Uniform Theory of Multiplicative Valued Difference Fields By

Uniform Theory of Multiplicative Valued Diﬀerence Fields

Koushik Pal

A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy

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

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 = {+, −, 0}, 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 = {+, −, ·, 0, 1}, 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 ∪ {σ}, where we treat σ as a unary function symbol, and consider the new theory Tσ = T ∪ {“σ is an L-automorphism”}. 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. (For other results in this direction, see [3, 15, 16].) Such a negative result is a big drawback because it implies that the existentially closed models of such theories are not first-order axiomatizable. Fortunately, as we show in this thesis, there are ways to get around this problem by putting restrictions on the automorphisms. With special kinds of automorphisms, model companions can exist even for theories with the strict order property. This is the content of Chapter 3, where we consider two special theories - the theory of linear orders and the theory of ordered abelian groups - both of which have the strict order property, and show that with certain class of restricted automorphisms on them, model companions do exist. For linear orders, the restricted class of automorphisms we consider is the class of increasing automorphisms σ, which satisfy, for all x, σ(x) > x. We first show that any linear order with an increasing injective endomorphism can be extended to a linear order with an increasing automorphism, which can be further extended to a non-trivial dense linear order with an increasing automorphism. It follows then that the theory of linear orders with an increasing automorphism has algebraically prime models (see Definition 2.1.25). Finally we show that the theory of non-trivial dense linear orders with an increasing automorphism is simply closed (see Definition 2.1.25). This shows that the theory of non-trivial dense linear orders with an increasing automorphism eliminates quantifiers (see Definition 2.1.20), and is thus the model companion of the theory of linear orders with an increasing automorphism. For ordered abelian groups, things are a little more tricky. One could try having a similar restriction (for all x > 0, σ(x) > x), but it is an open question whether the theory of an ordered abelian group with such an increasing automorphism has a model companion. The conjecture is that it probably does not exist. The restricted class of automorphisms we consider is the class of multiplicative automorphisms (see Definition 3.2.6). We call a group with a multiplicative automorphism σ a MODAG (Multiplicative Ordered Difference Abelian Group). Our axioms imply that Z[σ] takes on the structure of a quasi-ordered ring, with MODAG having the structure of an ordered Z[σ]-module. Here Z[σ] is the polynomial ring generated over the integers by the single invariant σ. An obvious candidate for a model CHAPTER 1. INTRODUCTION 3 companion then is a non-trivial “divisible” MODAG (which we call MODDAG), where divisibility in this context means divisible as an R-module (not just as a Z-module), where R is a ring closely related to Z[σ] (see precise definition in Definition 3.2.10). By abuse of terminology, we denote the theory of the class of all MODAGs (respectively, MODDAGs) also as MODAG (respectively, MODDAG). We first show that MODAG and MODDAG are co-theories (see Definition 2.1.28). We also show that MODDAG has algebraically prime models and is simply closed. It follows that MODDAG has quantifier elimination. Thus, we prove that the theory of non-trivial ordered divisible difference abelian groups (with a multiplicative automorphism) is the model companion of the theory of ordered abelian groups with a multiplicative automorphism. One could try doing similar sorts of things with other theories as well, for example, the theory of ordered fields or any o-minimal theory. It is not yet clear to us what class of restricted automorphisms will work in those contexts.

1.2 Study of the theory of (multiplicative) valued difference ﬁelds.

In Chapter 4, we consider the theory of valued fields with an automorphism. It is well known that the theory of fields with an automorphism has a model companion, namely ACF A (Algebraically Closed Fields with a Generic Automorphism) [7]. However, with a valued field, things are more complicated. For one thing, now we have an interpretable (see Definition 2.1.8) ordered set in the structure, namely the value group, which makes the base theory unstable. For another, it is not enough to consider only difference polynomials now. One has to take into account the restrictions on the valuation as well. In this thesis we are always in the equi-characteristic zero case, i.e., both the valued field and the residue field are always in characteristic zero. In the setting of valued fields, one defines the notion of a henselian valued field (see Definition 2.3.13), which is an elementary class. For this elementary class, Ax and Kochen, and independently Er˘sov, prove a classical result, which can be summarized as follows.

Theorem ([6, Theorem 5.4.12]). Let K and K0 be two unramified henselian valued fields with residue fields k and k0, and value groups Γ and Γ0 respectively. Then K ≡ K0 if and only if Γ ≡ Γ0, as ordered abelian groups, and k ≡ k0, as fields.

