kirby

Hilbert’s Nullstellensatz

2019-04-04

Jonathan Kirby

Linear Orders

2019-04-04

Jonathan Kirby

The Complex and Real Fields

2019-04-04

Jonathan Kirby

Theories and Axioms

2019-04-04

Jonathan Kirby

Vector Spaces and Categoricity

2019-04-04

Jonathan Kirby

Theories and Compactness

2019-04-04

Jonathan Kirby

Constructing Models from Syntax

2019-04-04

Jonathan Kirby

Real Vector Spaces and Parameters

2019-04-04

Jonathan Kirby

Axiomatisable Classes

2019-04-04

Jonathan Kirby

Definable Sets

2019-04-04

Jonathan Kirby

Compactness and New Constants

2019-04-04

Jonathan Kirby

Algebraic Closures of Fields

2019-04-04

Jonathan Kirby

Substructure Completeness

2019-04-04

Jonathan Kirby

Structures

2019-04-04

Jonathan Kirby

Fields and Their Extensions

2019-04-04

Jonathan Kirby

Formulas

2019-04-04

Jonathan Kirby

Countable Categoricity

2019-04-04

Jonathan Kirby

Elementary Extensions

2019-04-04

Jonathan Kirby

Omitting Types

2019-04-04

Jonathan Kirby

Saturated Models

2019-04-04

Jonathan Kirby

Bibliography

2019-04-04

Jonathan Kirby

Terms

2019-04-04

Jonathan Kirby

Definable Sets and Varieties

2019-04-04

Jonathan Kirby

Realising Types

2019-04-04

Jonathan Kirby

Cardinality Considerations

2019-04-04

Jonathan Kirby

Types

2019-04-04

Jonathan Kirby

Coauthor	Papers Together
Vahagn Aslanyan	11
Martin Bays	5
Sebastian Eterović	4
Robert Henderson	2
Mark Kamsma	2
Boris Zilber	2
Vincenzo Mantova	2
Tamara Servi	2
Gareth A. Jones	2
Olivier Le Gal	1
Tapani Hyttinen	1
A. J. Wilkie	1
Francesco Gallinaro	1
Bradd Hart	1
Francesco Gallinaro	1
Tim Zander	1
Alf Onshuus	1
David M. Evans	1
Levon Haykazyan	1
Meeri Kesälä	1
Angus Macintyre	1

Model theory: an introduction

2003-06-01

David Marker

This book is a modern introduction to model theory which stresses applications to algebra throughout the text. The first half of the book includes classical material on model construction techniques, … This book is a modern introduction to model theory which stresses applications to algebra throughout the text. The first half of the book includes classical material on model construction techniques, type spaces, prime models, saturated models, countable models, and indiscernibles and their applications. The author also includes an introduction to stability theory beginning with Morley's Categoricity Theorem and concentrating on omega-stable theories. One significant aspect of this text is the inclusion of chapters on important topics not covered in other introductory texts, such as omega-stable groups and the geometry of strongly minimal sets. The author then goes on to illustrate how these ingredients are used in Hrushovski's applications to diophantine geometry. David Marker is Professor of Mathematics at the University of Illinois at Chicago. His main area of research involves mathematical logic and model theory, and their applications to algebra and geometry. This book was developed from a series of lectures given by the author at the Mathematical Sciences Research Institute in 1998.

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Model Theory and Algebraic Geometry

1998-01-01

Geometric Stability Theory

1996-09-12

Anand Pillay

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.

Injective endomorphisms of varieties and schemes

1969-10-01

James Ax

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An introduction to stability theory

2018-04-06

Daniel Palacín

These lecture notes are based on the first section of Pillay's book \[3] and they cover fundamental notions of stability theory such as defi nable types, forking calculus and canonical … These lecture notes are based on the first section of Pillay's book \[3] and they cover fundamental notions of stability theory such as defi nable types, forking calculus and canonical bases, as well as stable groups and homogeneous spaces. The approach followed here is originally due to Hrushovski and Pillay \[2], who presented stability from a local point of view. Throughout the notes, some general knowledge of model theory is assumed. I recommend the book of Tent and Ziegler \[4] as an introduction to model theory. Furthermore, the texts of Casanovas \[1] and Wagner \[5] may also be useful to the reader to obtain a different approach to stability theory.

