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Ax-Schanuel Type Inequalities in Differentially Closed Fields

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Ax-Schanuel Type Inequalities in Differentially Closed Fields Ax-Schanuel Type Inequalities in Differentially Closed Fields Vahagn Aslanyan St Edmund Hall University of Oxford A thesis submitted for the degree of Doctor of Philosophy Hilary 2017 To my parents for their endless love and support. Acknowledgements Firstly, I would like to thank my supervisors Boris Zilber and Jonathan Pila for their constant support and guidance. They have had great influ- ence on my mathematical thinking and on this work in particular. It has been my great pleasure to write this thesis under their supervision. I am also grateful to all members of the Oxford logic group for being a wonderful team to work in. In particular, I thank Jamshid Derakhshan, Ehud Hrushovski, Jochen Koenigsmann and Alex Wilkie. I have learnt a lot from my fellow students and I want to thank them for that. Spe- cial thanks are due to Levon Haykazyan, Alfonso Ruiz Guido, Sebastian Eterovic and Felix Weitkamper for discussions on various topics and for their friendship. I would like to thank Jonathan Kirby for patiently answering my numerous questions over the course of my DPhil. I also thank my lecturers from Yerevan State University and in particu- lar my undergraduate dissertation supervisor and collaborator Yuri Mov- sisyan. It has been a great pleasure for me to be a member of the Oxford Uni- versity Armenian Society, and I would like to thank all the members of the society, especially David, Marianna, Varduhi and Ani for being my Armenian family in Oxford. The support and motivation I received from all my friends made my time in Oxford more enjoyable. I thank them all for that. Mher Safaryan, Tigran Bakaryan, Hayk Aleksanyan, Anush Tserunyan, Tigran Hakobyan and Hayk Saribekyan deserve a special mention. I am also very grateful to the Dulverton Trust, Luys Foundation and AGBU for funding my DPhil studies at the University of Oxford. Finally, I want to express my deepest and most sincere gratitude to my family and to Anahit for their endless love and encouragement. Without them this dream would not have come true. I would especially like to thank my father, for his role in all my achievements has been tremendous. Abstract In this thesis we study Ax-Schanuel type inequalities for abstract differ- ential equations. A motivating example is the exponential differential equation. The Ax-Schanuel theorem states positivity of a predimension defined on its solutions. The notion of a predimension was introduced by Hrushovski in his work from the 1990s where he uses an amalgamation- with-predimension technique to refute Zilber’s Trichotomy Conjecture. In the differential setting one can carry out a similar construction with the predimension given by Ax-Schanuel. In this way one constructs a limit structure whose theory turns out to be precisely the first-order theory of the exponential differential equation (this analysis is due to Kirby (for semiabelian varieties) and Crampin, and it is based on Zilber’s work on pseudo-exponentiation). One says in this case that the inequality is ade- quate. Thus, by an Ax-Schanuel type inequality we mean a predimension in- equality for a differential equation. Our main question is to understand for which differential equations one can find an adequate predimension inequality. We show that this can be done for linear differential equations with constant coefficients by generalising the Ax-Schanuel theorem. Further, the question turns out to be closely related to the problem of re- covering the differential structure in reducts of differentially closed fields where we keep the field structure (which is quite an interesting problem in its own right). So we explore that question and establish some crite- ria for recovering the derivation of the field. We also show (under some assumptions) that when the derivation is definable in a reduct then the latter cannot satisfy a non-trivial adequate predimension inequality. Another example of a predimension inequality is the analogue of Ax- Schanuel for the differential equation of the modular j-function due to Pila and Tsimerman. We carry out a Hrushovski construction with that predimension and give an axiomatisation of the first-order theory of the strong Fraïssé limit. It will be the theory of the differential equation of j under the assumption of adequacy of the predimension. We also show that if a similar predimension inequality (not necessarily adequate) is known for a differential equation then the fibres of the latter have interesting model theoretic properties such as strong minimality and geometric triv- iality. This, in particular, gives a new proof for a theorem of Freitag and Scanlon stating that the differential equation of j defines a trivial strongly minimal set. Contents 1 Introduction1 1.1 The main question............................1 1.2 Summary of the thesis..........................3 1.3 Notations.................................7 2 Preliminaries9 2.1 Differentially closed fields........................9 2.2 Differential algebraic curves....................... 