In other words, the elementary theory of an unramified (see Definition 2.3.7) henselian valued field is determined by the elementary theories of its value group and residue field. This result is popularly known as the AKE-principle. Valued fields with an automorphism (K, v, Γ, σ) are also called valued difference fields. The automorphism σ : K → K is a field automorphism, which, in addition, satisfies σ(O) = O, where O is the ring of integers. Our goal is to understand this theory and prove an CHAPTER 1. INTRODUCTION 4 analogous AKE-principle in this context. If we hope to have a model companion for the whole structure, we should have a model companion for the value group and the residue field. A first step to fulfilling this goal is to specify how the valuation v interacts with the automorphism σ. A second step is to have an analogous definition of henselianity in this context. Since we have to deal with difference polynomials or σ-polynomials (which are polynomials in the variables σ(x), σ2(x),...), this analogous concept is termed σ-henselianity. Two special cases have been worked out before. For similar results in a slightly different context, see [21]. The case of the isometric automorphism (where v(σ(x)) = v(x) for all x), has been worked out by Scanlon in [22], and later by Bélair,Macintyre and Scanlon in [5]. In this case, σ is trivial with respect to the valuation, and thus there is no additional structure on the value group. The value group is, thus, just an ordered abelian group, which has a model companion, namely, divisible ordered abelian group. The residue field is a difference field, which again has a model companion, namely ACF A. To see the definition of σ-henselianity in this context, see Definition 2.3.28. To obtain the AKE-principle in this context, one needs a technical condition about “existence of enough constants” (see Definition 2.3.29). To have relative quantifier elimination with respect to the theories of the value group and the residue field in this context, one further needs the residue field to be closed under taking roots. The second case, that of the contractive automorphism (where v(σ(x)) > nv(x) for all x ∈ K× with v(x) > 0 and all n), has been worked out by Azgin in [1]. In this case, σ increases the valuation infinitely. Again, the residue field is an ordinary difference field, which has a model companion, namely ACF A. However, the value group is no longer an ordinary ordered abelian group. The automorphism σ of the field K induces an automorphism on the value group as well. The value group, thus, has the structure of an ordered difference abelian group, and can be treated as an ordered module over the ring Z[σ]. Fortunately, since σ(x) is Pn i infinitely bigger than x for all x > 0, for any linear difference polynomial L(x) = i=0 aiσ (x) with ai ∈ Z for all i = 0, . . . , n and an 6= 0, it is easy to decide when L(x) > 0 (or L(x) < 0) by the rule: L(x) > 0 ⇐⇒ an > 0. It then follows that the theory of the value group, in this context, has a model companion. The notion of σ-henselianity in this context (see Definition 2.3.33) is quite strong because it forces the residue field to be linear difference closed. However, this strong notion is justified because there are counter-examples showing that for the AKE-principle to go through, the residue field has to be linear difference closed. Clearly, the above two cases are two ends of a spectrum. Our goal in this thesis is to bridge the gap between these two cases. For example, we want to deal with the cases, when v(σ(x)) = 2v(x), or v(σ(x)) = 3v(x), etc. To that end, we define the notion of multiplicative automorphism (see Definition 4.1.1), where we impose v(σ(x)) = ρ · v(x) for all x,√ where ρ is thought of as an element of a real-closed field. In particular, ρ can be 2, 3, 2, or even δ, where δ is an infinitesimal or an infinite element. Recall that such multiplication makes sense in a MODAG, and thus the value group has the structure of a MODAG in this context. By our results in Section 3.2 (see Theorem 3.2.15), the theory of MODAG has a model companion. The residue field, in this context, is again an ordinary difference field, CHAPTER 1. INTRODUCTION 5 whose model companion is ACF A. Different notions of σ-henselianity can be defined in this context. We will talk about two such notions in Chapter 4, including Azgin’s definition. The counter-examples from the contractive case, however, still hold in this context. And so we stick to Azgin’s strong definition. We also show that under the assumption that the residue field is linear difference closed, these two notions of σ-henselianity are equivalent. A key ingredient to the proof of the classical AKE-principle is the uniqueness of the algebraically maximal immediate algebraic extensions of an unramified valued field. A valued field extension K0 of K is called immediate if K and K0 have the same value group and the same residue field. A valued field is called maximal if it has no proper immediate extensions, and it is called algebraically maximal if it has no proper immediate algebraic extensions. It is well known that every algebraically maximal valued field is henselian, and that every valued field has algebraically maximal immediate algebraic extensions [13]. Kaplansky shows that

Theorem ([13]). All algebraically maximal immediate algebraic extensions of an unramiﬁed valued ﬁeld K are isomorphic over K.

In the valued difference field context, one has to deal with σ-polynomials instead of ordinary polynomials, and thus one is led to consider σ-algebraic extensions in place of algebraic extensions. A valued difference field is then called σ-algebraically maximal if it has no proper σ-algebraic immediate extension. It is easy to show that every σ-algebraically maximal valued difference field is σ-henselian, and that every valued difference field has σ-algebraically maximal immediate σ-algebraic extensions. Following Azgin’s approach, we show that for a multiplicative valued difference field K of equi-characteristic zero, the assumption that the residue field is linear difference closed implies that all σ-algebraically maximal immediate σ-algebraic extensions of K are isomorphic over K. This result then becomes a key ingredient in proving the AKE-principle in the multiplicative valued difference field context. 6

Chapter 2

Preliminaries

2.1 Basics from Model Theory

Model theory is the study of mathematical structures together with a specified language. The main reason for this study is the fact that the same mathematical structure can be looked at differently depending on the context, and then in different environments the same structure behaves differently. For example, the real numbers R can be considered as a dense linear order, or as an ordered abelian group, or even as an ordered field. We may also consider R as an ordinary abelian group or field without the order. Depending on how we look at it, R is behaves differently, and the theorems one can then prove vary accordingly. Thus, it is not only important to specify the structure we are going to talk about, but it is equally important to specify the language in which we are going to talk about that structure. All definitions and results in this section are taught in a basic graduate course in Model Theory and can be found in [19]. Definition 2.1.1. A language L is given by specifying a set F of function symbols, a set R of relation symbols, and a set C of constant symbols, together with a set of positive integers nF for each F ∈ F and a set of positive integers nR for each R ∈ R. The numbers nF and nR tell us that F is a function of nF variables, and R is a relation on nR-tuples. As a notation, we use bold letters (like x) to denote tuples in this thesis. Example 2.1.2. Some of the standard languages that we talk about in this thesis include:

−1 −1 1. The language of groups LG˜ = {·, , e}, where · is a binary function symbol, is a unary function symbol, and e is a constant symbol. For abelian groups, we use the convention LG = {+, −, 0}.

2. The language of ordered groups LOG = LG ∪{<}, where < is a binary relation symbol.

3. The language of rings LR = {+, −, ·, 0, 1}, where + and · are binary function symbols, − is a unary function symbol, and 0 and 1 are constant symbols. CHAPTER 2. PRELIMINARIES 7

Deﬁnition 2.1.3. An L-structure M is a non-empty set M, called the universe, together with a function F M : M nF → M for each F ∈ F, a set RM ⊆ M nR for each R ∈ R, and a special element cM ∈ M for each c ∈ C. We refer to F M,RM and cM as the interpretations of the symbols F,R and c.

