Uniform Theory of Multiplicative Valued Difference Fields By

Uniform Theory of Multiplicative Valued Difference Fields by Koushik Pal A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Logic and the Methodology of Science in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Thomas W. Scanlon, Chair Professor Leo A. Harrington Professor Theodore A. Slaman Professor W. Hugh Woodin Professor Lam Tsit-Yuen Spring 2011 Uniform Theory of Multiplicative Valued Difference Fields Copyright 2011 by Koushik Pal 1 Abstract Uniform Theory of Multiplicative Valued Difference Fields by Koushik Pal Doctor of Philosophy in Logic and the Methodology of Science University of California, Berkeley Professor Thomas W. Scanlon, Chair The first part of the thesis concerns the existence of model companions of certain unstable theories with automorphisms. Let T be a first-order theory with the strict order property. According to Kikyo and Shelah's theorem, the theory of models of T with a generic automorphism does not have a model companion. However, existence can be restored with some restriction on the automorphism. We show the existence of model companions of the theory of linear orders with increasing automorphisms and the theory of ordered abelian groups with multiplicative automorphisms. Both these theories have the strict order property. The second part of the thesis uses these results from the first part in the context of valued difference fields, which are valued fields with an automorphism on them. Understanding the theory of such structures requires one to specify how the valuation function interacts with the automorphism. Two special cases have been worked out before. The case of the isometric automorphism is worked out by Luc Bélair,Angus Macintyre and Thomas Scanlon; the case of the contractive automorphism is worked out by Salih Azgin. These two cases, however, are two ends of a spectrum. Our goal in this thesis is to fill this gap by defining the notion of a multiplicative valued difference field. We prove an Ax-Kochen-Ershov type of result, whereby we show that the theory of such structures is essentially controlled by the theory of their so-called \residue-valuation" structures (RVs). We also prove relative quantifier elimination theorem for such structures relative to their RVs. Finally we show that in the presence of a \cross-section", we can transfer these relative completeness and relative quantifier elimination results relative to their value groups and residue fields. i To my parents, without whose encouragement and support this would not have been a possibility. ii Contents 1 Introduction 1 1.1 Existence of model companion of certain theories with definable order and restricted automorphisms . 1 1.2 Study of the theory of (multiplicative) valued difference fields. 3 2 Preliminaries 6 2.1 Basics from Model Theory . 6 2.2 Basics from Difference Algebra . 13 2.3 Basics from Valuation Theory . 14 2.3.1 Valued Fields . 14 2.3.2 Valued Difference Fields . 19 3 Salvaging Kikyo-Shelah 22 3.1 Linear Order with Increasing Automorphism . 23 3.2 Ordered Abelian Group with Automorphism . 26 4 Multiplicative Valued Difference Field 34 4.1 Introduction . 34 4.2 Pseudoconvergence and Pseudocontinuity . 35 4.2.1 Basic Calculation . 35 4.2.2 Refinement of the Basic Calculation. 39 4.3 Around Newton-Hensel Lemma . 40 4.4 Immediate Extensions . 48 4.5 Example and Counter-example . 52 4.6 Extending the Residue Field and the Value Group . 53 4.7 Embedding Theorem . 56 4.8 Completeness and Quantifier Elimination Relative to RV . 59 4.9 Completeness and Quantifier Elimination Relative to (k; Γ) . 61 Bibliography 65 iii Acknowledgments First and foremost, I would like to express my deep gratitude to my advisor Thomas Scanlon. Despite his extremely busy schedule, he has always been easily available. His immense generosity with time, patience and ideas is a key to the realization of this thesis. His concern for his students, supportiveness and good sense of humor has made my graduate studies in general smooth and exciting. I owe him an enormous debt both intellectually and personally. Salih Azgin, Joseph Flenner, Lou van den Dries, Fran¸coiseDelon and Fran¸coisePoint have been very helpful with fruitful discussions on topics related to this thesis. I thank Leo Harrington for providing helpful suggestions from time to time. Ehud Hrushovski is also owed a mathematical debt for providing valuable insights and suggesting related research problems. I thank Lam Tsit-Yuen for careful reading of this thesis and for pointing out a serious error about the existence of \cross-sections" in Section 4.9. I also appreciate the time and effort invested by Alice Medvedev in going through this thesis meticulously and suggesting numerous changes. I owe special thanks to Dugald Macpherson, Anand Pillay and Boris Zilber for many helpful discussions and for their hospitality during (and after) my trips to Leeds and