Stable Domination and Independence in Algebraically Closed Valued Fields

2007-12-10

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

This book addresses a gap in the model-theoretic understanding of valued fields that had limited the interactions of model theory with geometry. It contains significant developments in both pure and … This book addresses a gap in the model-theoretic understanding of valued fields that had limited the interactions of model theory with geometry. It contains significant developments in both pure and applied model theory. Part I of the book is a study of stably dominated types. These form a subset of the type space of a theory that behaves in many ways like the space of types in a stable theory. This part begins with an introduction to the key ideas of stability theory for stably dominated types. Part II continues with an outline of some classical results in the model theory of valued fields and explores the application of stable domination to algebraically closed valued fields. The research presented here is made accessible to the general model theorist by the inclusion of the introductory sections of each part.

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Mathematical Logic

2000-09-07

René Cori , Daniel Lascar

Abstract Logic forms the basis of mathematics, and is hence a fundamental part of any mathematics course. In particular, it is a major element in theoretical computer science and has … Abstract Logic forms the basis of mathematics, and is hence a fundamental part of any mathematics course. In particular, it is a major element in theoretical computer science and has undergone a huge revival with the explosion of interest in computers and computer science. This book provides students with a clear and accessible introduction to this important subject. The concept of model underlies the whole book, giving the text a theoretical coherence whilst still covering a wide area of logic.

Zum Hilbertschen Nullstellensatz

1930-12-01

J. L. Rabinowitsch

Injective morphisms of real algebraic varieties

1962-04-01

Andrzej Białynicki-Birula , Maxwell Rosenlicht

1. A recent note of D. J. Newman [3] shows that for polynomial maps of the real plane into itself injectivity implies surjectivity. The present note combines two independently obtained … 1. A recent note of D. J. Newman [3] shows that for polynomial maps of the real plane into itself injectivity implies surjectivity. The present note combines two independently obtained corrections and generalizations of Newman's proof. We need some preliminaries. Take the complex numbers as universal domain and define a real algebraic set to be the set of real points of an algebraic set that is defined over the reals. A real algebraic set V is Zariski-dense in its Zariski closure V, which is an algebraic set defined over the reals, and V is the set of real points of V. Any algebraic subset of V meets V in a real algebraic subset. V is irreducible (in its Zariski topology), i.e. V is a real algebraic variety, if and only if V is irreducible (over the complex numbers), we define dim V =dim V, and we call PC V simple if P is a simple point of V; such points P exist, and there exist uniformizing parameters that are defined over the reals for 7 at such a point P, hence real local power series expansions, so that at each of its simple points V, in its ordinary topology, is locally a real analytic manifold of dimension dim V. For real algebraic sets V, W define a rational map f: V-*W to be the restriction to VX W of a rational map f: V--W that is defined over the reals; the rational map f: V-*W is a morphism if f is defined at each point of V. Supposing the morphism of real algebraic varieties f: V-*W to be such that f(V) is Zariski-dense in W, a simple point PE V may be found such that f(P) is simple on Wand df has the correct rank dim V-dim W at P, implying that, for the real analytic structure of V, W at P, f(P) respectively, f is locally a projection onto a direct factor; in particular, if f is finite-to-one, then dim V=dim W

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Pseudo-exponentiation on algebraically closed fields of characteristic zero

2004-08-25

Boris Zilber

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Exponential algebraicity in exponential fields

2010-06-17

Jonathan Kirby

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On Schanuel's Conjectures

1971-03-01

James Ax

The theory of the exponential differential equations of semiabelian varieties

2009-07-16

Jonathan Kirby

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On quasiminimal excellent classes

2010-03-18

Jonathan Kirby

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Finitely presented exponential fields

2013-08-29

Jonathan Kirby

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A remark on Zilber's pseudoexponentiation

2006-09-01

David Marker

When studying the model theory of the first observation is that the integers can be defined as Since ∂ exp is subject to all of Gödel's phenomena, this is often … When studying the model theory of the first observation is that the integers can be defined as Since ∂ exp is subject to all of Gödel's phenomena, this is often also the last observation. After Wilkie proved that ℝ exp is model complete, one could ask the same question for ∂ exp , but the answer is negative. P roposition 1.1. ∂ exp is not model complete P roof . If ∂ exp is model complete, then every definable set is a projection of a closed set. Since ∂ is locally compact, every definable set is F σ . The same is true for the complement, so every definable set is also G δ . But, since ℤ is definable, ℚ is definable and a standard corollary of the Baire Category Theorem tells us that ℚ is not G δ . Still, there are several interesting open questions about ∂ exp . • Is ℝ definable in ∂ exp ? • (quasiminimality) Is every definable set countable or co-countable? (Note that this is true in the structure (∂, ℤ, +, ·) where we add a predicate for ℤ). • (Mycielski) Is there an automorphism of ∂ exp other than the identity and complex conjugation? 1 A positive answer to the first question would tell us that ∂ exp is essentially second order arithmetic, while a positive answer to the second would say that integers are really the only obstruction to a reasonable theory of definable sets. A fascinating, novel approach to ∂ exp is provided by Zilber's [6] pseudoexponentiation. Let L be the language {+, · E , 0, 1}.