Example 2.1.4. (R, +, −, 0) is an LG-structure, where + is interpreted as addition on real numbers, − is interpreted as the additive inverse, and 0 is interpreted as the real number zero. Similarly, (R, +, −, ·, 0, 1) is an LR-structure, where · is interpreted as multiplication of real numbers, and 1 is interpreted as the real number one. Definition 2.1.5. A morphism of L-structures η : M → N is a map from M to N respecting all the structures given by L. That is, if we let η also denote the induced map on the Cartesian powers of M, then η(RM) ⊆ RN , η(F M(x)) = F N (η(x)), and η(cM) = cN for all relation symbols R, function symbols F , and constant symbols c in L. A bijective L-morphism, whose inverse is also an L-morphism, is called an L-isomorphism. A morphism of L-structures that is one-to-one and an L-isomorphism onto its image is called an L-embedding. If M ⊆ N and the inclusion map is an L-embedding, we say either that M is a substructure of N or that N is an extension of M. An L-morphism η : M → M is called an L-endomorphism of M. An L-isomorphism η : M → M is called an L-automorphism of M. Given a language L, a key thing to consider are first-order L-formulas, which are constructed in an inductive way. First one constructs terms of L by composing a set of variables ∞ {xi}i=1 and the function and constant symbols of L in a meaningful way, e.g. +(x1, 0) (com- monly written as, x1 + 0) is a term of LG. Using the symbol ‘=’ (for equality) and other relation symbols of L on these terms, one then constructs basic (or atomic) formulas, e.g. x1 + 0 = x1 is a basic formula in LG. Finally one constructs formulas of L from the basic formulas by finite Boolean combinations (“not”, “and”, “or”) and existential and universal quantification over the variables xi, e.g. ∀x1(x1 + 0 = x1 ∨ ¬(x2 = 0)) is an LG-formula. If an occurrence of a variable xi in a formula lies in the “scope” of a quantifier (∀xi or ∃xi), it is called a bound occurrence; otherwise it is called a free occurrence. In the last example, x1 is bound whereas x2 is free. An L-formula where all occurrences of all the variables are bound is called an L-sentence. An L-formula where there is no occurrence of quantifiers is called a quantifier-free formula. For precise definitions of these notions, see [19, Definition 1.1.4–1.1.6]. Remark 2.1.6. In this thesis, we deal with multi-sorted languages, where we have a set S of sorts, in addition to the sets R of relation symbols, F of function symbols, and C of constant symbols. We also have a function called sort, that associates to each R ∈ R an nR-tuple, sort(R) = hS ,...,S i, with the intention that the question of whether R(x , . . . , x ) R1 RnR 1 nR

holds or not can be asked only if xj is of sort SRj . Similarly for each F ∈ F, sort(F ) is a (n + 1)-tuple hS ,S ,...,S i of sorts deﬁning the sorts both of the domain and the F F0 F1 FnF range of F . Finally, for each c ∈ C, sort(c) = Sc gives the sort of c. CHAPTER 2. PRELIMINARIES 8

A many-sorted structure N with sorts S is then a S-sequence of non-empty sets N := N hNS : S ∈ Si. For each R ∈ R, then R ⊂ NSR × · · · × NSR ; for each F ∈ F, we have 1 nR N N F : NSF × · · · × NSF → NSF ; and ﬁnally, for each c ∈ C, we have c ∈ NSc . 1 nF 0 The conditions for η : M → N to be a morphisms of multi-sorted structures are identical to those in the one-sorted case, with the additional restriction that for each S ∈ S and m ∈ M, m ∈ MS =⇒ η(m) ∈ NS. Another feature of multi-sorted languages is the assignment of variables to speciﬁc sorts. That is, there is a set V such that for each S ∈ S there is a set of variables VS ∈ V, with the intention that any variable xi ∈ VS is allowed to vary only over domain S. Whatever follows in this section can be very naturally generalized to the multi-sorted case. So for the rest of this section we will concentrate on one-sorted languages and structures.

Definition 2.1.7. Let L be a language and M be an L-structure. Let φ(x1, . . . , xn) be an L-formula in the free variables x1, . . . , xn. Let a1, . . . , an ∈ M (not necessarily distinct). Then we write M |= φ(a1, . . . , an) if φ is true of (a1, . . . , an) in M. If φ is an L-sentence, then there are no free variables to be substituted, and hence φ is either true or false in M, and we write M |= φ (read as “M models φ”). The solution set of φ in M, written φ(M), is the set {a ∈ M n | M |= φ(a)}. A subset of D of M n is called definable if it is the solution set of some L-formula in M. A function is said to be definable if its graph is definable. For example, in any M, the whole universe M is definable by the formula x = x, and the empty set√ is√ definable by the formula x 6= x.A more nontrivial example is the fact that the set {− 2, 2} is definable in (R, +, −, ·, 0, 1) by the formula x · x = 1 + 1. If A ⊆ M is any subset, then we can expand the language to LA by adding new constant symbols for each a ∈ A. M, then, naturally becomes an LA-structure by interpreting each new constant symbol by its corresponding element in A. We call the LA-definable sets in M A-definable or definable with parameters from A. For example, π2 being transcendental over the rationals is not definable in (R, +, −, ·, 0, 1), but is definable in the expanded language (R, +, −, ·, 0, 1, π) by the formula x = π · π. Definition 2.1.8. Let L and L0 be two languages. We say that an L0-structure M0 is interpretable in an L-structure M if there is a definable X ⊆ M n, a definable equivalence relation E on X, and for each symbol of L0 there is a definable E-invariant set on X (where “definable” means definable in L) such that X/E with the induced structure is L0-isomorphic to M0.

Deﬁnition 2.1.9. A set Σ of L-sentences is said to be satisﬁable (or equivalently, consistent) if there is an L-structure M such that M |= Σ, i.e., M |= φ for each sentence φ ∈ Σ. An (L-) theory T is a consistent set of L-sentences. A model of T is an L-structure M such that M |= T . For any sentence φ, we say T ` φ (read as, T proves φ) if M |= φ for all models M |= T . CHAPTER 2. PRELIMINARIES 9

T is said to be complete if for all L-sentences φ, either T ` φ or T ` ¬φ. For example, the theory of algebraically closed fields of characteristic zero is a complete theory. T0 is a set of axioms for T if M |= T0 =⇒ M |= T . For example, the theory of algebraically closed fields is axiomatized by the theory of fields together with a countably infinite list of sentences saying for each integer k ≥ 1,

k ∀a0, . . . , ak ak 6= 0 =⇒ ∃x a0 + a1x + ··· + akx = 0 .

If M is an L-structure, the theory of M is T h(M) := {φ | M |= φ}. Clearly, T h(M) is a complete theory.

Deﬁnition 2.1.10. Two L-structures M and N are called elementarily equivalent, written M ≡ N , if for all L-sentences φ,

M |= φ ⇐⇒ N |= φ.

Clearly M ≡ N if and only if T h(M) = T h(N ).

Theorem 2.1.11 ([19, Theorem 1.1.10]). Suppose that j : M → N is an isomorphism. Then, M ≡ N .

Theorem 2.1.12 (Compactness Theorem, [19, Theorem 2.1.4]). A theory T is satisfiable if and only if every finite subset of T is satisfiable.

Allowing for formulas with free variables instead of sentences gives us types. Types are in essence theories, with free variables thought of as new constant symbols. Let L be a language, M an L-structure, and A ⊆ M. Let LA be the language obtained by adding to L constant symbols for each a ∈ A. Let T hA(M) be the set of all LA-sentences true in M.

