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Algebraic and Model Theoretic Properties of O-Minimal Exponential Fields

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Algebraic and Model Theoretic Properties of O-Minimal Exponential Fields Algebraic and Model Theoretic Properties of O-minimal Exponential Fields Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr.rer.nat.) vorgelegt von Lothar Sebastian Krapp an der Mathematisch-Naturwissenschaftliche Sektion Fachbereich Mathematik und Statistik Konstanz, 2019 Tag der m¨undlichen Pr¨ufung: 21. November 2019 1. Referentin: Prof. Dr. Salma Kuhlmann 2. Referent: Prof. Dr. Tobias Kaiser Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-2-166bghaubh8tf9 F¨ur Svea, der ich diese Widmung vor acht Jahren versprochen habe. Abstract An exponential exp on an ordered field ( K, +, , , 0, 1, < ) is an order-preserving − · isomorphism from the ordered additive group ( K, +, 0, < ) to the ordered multiplica- tive group of positive elements K>0, , 1, < . The structure ( K, +, , , 0, 1, <, exp) · − · is then called an ordered exponential field. A linearly ordered structure ( M,<,... ) is called o-minimal if every parametrically definable subset of M is a finite union of points and open intervals of M. The main subject of this thesis is the algebraic and model theoretic examination of o-minimal exponential fields ( K, +, , , 0, 1, <, exp) whose exponential satisfies − · the differential equation exp ′ = exp with initial condition exp(0) = 1. This study is mainly motivated by the Transfer Conjecture, which states as follows: Any o-minimal exponential field (K, +, , , 0, 1, <, exp) whose exponential satis- − · fies the differential equation exp ′ = exp with initial condition exp(0) = 1 is elemen- tarily equivalent to Rexp . Here, R denotes the real exponential field ( R, +, , , 0, 1, <, exp), where exp exp − · denotes the standard exponential x ex on R. Moreover, elementary equivalence 7→ means that any first-order sentence in the language = +, , , 0, 1, <, exp Lexp { − · } holds for ( K, +, , , 0, 1, <, exp) if and only if it holds for R . − · exp The Transfer Conjecture, and thus the study of o-minimal exponential fields, is of particular interest in the light of the decidability of Rexp . To the date, it is not known if Rexp is decidable, i.e. whether there exists a procedure determining for a given first-order -sentence whether it is true or false in R . However, under Lexp exp the assumption of Schanuel’s Conjecture – a famous open conjecture from Transcen- dental Number Theory – a decision procedure for Rexp exists. Also a positive answer to the Transfer Conjecture would result in the decidability of Rexp . Thus, we study o-minimal exponential fields with regard to the Transfer Conjecture, Schanuel’s Con- jecture and the decidability question of Rexp . Overall, we shed light on the valuation theoretic invariants of o-minimal exponen- tial fields – the residue field and the value group – with additional induced struc- ture. Moreover, we explore elementary substructures and extensions of o-minimal exponential fields to the maximal ends – the smallest elementary substructures be- ing prime models and the maximal elementary extensions being contained in the surreal numbers. Further, we draw connections to models of Peano Arithmetic, in- teger parts, density in real closure, definable henselian valuations and strongly NIP ordered fields. Deutsche Zusammenfassung (German Summary) Eine Exponentialfunktion exp auf einem angeordneten K¨orper ( K, +, , , 0, 1, < ) − · ist ein ordnungserhaltender Isomorphismus von der angeordneten additiven Gruppe v (K, +, 0, < ) auf die angeordnete multiplikative Gruppe positiver Elemente K>0, , 1, · < ). Die Struktur ( K, +, , , 0, 1, <, exp) nennen wir einen angeordneten Exponen- − · tialk¨orper. Eine linear angeordnete Struktur ( M,<,... ) ist o-minimal, falls jede parametrisch definierbare Teilmenge von M eine endliche Vereinigung von Punkten und offenen Intervallen in M ist. Der Fokus dieser Dissertation liegt auf der algebraischen und modelltheoreti- schen Untersuchung o-minimaler Exponentialk¨orper ( K, +, , , 0, 1, <, exp), deren − · Exponentialfunktion die Differentialgleichung exp ′ = exp mit Anfangsbedingung exp(0) = 1 erf¨ullt. Diese Untersuchung ist haupts¨achlich durch die Transfervermu- tung motiviert, die wie folgt lautet: Jeder o-minimale Exponentialk¨orper (K, +, , , 0, 1, <, exp) , dessen Exponential- − · funktion die Differentialgleichung exp ′ = exp mit Anfangsbedingung exp(0) = 1 erf¨ullt, ist zu Rexp elementar ¨aquivalent. Dabei bezeichnet R den reellen Exponentialk¨orper ( R, +, , , 0, 1, <, exp) mit exp − · der Standardexponentialfunktion exp: x ex auf R. Elementare Aquivalenz¨ 7→ herrscht vor, wenn jede Aussage erststufiger Logik in der Sprache = +, , , 0, 1, Lexp { − · <, exp genau dann in ( K, +, , , 0, 1, <, exp) gilt, wenn sie auch in R