Pseudo-exponential maps, variants, and quasiminimality

2018-06-12

Martin Bays , Jonathan Kirby

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Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function

1996-01-01

Andrew O.M. Wilkie

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A new strongly minimal set

1993-07-01

Ehud Hrushovski

Exponentially closed fields and the conjecture on intersections with tori

2014-07-08

Jonathan Kirby , Boris Zilber

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EXPONENTIAL SUMS EQUATIONS AND THE SCHANUEL CONJECTURE

2002-02-01

Boris Zilber

A uniform version of the Schanuel conjecture is discussed that has some model-theoretical motivation. This conjecture is assumed, and it is proved that any 'non-obviously-contradictory' system of equations in the … A uniform version of the Schanuel conjecture is discussed that has some model-theoretical motivation. This conjecture is assumed, and it is proved that any 'non-obviously-contradictory' system of equations in the form of exponential sums with real exponents has a solution.

COVERS OF THE MULTIPLICATIVE GROUP OF AN ALGEBRAICALLY CLOSED FIELD OF CHARACTERISTIC ZERO

2006-08-01

Boris Zilber

We consider the classical universal covering exp: C → C* of the complex torus as an algebraic structure. The exponentiation is seen here as an abstract homomorphism from a divisible … We consider the classical universal covering exp: C → C* of the complex torus as an algebraic structure. The exponentiation is seen here as an abstract homomorphism from a divisible torsion-free group onto the multiplicative group of an algebraically closed field of characteristic zero, with cyclic kernel. We prove that any structure satisfying this description is isomorphic to the classical one provided that the cardinality of the underlying field is equal to that of C. This can also be seen as a model-theoretic statement on the categoricity of a corresponding Lω1,ω-sentence. The proof is a combination of arithmetic and model-theoretic methods.

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Some Topics in Differential Algebraic Geometry I: Analytic Subgroups of Algebraic Groups

1972-10-01

James Ax

A categoricity theorem for quasi-minimal excellent classes

2005-01-01

Boris Zilber

Introduction to Transcendental Numbers

2000-01-01

Serge Lang

Périodes de 1-motifs et transcendance

2002-12-01

Cristiana Bertolin

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Covers of multiplicative groups of algebraically closed fields of arbitrary characteristic

2011-02-26

Martin Bays , Boris Zilber

We show that algebraic analogues of universal group covers, surjective group homomorphisms from a $\mathbb{Q}$-vector space to $F^{\times}$ with "standard kernel", are determined up to isomorphism of the algebraic structure … We show that algebraic analogues of universal group covers, surjective group homomorphisms from a $\mathbb{Q}$-vector space to $F^{\times}$ with "standard kernel", are determined up to isomorphism of the algebraic structure by the characteristic and transcendence degree of $F$ and, in positive characteristic, the restriction of the cover to finite fields. This extends the main result of "Covers of the Multiplicative Group of an Algebraically Closed Field of Characteristic Zero" (B. Zilber, JLMS 2007), and our proof fills a hole in the proof given there.

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Excellence and uncountable categoricity of Zilber's exponential fields

2013-01-01

Martin Bays , Jonathan Kirby

Some local definability theory for holomorphic functions

2008-05-22

A. J. Wilkie

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

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Solutions of equations involving the modular 𝑗 function

2020-07-29

Sebastian Eterović , Sebastián Herrero

Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j"> <mml:semantics> <mml:mi>j</mml:mi> <mml:annotation encoding="application/x-tex">j</mml:annotation> </mml:semantics> </mml:math> </inline-formula> … Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j"> <mml:semantics> <mml:mi>j</mml:mi> <mml:annotation encoding="application/x-tex">j</mml:annotation> </mml:semantics> </mml:math> </inline-formula> function. We show general cases in which these systems have solutions, and then we look at certain situations in which the modular Schanuel conjecture implies that these systems have generic solutions. An unconditional result in this direction is proven for certain polynomial equations on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j"> <mml:semantics> <mml:mi>j</mml:mi> <mml:annotation encoding="application/x-tex">j</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with algebraic coefficients.