Definition 2.1.13. A set p of LA-formulas in free variables v1, . . . , vn is called an n-type over A if p ∪ T hA(M) is satisfiable, or equivalently, by compactness theorem, finitely satisfiable. p(x) ` φ(x) if for all N |= T hA(M) and a ∈ N , we have N |= p(a) ⇐⇒ N |= φ(a). p is called a complete n-type over A if p ` φ or p ` ¬φ for all LA-formulas φ with free variables from v1, . . . , vn; otherwise we say p is a partial type. Example 2.1.14. We give two examples - the first one is a partial type, and the second one is a complete type.

1. Consider M = (Z, <, +) and A = Z. Let p(v) = {v > 1, v > 2, v > 3 ...}. Any finite subset of p(v) ∪ T hA(M) is satisfiable by interpreting v as a sufficiently large element of Z. Hence p(v) is a 1-type over Z. It is partial because neither ∃y(v = y + y) ∈ p(v), nor ¬∃y(v = y + y) ∈ p(v). CHAPTER 2. PRELIMINARIES 10

n 2. Let M be an LA-structure and a ∈ M . Consider p(x) = {φ(x) ∈ LA | M |= φ(a)}. Clearly p is an n-type over A. Moreover, since for every LA-formula φ(x), either M |= φ(a) or M |= ¬φ(a), p(x) is a complete n-type over A.

We denote such a type as the complete type of a over A in M, written tpM(a/A). When A = φ, we just write it as tpM(a). Deﬁnition 2.1.15. For L-structures M and N and B ⊆ M, η : B → N is a partial elementary map if and only if for all L-formulas φ and all ﬁnite sequences b from B,

M |= φ(b) ⇐⇒ N |= φ(η(b)).

Clearly, if η is a partial elementary map, then tpM(b) = tpN (η(b)). If B = M, η is called an elementary map. If M ⊆ N and the inclusion map is elementary, then N is said to be an elementary extension of M, written as M ≺ N .

Saturated models are thus very important because they have realizations to all types over small parameter sets. So question arises: do such models exist?

+ Theorem 2.1.17 ([19, Theorem 4.3.12]). Let κ > ℵ0. For all M, there is a κ -saturated extension M ≺ N with |N| ≤ |M|κ.

Corollary 2.1.18 ([19, Corollary 4.3.13]). Assuming GCH (2κ = κ+), there are saturated models of T of size κ+ for all κ.

Remark 2.1.19. Because of the existence of such saturated models of arbitrarily large cardinality of any ﬁxed theory T , model theorists prefer working inside a big saturated model C, called the monster model, which is thought of as the entire universe, and then any other model M of the theory T is considered to be an elementary substructure of C.

Definition 2.1.20. A theory T is said to eliminate quantifiers (or have quantifier elimination) if for every model M |= T every definable set in M is a boolean combination of basic definable sets. More precisely, for every formula φ(x) in the language, there is a quantifier- free formula ψ(x), such that in every model M |= T , we have M |= ∀x(φ(x) ↔ ψ(x)).

Example 2.1.21. We give examples of two theories - one which eliminates quantiﬁers and one which does not.

1. Tarski proves that the theory of algebraically closed ﬁelds eliminates quantiﬁers in the language LR (see [19, Theorem 3.2.2]). CHAPTER 2. PRELIMINARIES 11

2. The theory (Q, +, −, ·, 0, 1) does not eliminate quantifiers. For example, the formula ∃z(x = z · z) defines an infinite co-infinite set in the rational numbers, and hence is not equivalent to any quantifier-free formula. There are different tests for quantifier elimination, and we use two of them in this thesis. Test 2.1.22 ([19, Proposition 4.3.28]). Let T be a theory in a first order language L. Suppose that T has no finite models, that T is complete with respect to the atomic theory (i.e., for each atomic sentence ψ of L, ψ ∈ T or ¬ψ ∈ T ), and that L has at least one constant symbol. Then the following are equivalent: • T is complete and eliminates quantifiers. • If M, N |= T are κ+-saturated for some infinite cardinal κ ≥ |T |, A ⊆ M is a substructure of M of cardinality at most κ, f : A,→ N is an L-embedding, and a ∈ M, then there is a substructure A0 of M containing A and a and an extension of f to an L-embedding g : A0 ,→ N . To state the other test, we need a couple of definitions.

Deﬁnition 2.1.23. For a theory T , we denote by T∀ the set of all universal consequences of T .

Remark 2.1.24. It is an easy exercise to show that A |= T∀ if and only if there is M |= T with A ⊆ M. For example, if T is the theory of ﬁelds, then T∀ is the theory of integral domains.

Deﬁnition 2.1.25. A theory T has algebraically prime models if for any A |= T∀ there is M |= T and an embedding i : A → M, such that for all N |= T and embeddings j : A → N , there is h : M → N such that j = h ◦ i. If M, N |= T and M ⊆ N , then M is said to be simply closed in N , written as M ≺s N , if, for any quantiﬁer-free formula φ(w) in 1-variable with parameters from M there is a ∈ N such that N |= φ(a), then there is b ∈ M such that M |= φ(b). Test 2.1.26 ([19, Corollary 3.1.12]). Suppose that T is an L-theory such that 1.T has algebraically prime models and

2. M ≺s N whenever M ⊆ N are models of T. Then, T has quantifier elimination. For theories we consider in this thesis, for example theory of fields, theory of valued fields, theory of difference fields, etc., an important question is to decide when certain equations have solutions. For example, in the context of fields, we ask when certain polynomial equations have solutions; in the context of difference fields, we ask when certain difference polynomial equations have solutions, etc. It is easy to see that, in these contexts, such equations are actually basic formulas. So we define the following. CHAPTER 2. PRELIMINARIES 12