gilt. } − · exp Die Transfervermutung und daher auch die Untersuchung o-minimaler Exponen- tialk¨orper ist insbesondere mit Blick auf die Entscheidbarkeit von Rexp von Interesse. Bis heute ist nicht bekannt, ob Rexp entscheidbar ist, also ein Algorithmus exis- tiert, der f¨ur jede erststufige -Aussage bestimmt, ob diese wahr oder falsch in Lexp Rexp ist. Unter Annahme der Vermutung von Schanuel – einer bekannten offenen Vermutung der transzendenten Zahlentheorie – existiert jedoch ein Entscheidungs- algorithmus f¨ur Rexp . Ebenso w¨urde aus der Richtigkeit der Transfervermutung die Entscheidbarkeit von Rexp folgen. Daher untersuchen wir o-minimale Expo- nentialk¨orper vor dem Hintergrund der Transfervermutung, der Vermutung von Schanuel und der Entscheidbarkeitsfrage f¨ur Rexp . Wir beleuchten die bewertungstheoretischen Invarianten o-minimaler Exponen- tialk¨orper – den Restklassenk¨orper und die Wertegruppe – mit zus¨atzlich induzierter Struktur. Weiterhin erforschen wir minimale elementare Substrukturen und maxi- male elementare Erweiterungen o-minimaler Exponentialk¨orper – erstere sind Prim- modelle, zweitere haben die Klasse der surrealen Zahlen als Tr¨ager. Schließlich stellen wir Beziehungen zu Modellen der Peano-Arithmetik, ganzzahligen Anteilen, Dichtheit im reellen Abschluss, definierbaren henselschen Bewertungen und stark NIP angeordneten K¨orpern her. vi Acknowledgements I would like to express my deepest gratitude to my supervisor Salma Kuhlmann for her constant guidance and continuous encouragement throughout my doctoral studies. In our long discussions, her inspiring enthusiasm always managed to arouse my curiosity for each and every detail. Beyond that, she taught me that writing mathematics is more than just listing results and proofs but an art. I would like to thank Merlin Carl for involving me in his project on integer parts leading to Chapter 7 of this thesis. Moreover, I wish to thank Salma Kuhlmann and Gabriel Leh´ericy for working with me on strongly NIP ordered fields and definable convex valuations, which are treated in Chapter 8. For long and short discussions and helpful advice during conferences, seminars or via email, which helped me progress with my research, I would also like to express my thanks to Sylvy Anscombe, Vincent Bagayoko, Stein Baluyot, Philip Dittmann, Antongiulio Fornasiero, Yatir Halevi, Immanuel Halupczok, Deirdre Haskell, As- saf Hasson, Franziska Jahnke, Tobias Kaiser, Jonathan Kirby, Vincenzo Mantova, Joris Roos, Florian Severin, Athipat Thamrongthanyalak, Alex Wilkie as well as an anonymous referee for the Journal of Logic and Analysis. For proofreading several iterations of this thesis, I would like to thank Franziska Schropp. I was supported by a doctoral scholarship of Studienstiftung des deutschen Volkes from November 2016 until the completion of my thesis in July 2019 and of Carl- Zeiss-Stiftung from July 2016 to June 2018. For further financial and administrative support during workshops and confer- ences I would also like to thank Centre International de Rencontres Math´ematiques, Deutscher Akademischer Austauschdienst, Deutsche Vereinigung f¨ur mathematische Logik und f¨ur Grundlagenforschung der exakten Wissenschaften, Institut Henri Poincar´e, Mathematisches Forschungsinstitut Oberwolfach, Universit¨atKonstanz International Office, Westf¨alische Wilhelms-Universit¨atas well as all people in- volved in the organisation. Moreover, I wish to thank the Fachbereich Mathematik und Statistik of Universit¨atKonstanz and in this regard Rainer Janßen for always providing additional funding when needed. vii Contents 1. Introduction 1 2. Preliminaries 7 3. General Ordered Exponential Fields 15 3.1. Archimedean Exponential Fields . 15 3.2. Differentiability and v-Compatibility .................. 17 3.3. Growth and Taylor Axiom Schemes . 34 3.4. Real Exponential Residue Fields . 37 4. O-minimal Exponential Fields and the Decidability of Rexp 53 4.1. O-minimalEXP-Fields.......................... 53 4.1.1. O-minimality of Rexp ....................... 53 4.1.2. Growth Dichotomy . 57 4.1.3. Recursive Axiomatisation . 58 4.1.4. Archimedean Case . 63 4.1.5. Residue Exponential Field . 64 4.2. Existential and Universal Theory of Real Exponentiation . 70 4.3. TransferConjecture ........................... 74 4.4. Exponential Boundedness and T -convexity............... 77 4.5. PrimeModels............................... 85 4.6. Schanuel’s Conjecture in O-minimal EXP-fields . 92 5. Exponential Groups 97 5.1. Theory of Restricted Real Exponentiation . 98 5.2. Countable Groups . 102 5.3. Uncountable Models of Real Exponentiation . 110 5.4. κ-Saturated Groups . 117 5.5. Contraction Groups . 122 6. Surreal Exponential Fields 125 6.1. Universal Domain for Real Exponentiation . 125 6.2. O-minimal Surreal Exponential Fields . 132 6.3. Applications to Conjectures
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