Oxford. Special thanks are also due to Maryanthe Malliaris, Alice Medvedev, Lynn Scow, John Goodrick and Uri Andrews for always being available for discussion and providing valuable logistic information. During my first two years at Berkeley, I was generously supported by the Berkeley Fel- lowship. I was also supported by Thomas Scanlon's NSF grant for another two semesters. My visit to Leeds was supported by MATHLOGAPS (Marie Curie) Fellowship. I have also received much-appreciated financial support from Association of Symbolic Logic, National Science Foundation and American Mathematical Society to attend numerous conferences and workshops. I would like to thank the department stuff, especially Barbara Peavy and Barb Waller, for keeping the administrative side of my graduate studies run smoothly. Thanks are also due to numerous graduate students and friends who have made my life and stay at Berkeley enjoyable and productive. Special mention must be made of Anurag Gupta, Ambuj Tewari, Nikhil Shetty, Akshatha Shetty, Vinayak Nagpal, Bhavna Kapoor, Arnab Sen, Partha Sarathi Dey, Subhroshekhar Ghosh, Sayak Ray and Sharmodeep Bhat- tacharyya. The list is no way complete and I apologize to people who I inadvertently omit. And last but not the least, I am immensely indebted to my parents and my brother for their support, encouragement and faith in me throughout the ups and downs in my graduate life in particular, and life in general. 1 Chapter 1 Introduction 1.1 Existence of model companion of certain theories with definable order and restricted automorphisms The existence of model companion (see Definition 2.1.28) of a theory is an important question to model theorists because, often, it is equivalent to axiomatizing the existentially closed models of the theory. Existentially closed models of a theory are the models where quantifier- free formulas, consistent with the theory (in our context, consistent systems of equations), have solutions. For example, for the theory of abelian groups in the language LG = f+; −; 0g, the existentially closed models are the divisible abelian groups, where any consistent system of linear equations has a solution. For the theory of fields in the language LR = f+; −; ·; 0; 1g, the existentially closed models are the algebraically closed fields, where any consistent system of polynomial equations has a solution. It is important to specify the language we are talking about because that specifies the kind of equations we are interested in. Often we have to deal with theories or models of theories where there is a definable automorphism in the structure. Roughly speaking, an L-automorphism of a structure is a bijective map that preserves the basic operations of the structure in the language L (for a precise definition, see Definition 2.1.5). Because of the close connection of automorphisms with difference operators, we refer to automorphisms as difference operators in this thesis. For example, an LG-automorphism of an abelian group G is a group isomorphism σ : G ! G, a bijective map that preserves addition and subtraction. Similarly an LR-automorphism of a field F is a field isomorphism σ : F ! F , a bijective map that preserves addition, subtraction and multiplication. The question naturally arises: can we classify the existentially closed models of such a theory with an automorphism? Let T be a first-order theory in a language L. To study the theory T together with an automorphism σ, we pass on to an expanded language Lσ = L [ fσg, where we treat σ as a unary function symbol, and consider the new theory Tσ = T [ f\σ is an L-automorphism"g. We are interested in finding out whether Tσ has a model companion in the language Lσ. In CHAPTER 1. INTRODUCTION 2 case it exists, it is referred to as TA. Clearly, to find out whether TA exists, we have to deal with formulas (or equations) involving σ. For example, in the context of abelian groups, we need to know which abelian groups with distinguished automorphisms have solutions to consistent systems of linear difference equations. In the case of fields, we need to know which fields with distinguished automorphisms have solutions to consistent systems of difference polynomials. It is well-known that for these theories, TA does exist. However, things are problematic when there is a definable order in the theory. It is well- known that first-order theories can be classified into stable and unstable theories [24]. In a weak sense, an unstable theory is one where there is an order around. Further, unstable theories either have the Strict Order Property or have the Independence Property. For definitions of these notions, see Definition 3.0.1. In [15] Kikyo shows that, for unstable theories T without the independence property, TA does not exist. Later Kikyo and Shelah improves this result in [17] to showing that, for theories T with the strict order property, TA does not exist.

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