Definition 2.1.27. A model M |= T is existentially closed in the class of models of T if, for any quantifier-free formula φ(x) with parameters from M, if there exists a model N |= T , M ⊆ N and a ∈ N such that N |= φ(a), then there exists b ∈ M such that M |= φ(b). It is then an important question to classify such existentially closed models of a theory T . To that end, we define the following. Definition 2.1.28. A theory T is model-complete if whenever M, N |= T and M ⊆ N , we have M ≺ N . That is, T is model-complete if and only if all embeddings are elementary. A companion of a theory T is a theory T ∗ such that every model of T can be embedded in a model of T ∗ and vice versa. Such T and T ∗ are also called co-theories. A model companion of T is a companion theory (or co-theory) T ∗ that is model-complete. If T ∗ eliminates quantifiers as well, we say that T ∗ is the model completion of T . Example 2.1.29. The theory of divisible abelian groups is the model completion of the theory of abelian groups. The theory of algebraically closed fields is the model completion of the theory of fields. Theorem 2.1.30 ([19, Exercise 3.4.13]). Let T be an ∀∃-theory, i.e., T is axiomatized only by sentences of the form ∀x1 · · · ∀xn∃y1 · · · ∃ymφ(x1, . . . , xn, y1, . . . , ym), where φ is a quantifier- ∗ free formula in the free variables x1, . . . , xn, y1, . . . , ym. If T is a model companion of T , then T ∗ is the theory of the existentially closed models of T . A model complete theory need not be complete, or vice versa. For example, the theory of dense linear orders with a least and a greatest element is complete, but not model complete. Conversely, the theory of algebraically closed fields is model complete, but not complete. However, Theorem 2.1.31 ([19, Proposition 3.1.15]). Let T be a model-complete theory. Suppose that there is M0 |= T such that M0 embeds into every model of T . Then, T is complete. Moreover, Theorem 2.1.32 ([19, Proposition 3.1.14]). If T has quantifier elimination, then T is model- complete. Model complete theories have a nice syntactic characterization. Theorem 2.1.33. An L-theory T is model complete if and only if every L-formula φ(x) is equivalent modulo T to a ∀-formula ψ(x) of L. Definition 2.1.34. T is said to be decidable if there is an algorithm that, given any L- sentence φ, decides whether T ` φ. For example, T h(C), the first-order theory of the field of complex numbers in the language LR, is decidable. Theorem 2.1.35 ([19, Lemma 2.2.8]). Let T be a recursive complete theory in a recursive language L. Then T is decidable. CHAPTER 2. PRELIMINARIES 13

2.2 Basics from Diﬀerence Algebra

Definition 2.2.1. A pair K = (K, σ) is called a difference ring if K is a ring, and σ : K → K is a ring automorphism. If K is a field, (K, σ) is called a difference field.

Remark 2.2.2. In literature a difference ring generally means a ring with an endomorphism. For our case, a difference ring always means a ring with an automorphism. This is known as inversive difference ring in literature.

Definition 2.2.3. Let K ≺ K0 be an extension of difference rings. For any a ∈ K0, Khai denotes the smallest difference subring of K0 containing K and a. The underlying ring of Khai is K[σi(a): i ∈ Z].

Definition 2.2.4. For any (n + 1)-variable polynomial P (X0,...,Xn) ∈ K[X0,...,Xn], we define a corresponding 1-variable σ-polynomial f(x) = P (x, σ(x), σ2(x), . . . , σn(x)). We define the degree of f to be the total degree of P ; and the order d of f to be the largest integer 0 ≤ d ≤ n such that Xd appears nontrivially in P . If f ∈ K, then order(f) := −∞. Finally we define the complexity of f as

complexity(f) := (d, deg f, deg f) ∈ ( ∪ {−∞})3, xd N where complexity(0) := (−∞, −∞, −∞) and for f ∈ K, f 6= 0, complexity(f) := (−∞, 0, 0). We order complexities lexicographically.

Let x = (x0, . . . , xn), y = (y0, . . . , yn) be tuples of indeterminates and a = (a0, . . . , an) n+1 be a tuple of elements from some ring K. Let I = (i0, . . . , in) be a multi-index (I ∈ N ).

I i0 in Definition 2.2.5. We define a := a0 ··· an and the length of I as |I| := i0 + ··· + in. Clearly |I| ∈ N. It is well-known that for any polynomial P (x) over a ring K, we have a unique Taylor expansion in K[x, y]: X I P (x + y) = P(I)(x) · y , I n+1 where the sum is over all I = (i0, . . . , in) ∈ N , each P(I)(x) ∈ K[x], with P(I) = 0 I i0 in for |I| > deg(P ), and y := y0 ··· yn . Thus I!P(I) = ∂I P where ∂I is the operator i0 in (∂/∂x0) ··· (∂/∂xn) on K[x], and I! := i0! ··· in!. When |I| = 1, we denote P(I) by P(k) where k is the unique position such that ik = 1. We construe Nn+1 as a monoid under + (componentwise addition), and let ≤ be the product (partial) ordering on Nn+1 induced by the natural order on N. Define for I ≤ J ∈ Nn+1, J j j := 0 ··· n , I i0 in CHAPTER 2. PRELIMINARIES 14

j j! where = . Then it is easy to check that for I, J ∈ n+1, i i!(j − i)! N I + J (f ) = f . (I) (J) I (I+J)

n+1 Let x be an indeterminate and a be an element of a ring K. Let I = (i0, . . . , in) ∈ N . When n is clear from the context, we define the following. Note that this definition makes sense even when K is an abelian group. Definition 2.2.6. σ(x) := (x, σ(x), . . . , σn(x)), and σ(a) = (a, σ(a), . . . , σn(a)). We also I Pn k define σ (x) := k=0 ikσ (x).

Then it follows from above that for P ∈ K[x0, . . . , xn] and f(x) = P (σ(x)), we have f(x + y) = P (σ(x + y)) = P (σ(x) + σ(y)) X I X I = P(I)(σ(x)) · σ(y) = f(I)(x) · σ(y) , I I

where f(I)(x) := P(I)(σ(x)). Definition 2.2.7. Let K ≺ K0 be a difference field extension and a ∈ K0. Then a is said to be σ-algebraic over K if there is a non-zero σ-polynomial f(x) with coefficients from K such that f(a) = 0; otherwise, a is said to be σ-transcendental over K. If a is σ-algebraic, then there is a σ-polynomial of the least complexity (not necessarily unique) that a satisfies, called a minimal σ-polynomial of a over K.