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Generic solutions of equations with iterated exponentials

2017-03-06

Paola D’Aquino , Antongiulio Fornasiero , Giuseppina Terzo

We study solutions of exponential polynomials over the complex field. Assuming Schanuel's Conjecture we prove that certain polynomials of the form \[ p(z, e^z, e^{e^z}, e^{e^{e^{z}}}) = 0 \] have … We study solutions of exponential polynomials over the complex field. Assuming Schanuel's Conjecture we prove that certain polynomials of the form \[ p(z, e^z, e^{e^z}, e^{e^{e^{z}}}) = 0 \] have generic solutions in $\mathbb C$.

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Introduction to Transcendental Numbers

2000-01-01

Serge Lang

Adequate Predimension Inequalities in Differential Fields

2018-03-13

Vahagn Aslanyan

In this paper we study predimension inequalities in differential fields and define what it means for such an inequality to be \emph{adequate}. Adequacy was informally introduced by Zilber, and here … In this paper we study predimension inequalities in differential fields and define what it means for such an inequality to be \emph{adequate}. Adequacy was informally introduced by Zilber, and here we give a precise definition in a quite general context. We also discuss the connection of this problem to definability of derivations in the reducts of differentially closed fields. The Ax-Schanuel inequality for the exponential differential equation (proved by Ax) and its analogue for the differential equation of the $j$-function (established by Pila and Tsimerman) are our main examples of predimensions. We carry out a Hrushovski construction with the latter predimension and obtain a natural candidate for the first-order theory of the differential equation of the $j$-function. It is analogous to Kirby's axiomatisation of the theory of the exponential differential equation (which in turn is based on the axioms of Zilber's pseudo-exponentiation), although there are many significant differences. In joint work with Sebastian Eterovic and Jonathan Kirby we have recently proven that the axiomatisation obtained in this paper is indeed an axiomatisation of the theory of the differential equation of the $j$-function, that is, the Ax-Schanuel inequality for the $j$-function is adequate.

Zero estimates with moving targets

2017-01-09

W. Dale Brownawell , David Masser

Zero estimates have a classical history in diophantine approximation and transcendence, as well as more recent applications to counting rational points on analytic sets. We give some examples showing that … Zero estimates have a classical history in diophantine approximation and transcendence, as well as more recent applications to counting rational points on analytic sets. We give some examples showing that the sharpest conceivable results can be false, and in some cases the natural guess has even to be doubled. A by-product is a 'non-degenerate' case of the (unproved) 'Zilber Nullstellensatz' in connection with 'strong exponential closure'.

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Anomalous Subvarieties—Structure Theorems and Applications

2007-01-01

Enrico Bombieri , David Masser , Umberto Zannier

Journal Article Anomalous Subvarieties—Structure Theorems and Applications Get access E. Bombieri, E. Bombieri 1 School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540, USA Correspondence to be sent … Journal Article Anomalous Subvarieties—Structure Theorems and Applications Get access E. Bombieri, E. Bombieri 1 School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540, USA Correspondence to be sent to: E. Bombieri, School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540, USA. e-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar D. Masser, D. Masser 2 Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Switzerland Search for other works by this author on: Oxford Academic Google Scholar U. Zannier U. Zannier 3 Scuola Normale, Piazza Cavalieri 7, 56126 Pisa, Italy Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2007, 2007, rnm057, https://doi.org/10.1093/imrn/rnm057 Published: 01 January 2007 Article history Published: 01 January 2007 Received: 09 May 2007 Accepted: 05 July 2007

A theory of a generic function with derivations

2002-01-01

Boris Zilber

Generalized analytic sets

1997-07-01

Boris Zilber

Introduction to Complex Analytic Geometry

1991-01-01

Stanisław Łojasiewicz

facts. An elementary acquaintance with topology, algebra, and analysis (in cluding the notion of a manifold) is sufficient as far as the understanding of this book is concerned. All the … facts. An elementary acquaintance with topology, algebra, and analysis (in cluding the notion of a manifold) is sufficient as far as the understanding of this book is concerned. All the necessary prop

Fields with pseudo-exponentiation

2000-01-01

Boris Zilber

We study the structure of an algebraically closed field with extra function resembling the classical exponentiation on complex numbers. We study the structure of an algebraically closed field with extra function resembling the classical exponentiation on complex numbers.

Ax–Schanuel for the j-function

2016-06-10

Jonathan Pila , Jacob Tsimerman

In this paper we prove a functional transcendence statement for the j-function which is an analogue of the Ax-Schanuel theorem for the exponential function. It asserts, roughly, that atypical algebraic … In this paper we prove a functional transcendence statement for the j-function which is an analogue of the Ax-Schanuel theorem for the exponential function. It asserts, roughly, that atypical algebraic relations among functions and their compositions with the j-function are governed by modular relations.