11 2.3 Two theorems on dimension of algebraic varieties........... 12 2.4 Pregeometries............................... 13 2.5 Trivial strongly minimal sets....................... 14 3 Predimensions and Hrushovski Constructions 16 3.1 Predimensions............................... 17 3.2 Amalgamation with predimension.................... 20 3.3 Examples................................. 25 3.3.1 Complex exponentiation..................... 25 3.3.2 Exponential differential equation................ 26 3.4 Predimensions in the differential setting................. 27 4 Definability of Derivations in the Reducts of Differentially Closed Fields 31 4.1 Setup and the main question....................... 31 4.2 Definable derivations........................... 33 4.3 Model theoretic properties of the reducts................ 35 4.4 An example................................ 37 4.5 Generic points............................... 41 4.6 Further examples............................. 44 4.7 Connection to predimensions....................... 51 4.8 Model completeness............................ 52 4.9 Pregeometry on differentially closed fields and reducts......... 57 5 Ax-Schanuel for Linear Differential Equations 59 5.1 The exponential differential equation.................. 59 5.2 Higher order linear differential equations................ 62 i 5.3 The complete theory........................... 66 5.4 Rotundity and freeness.......................... 69 5.5 Adequacy................................. 70 6 The j-function 73 6.1 Background on the j-function...................... 73 6.2 Ax-Schanuel and weak modular Zilber-Pink.............. 75 6.3 The universal theory and predimension................. 76 6.4 Amalgamation............................... 78 6.5 Normal and free varieties......................... 80 6.6 Existential closedness........................... 82 6.7 The complete theory........................... 85 6.8 The general case............................. 89 7 Ax-Schanuel Type Theorems and Geometry of Strongly Minimal Sets in DCF0 94 7.1 Setup and main results.......................... 95 7.2 Proofs of the main results........................ 98 7.3 The modular j-function......................... 100 7.4 Concluding remarks............................ 101 Bibliography 104 ii Chapter 1 Introduction 1.1 The main question In [Lan66] Serge Lang mentions that Stephen Schanuel conjectured that for any Q- linearly independent complex numbers z1; : : : ; zn one has z1 zn tdQ Q(z1; : : : ; zn; e ; : : : ; e ) ≥ n: (1.1) This is now known as Schanuel’s conjecture. It generalises many results (e.g. the Lindemann-Weierstrass theorem) and conjectures in transcendental number theory and is widely open. For example, a simple consequence of Schanuel’s conjecture is algebraic independence of e and π which is a long standing open problem. Schanuel’s conjecture (and its real version) is closely related to the model theory of the complex (real) exponential field Cexp = (C; +; ·; exp) (respectively, Rexp = (R; +; ·; exp), see [MW96]). Most notably, Boris Zilber noticed that the inequality (1.1) states the positivity of a predimension. The notion of a predimension was defined by Ehud Hrushovski in [Hru93] where he uses an amalgamation-with-predimension technique (which is a variation of Fraïssé’s amalgamation construction) to refute Zilber’s Trichotomy Conjecture. More precisely, Schanuel’s conjecture is equivalent to the following statement: for any z1; : : : ; zn 2 C the inequality δ(¯z) = tdQ Q(¯z; exp(¯z)) − ldimQ(¯z) ≥ 0 (1.2) holds, where td and ldim stand for transcendence degree and linear dimension re- spectively. Here δ satisfies the submodularity law1 which allows one to carry out a Hrushovski construction. In this way Zilber constructed pseudo-exponentiation on algebraically closed fields of characteristic zero. He proved that there is a unique model of that (non first-order) theory in each uncountable cardinality and conjec- @0 tured that the model of cardinality 2 is isomorphic to Cexp. Since (1.2) holds for pseudo-exponentiation (it is included in the axiomatisation given by Zilber), Zilber’s conjecture implies Schanuel’s conjecture. For details on pseudo-exponentiation see [Zil04b, Zil05, Zil02, Zil16, KZ14, Kir13]. 1This is not quite right as we will see in Chapter3. It will be explained there in what sense δ is submodular. 1 Zilber’s work also gave rise to a diophantine conjecture (Conjecture on Intersection with Tori) which was later generalised by Pink and is now known as the Zilber- Pink conjecture ([Zil02, KZ14, Pin05]). It generalises many
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