2.3 Basics from Valuation Theory

2.3.1 Valued Fields Definition 2.3.1. A triple K = (K, v, Γ) is called a valued field if K is a field, Γ is an ordered abelian group and v : K× → Γ is a surjective map satisfying v(x · y) = v(x) + v(y) v(x + y) ≥ min{v(x), v(y)}, whenever x + y 6= 0 The second condition is called ultra-metric triangle inequality. We introduce a new symbol ∞ and extend the valuation to 0 by defining v(0) = ∞, with the condition that ∞ > γ for all γ ∈ Γ γ + ∞ = ∞ + γ = ∞ for all γ ∈ Γ ∪ {∞} Γ is called the value group and v is called the valuation function. It is worthwhile to mention here that the value group can be really big and is not, in general, a subset of the reals. Also, the valuation function we consider is sometimes known in literature as Krull valuation. CHAPTER 2. PRELIMINARIES 15

Deﬁnition 2.3.2. The set OK := {x ∈ K | v(x) ≥ 0} is called the ring of integers. It is easy to check that

Proposition 2.3.3 ([18, Chapter XII, §4]). OK is a valuation ring, i.e., it is an integral domain with fraction ﬁeld K, has a unique maximal ideal, and for every non-zero x in the −1 fraction ﬁeld K of OK , either x ∈ OK or x ∈ OK . Also the unique maximal ideal of OK is given by mK := {x ∈ K | v(x) > 0}.

When K is clear from the context, we denote OK and mK by O and m respectively.

Definition 2.3.4. Since mK is a maximal ideal of OK , the quotient k := OK /mK is a field, called the residue field. The natural quotient map π : OK → k is called the residue map. Example 2.3.5. Some examples of valued fields with value groups and residue fields:

• Any ﬁeld K with the trivial valuation deﬁned by v(x) = 0 for all x ∈ K×. Here Γ = {0} and k = K.

• Fix a prime number p. Then the p-adic valuation on Q is defined by vp(x) = r, where a x ∈ × can be written as x = pr , with gcd(p, ab) = 1. Here Γ = and k = . Q b Z Fp • For any field k, the field k((t)) of Laurent series over k is a valued field via

X n v(f) = N ⇐⇒ f = ant with aN 6= 0. n≥N

Here the value group is Z and the residue field is k. Definition 2.3.6. A valued field (K, v, Γ), with residue field k, is said to be of equi- characteristic zero if char(K) = char(k) = 0; it is said to be of equi-characteristic p, for some positive prime p, if char(K) = char(k) = p. If char(K) = 0 but char(k) = p > 0, it is said to be of mixed characteristic. These are the only three possibilities for a valued field.

Definition 2.3.7. A valued field (K, v, Γ) with residue field k is called unramified if

• char(K) = char(k) = 0, or

• char(K) = 0, char(k) = p > 0 and v(p) is the least positive element in Γ.

(K, v, Γ) with residue field k is said to be finitely ramified if

• char(K) = char(k) = 0, or

• char(K) = 0, char(k) = p > 0 and {γ ∈ Γ | 0 < γ < v(p)} is ﬁnite. CHAPTER 2. PRELIMINARIES 16

Deﬁnition 2.3.8. A limit-ordinal indexed sequence {aη}η<κ in K is called a Cauchy sequence 0 00 if, for every γ ∈ Γ, there exists η0 < κ such that for all η , η ≥ η0, v(aη00 − aη0 ) > γ. We say that {aη}η<κ converges to a ∈ K if, for every γ ∈ Γ, there exists η0 < κ such that for all η > η0, v(a − aη) > γ; this is also written as v(a − aη) → ∞. A valued ﬁeld (K, v, Γ) is called complete if every Cauchy sequence in K converges to an element of K.

It is well known that

Theorem 2.3.9 ([10, Theorem 2.4.3]). Every valued ﬁeld has a unique (up to isomorphism of valued ﬁelds) complete extension in which it is dense, namely its completion.

Remark 2.3.10. A completion of a valued field is also a valued field with the same value group and the same residue field. For example, for any given prime p > 0, the field Qp of p-adic numbers is the completion of the rationals with the p-adic valuation, and is a valued field with Γ = Z and k = Fp.

Definition 2.3.11. A value group Γ is said to be archimedian if for any 0 < γ1, γ2 ∈ Γ, there exists n ∈ Z such that nγ1 > γ2. For example, Γ = Z is archimedian. Theorem 2.3.12 (Hensel’s Lemma, [10, Theorem 4.1.3]). Let K be a complete valued field with an archimedian value group. Let P (X) ∈ OK [X] be a polynomial with integral coeffi- 0 cients. Suppose that a ∈ OK is a point for which v(P (a)) > 0 = v(P (a)). Then there is some b ∈ K with v(a − b) = v(P (a)) and P (b) = 0.

Deﬁnition 2.3.13. A valued ﬁeld K is called henselian if the conclusion of Hensel’s Lemma 0 holds in K, i.e., for any P (X) ∈ OK [X] and a ∈ OK with v(P (a)) > 0 = v(P (a)), there is b ∈ K with v(a − b) = v(P (a)) and P (b) = 0.

Definition 2.3.14 (3-sorted language of valued fields). The 3-sorted language of valued fields is the language L3vdf := hK, Γ, k; v, πi, where K is the field sort and k is the residue field sort, both considered as structures in the language LR := {+, −, ·, 0, 1}, and Γ is the value group sort considered as a structure in the language LOG := {+, −, <, 0}. These sorts are connected by the valuation function v : K → Γ ∪ {∞} and the residue map π : O → k.

Theorem 2.3.15 (Ax-Kochen-Er˘sov, [6, Theorem 5.4.12]). If K and L are two henselian ﬁelds with residue ﬁelds kK and kL of characteristic zero, and value groups ΓK and ΓL, then

K ≡L3vf L if and only if kK ≡LR kL and ΓK ≡LOG ΓL. In other words, the theory of K, as a valued ﬁeld, is determined by the theory of Γ, as an ordered abelian group, and the theory of k, as a ﬁeld. This is known as the AKE-principle. This theorem has some nice consequences. CHAPTER 2. PRELIMINARIES 17

Corollary 2.3.16 ([6, Corollary 5.4.18]).

• Qp and Fp((t)) have the same limit theories as p → ∞. • If K is a henselian field of equi-characteristic zero whose residue field and value group have decidable theories, then so does K as a valued field.

Definition 2.3.17 ([13, 14]). An extension K ≺ K0 of valued fields is said to be immediate if K and K0 have the same value group and the same residue field. A valued field K is called maximally complete if it has no proper immediate extensions.

For example k((t)) is maximally complete for any field k. It is well known that a maximally complete valued field is both complete and henselian. A key notion in the study of immediate extensions of valued fields is that of pseudo-cauchy sequence.

Deﬁnition 2.3.18. A pseudo-cauchy sequence (henceforth, pc-sequence) from K is a limit ordinal indexed sequence {aη}η<λ of elements of K such that for some index η0,

00 0 η > η > η ≥ η0 =⇒ v(aη00 − aη0 ) > v(aη0 − aη).