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A Schanuel property for exponentially transcendental powers

2010-08-04

Martin Bays , Jonathan Kirby , A. J. Wilkie

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RAISING TO POWERS IN ALGEBRAICALLY CLOSED FIELDS

2003-11-01

Boris Zilber

We study structures on the fields of characteristic zero obtained by introducing (multi-valued) operations of raising to power. Using Hrushovski–Fraisse construction we single out among the structures exponentially-algebraically closed once … We study structures on the fields of characteristic zero obtained by introducing (multi-valued) 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.

Independence in finitary abstract elementary classes

2006-05-05

Tapani Hyttinen , Meeri Kesälä

Locally polynomially bounded structures

2008-03-20

G. O. Jones , A. J. Wilkie

We prove a theorem which provides a method for constructing points on varieties defined by certain smooth functions. We require that the functions are definable in a definably complete expansion … We prove a theorem which provides a method for constructing points on varieties defined by certain smooth functions. We require that the functions are definable in a definably complete expansion of a real closed field and are locally definable in a fixed o-minimal and polynomially bounded reduct. As an application we show that in certain o-minimal structures definable functions are piecewise implicitly defined over the basic functions in the language.

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Quasiminimal structures and excellence

2013-10-24

Martin Bays , Bradd Hart , Tapani Hyttinen +2 more

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Strong minimality and the $j$-function

2017-12-20

James Freitag , Thomas Scanlon

We show that the order three algebraic differential equation over {\mathbb Q} satisfied by the analytic j -function defines a non- \aleph_0 -categorical strongly minimal set with trivial forking geometry … We show that the order three algebraic differential equation over {\mathbb Q} satisfied by the analytic j -function defines a non- \aleph_0 -categorical strongly minimal set with trivial forking geometry relative to the theory of differentially closed fields of characteristic zero answering a long-standing open problem about the existence of such sets. The theorem follows from Pila's modular Ax–Lindemann–Weierstrass with derivatives theorem using Seidenberg's embedding theorem. As a by product of this analysis, we obtain a more general version of the modular Ax-Lindemann-Weierstrass theorem, which, in particular, applies to automorphic functions for arbitrary arithmetic subgroups of SL_2 (\mathbb Z) . We then apply the results to prove effective finiteness results for intersections of subvarieties of products of modular curves with isogeny classes. For example, we show that if \psi:{\mathbb P}^1 \to {\mathbb P}^1 is any non-identity automorphism of the projective line and t \in {\mathbb A}^1(\mathbb C) \smallsetminus {\mathbb A}^1({\mathbb Q}^\mathrm {alg}) , then the set of s \in {\mathbb A}^1(\mathbb C) for which the elliptic curve with j -invariant s is isogenous to the elliptic curve with j -invariant t and the elliptic curve with j -invariant \psi(s) is isogenous to the elliptic curve with j -invariant \psi(t) has size at most 2^{38} \cdot 3^{14} . In general, we prove that if V is a Kolchin-closed subset of \mathbb A^n , then the Zariski closure of the intersection of V with the isogeny class of a tuple of transcendental elements is a finite union of weakly special subvarieties. We bound the sum of the degrees of the irreducible components of this union by a function of the degree and order of V .

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Model theory of partial differential fields: From commuting to noncommuting derivations

2007-01-12

Michael F. Singer

McGrail (2000) has shown the existence of a model completion for the universal theory of fields on which a finite number of commuting derivations act and, independently, Yaffe (2001) has … McGrail (2000) has shown the existence of a model completion for the universal theory of fields on which a finite number of commuting derivations act and, independently, Yaffe (2001) has shown the existence of a model completion for the univeral theory of fields on which a fixed Lie algebra acts as derivations. We show how to derive the second result from the first.

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On algebraic differential equations satisfied by automorphic functions

1969-11-01

K. Mahler

In the present note, I investigate the question whether automorphic functions of the Fuchsian type can satisfy algebraic differential equations of the second order. Under rather general restrictions on the … In the present note, I investigate the question whether automorphic functions of the Fuchsian type can satisfy algebraic differential equations of the second order. Under rather general restrictions on the corresponding group, it is proved that there is no such differential equation.

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LOCAL INTERDEFINABILITY OF WEIERSTRASS ELLIPTIC FUNCTIONS

2014-11-24

Gareth A. Jones , Jonathan Kirby , Tamara Servi

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Basic Algebraic Geometry 2

1994-01-01

Igor R. Shafarevich