We say that a property P holds eventually for a sequence {aη} if there is some η0 such that for all η ≥ η0, aη satisﬁes P . We say a is a pseudo-limit of a limit ordinal indexed sequence {aη} from K (denoted aη a) if there is some index η0 such that

0 η > η ≥ η0 =⇒ v(a − aη0 ) > v(a − aη).

Such a sequence is necessarily a pc-sequence in K. For a pc-sequence {aη} as above, let 0 γη := v(aη0 − aη) for η > η ≥ η0; it follows by ultra-metric triangle inequality that γη

depends only on η. Then {γη}η≥η0 is strictly increasing, or alternatively, {γη} is eventually increasing. We deﬁne the width of {aη} as the set

{γ ∈ Γ ∪ {∞} : γ > γη for all η ≥ η0}.

We say two pc-sequences {aη} and {bη} from K are equivalent if they have the same pseudo-limits in all valued ﬁeld extensions of K. Equivalently, {aη} and {bη} are equivalent iﬀ they have the same width and a common pseudo-limit in some extension of K.

We now mention another important structure that is deﬁnable in a valued ﬁeld. Let K×, O× and k× denote the set of units of K, O and k respectively. It is easy to see that (1+m) is the kernel of the residue map π restricted to O×. Since π is a group homomorphism, (1 + m) is a subgroup of O×, and hence a subgroup of K× under multiplication. Thus taking the quotient K×/(1 + m) makes sense. CHAPTER 2. PRELIMINARIES 18

Definition 2.3.19. We denote the factor group as RV := K×/(1 + m) and the natural quotient map as rv : K× → RV (short for “residual-valuation”). To extend the map to whole of K, we introduce a new symbol “∞” (as we do with value groups) and define rv(0) = ∞. Though RV is defined merely as a group, it inherits much more structure from K. Since the valuation v on K is given by the exact sequence ι v 1 / O× / K× / Γ / 0 and since 1 + m ≤ O×, the valuation descends to RV giving the following exact sequence

ι vrv 1 / k× / RV / Γ / 0 (note that O×/(1 + m) = (O/m)× = k×). It follows straight from the deﬁnitions that, Lemma 2.3.20. For all non-zero x, y ∈ K, the following are equivalent: 1. rv(x) = rv(y) 2. v(x − y) > v(y) 3. π(x/y) = 1 Proof. See [11, Proposition 1.3.3]. Also note that if x, y ∈ O×, then the last condition is equivalent to saying π(x) = π(y). And both (2) and (3) imply that v(x) = v(y). RV also inherits an image of addition from K via the relation

1 n 1 n 1 1 n n ⊕(xrv, . . . , xrv, zrv) = ∃x , . . . , x , z ∈ K xrv = rv(x ) ∧ · · · ∧ xrv = rv(x ) ∧

1 n zrv = rv(z) ∧ x + ··· + x = z .

1 n The sum xrv + ··· + xrv is said to be well-deﬁned (and = zrv) if there is exactly one zrv such 1 n that ⊕(xrv, . . . , xrv, zrv) holds. Unfortunately this is not always the case. Fortunately, the sum is well-deﬁned when it is “expected” to be. Formally,

Lemma 2.3.21 ([11, Proposition 1.3.6, 1.3.7 and 1.3.8]). rv(x1)+···+rv(xn) is well-deﬁned (and is equal to rv(x1 + ··· + xn)) if and only if v(x1 + ··· + xn) = min{v(x1), . . . , v(xn)}.

Definition 2.3.22 (Languages). We consider RV as a structure in the language Lrv := −1 {·, , ⊕, 1}. This leads to a 4-sorted language for valued fields L4vf := hK, Γ, k, RV ; v, π, vrv, ι, rvi, where the 4th sort is that of the RV and it is connected to the other sorts by the rv map rv : K → RV ∪ {∞}, the valuation map vrv : RV → Γ, and the inclusion map ι : k → RV ∪ {∞}. And finally, we have Proposition 2.3.23 ([11, Proposition 3.1.4]). Let K be a valued field. Then Γ and k are interpretable in RV . CHAPTER 2. PRELIMINARIES 19

2.3.2 Valued Difference Fields Definition 2.3.24. A valued difference field is a valued field K together with a map σ : K → K, which is a field automorphism that, in addition, satisfies σ(O) = O. It is easy to see that the automorphism of the big field induces automorphisms of the value group, the residue field and the RV. On the value group, σ : K → K induces an automorphismσ ˜ :Γ → Γ as follows: For γ ∈ Γ, choose a ∈ K such that v(a) = γ. This is possible to do because v is assumed to be surjective. Then defineσ ˜(γ) := v(σ(a)). Since σ(O) = O implies σ(O×) = O×, it follows thatσ ˜(0) = 0. Moreover,σ ˜(γ1 + γ2) =σ ˜(v(a1) + v(a2)) =σ ˜(v(a1 · a2)) = v(σ(a1 · a2)) = −1 v(σ(a1) · σ(a2)) = v(σ(a1)) + v(σ(a2)) = γ1 + γ2. We can do a similar thing with σ to getσ ˜−1. Thus,σ ˜ is an automorphism of abelian groups. Since σ(O) = O,σ ˜ also preserves order, and hence is an automorphism of ordered abelian groups. Thus the value group has the structure of an ordered difference abelian group. Similarly, σ : K → K induces an automorphismσ ¯ : k → k as follows: For x ∈ k, choose y ∈ O such that π(y) = x. Then defineσ ¯(x) := π(σ(y)). Thus the residue field has the structure of a difference field. σ(O) = O implies σ(m) = m: v(x) > 0 ⇐⇒ v(x−1) < 0 ⇐⇒ v(σ(x−1)) < 0 ⇐⇒ −v(σ(x)) < 0 ⇐⇒ v(σ(x)) > 0. Thus, the automorphism σ : K → K also induces an automorphism σrv : RV → RV as follows: For r ∈ RV , choose a ∈ K such that rv(a) = r. Then define σrv(r) = rv(σ(a)). Thus RV also has a difference structure on it. Definition 2.3.25 (Languages). The 4-sorted language of the valued difference fields is the language L4vdf := hK, Γ, k, RV ; v, π, vrv, ι, rvi, where we consider K as a structure in the language LR,σ := {+, −, ·, 0, 1, σ}, k as a structure in the language LR,σ¯ := {+, −, ·, 0, 1, σ¯},Γ as a structure in the language LOG,σ˜ := {+, −, <, 0, σ˜}, and RV as a structure in the language −1 Lrv,σrv := {·, , ⊕, 1, σrv}. These sorts are connected by the valuation maps v : K → Γ∪{∞} and vrv : RV → Γ, the residue map π : O → k, the rv map rv : K → RV ∪ {∞} and the inclusion map ι : k → RV ∪ {∞}. The corresponding language L3vdf := hK, Γ, k; v, πi is called the 3-sorted language of valued difference fields. It follows that Proposition 2.3.23 is still valid in the valued difference field context. Proposition 2.3.26. Let K be a valued difference field. Then Γ and k are interpretable in RV . Proof. By Proposition 2.3.23, it is enough to show that the difference structures can be properly interpreted. The difference operator on Γ is interpreted in terms of the difference operator on RV asσ ˜(v(x)) = v(σrv(x)). And since the nonzero elementsx ¯ of the residue field are in bijection with x ∈ RV such that v(x) = 0,σ ¯(¯x) is interpreted in the obvious way as σrv(x). To understand the theory of such a structure, we need to specify how the valuation v interacts with the automorphism σ. Two such cases have been worked out before. CHAPTER 2. PRELIMINARIES 20

Case I: Isometric Automorphism. Definition 2.3.27. σ is called isometric if v(σ(x)) = v(x) for all x ∈ K. That is, σ is trivial with respect to the valuation. As a result the automorphismσ ˜ on the value group is the identity map. Thus, the value group has the structure of an ordered abelian group, which has a model companion, namely, divisible ordered abelian group. Definition 2.3.28. K is called σ-henselian if, for any σ-polynomial P (x) ∈ Ohxi and a ∈ O with v(P (a)) > 0 = v(P(i)(a)) for some i, there is b ∈ K such that v(a − b) ≥ v(P (a)) and P (b) = 0. Definition 2.3.29. K is said to have enough constants if, for all γ ∈ Γ, there exists a ∈ K such that v(a) = γ and σ(a) = a. In [22], Scanlon proves the following AKE-principle. Bélair,Macintyre and Scanlon reprove it in [5]. Theorem 2.3.30. Let K = (K, Γ, k; v, π) and K0 = (K0, Γ0, k0; v0, π0) be σ-henselian valued difference fields with isometric automorphism in equi-characteristic zero and having enough 0 0 constants. Then K ≡L3vdf K (as valued difference fields) if and only if k ≡LR,σ¯ k (as 0 difference fields) and Γ ≡LOG Γ (as ordered abelian groups). Moreover, under a stronger assumption that the residue field be closed under taking roots, the theory of K eliminates quantifiers relative to the theories of the value group and the residue field.

Case II: Contractive Automorphism. Definition 2.3.31. σ is called contractive if v(σ(x)) > nv(x) for all x ∈ K× with v(x) > 0 and for all n. Thus in this case, σ is highly non-trivial with respect to the valuation. In particular, it increases the valuation infinitely. The induced automorphismσ ˜ on the value group Γ has the property thatσ ˜(γ) > nγ for all n and all 0 < γ ∈ Γ. This gives Γ the structure of an ordered module over the ring Z[˜σ]. Moreover, it helps to order the ring Z[˜σ] as follows: Pn i for any L = i=0 aiσ˜ ∈ Z[˜σ] with an 6= 0, we have L > 0 ⇐⇒ an > 0. It follows that L(γ) > 0 ⇐⇒ L > 0, for all 0 < γ ∈ Γ. Thus, Γ becomes an ordered module over an ordered ring, and this theory has a model companion [25]. Let K be a valued difference field with a contractive automorphism. Let P (x) be a σ-polynomial over K of order ≤ n, and a ∈ K. Let I, J, L ∈ Nn+1. Definition 2.3.32. We say (P, a) is in σ-hensel configuration if P is not a constant, and there is 0 ≤ i ≤ n and γ = γ(P, a) ∈ Γ such that

i j (i) v(P (a)) = v(P(i)(a)) +σ ˜ (γ) ≤ v(P(j)(a)) +σ ˜ (γ) whenever 0 ≤ j ≤ n, J L (ii) v(P(J)(a)) + σ˜ (γ) < v(P(L)(a)) + σ˜ (γ) whenever 0 6= J < L and P(J) 6= 0. CHAPTER 2. PRELIMINARIES 21

Deﬁnition 2.3.33. K is called σ-henselian if, for any (P, a) in σ-hensel conﬁguaration, there is b ∈ K such that v(b − a) = γ(P, a) and P (b) = 0.

In [1], Azgin proves the following.

Theorem 2.3.34. Let K = (K, Γ, k; v, π) and K0 = (K0, Γ0, k0; v0, π0) be σ-henselian valued 0 difference fields with contractive automorphism in equi-characteristic zero. Then K ≡L3vdf K 0 0 (as valued difference fields) if and only if k ≡LR,σ¯ k (as difference fields) and Γ ≡LOG,σ˜ Γ (as ordered Z[˜σ]-modules). The theory of K does not eliminate quantifiers relative to the theories of the value group and the residue field in this language. However, it does so in an expanded language with a symbol for a cross-section (see Definition 4.9.1). 22

Chapter 3

Salvaging Kikyo-Shelah

Let T be a model complete theory in a first-order language L. We use σ to denote a new unary function symbol for an automorphism of a model of T . We define a new language Lσ = L ∪ {σ}, and a new theory Tσ = T ∪ {“σ is an L-automorphism”}. Now any model complete theory is ∀∃-axiomatizable [12, Theorem 7.3.3]. Moreover it is clear from Definition 2.1.5 that axioms mentioning “σ is an L-automorphism” are all universal sentences with only one ∀∃-condition, namely the existence of an inverse. Thus, Tσ becomes an ∀∃-theory in the language Lσ. If Tσ has a model companion, it is denoted by TA. As noted in Theorem 2.1.30, if a first-order theory T is ∀∃-axiomatizable, then the model companion of T is the theory of the existentially closed models of T . Unfortunately, as mentioned in the introduction, if T is a theory with the Strict Order Property, TA does not exist. However, even though this negative result is unfortunate, not everything is lost. All this result says is that one cannot hope to have a nice theory with a generic automorphism. But with some restriction on the automorphism, things can be nice again. This is the content of this chapter. We are going to show that with certain restrictions on the automorphism, one can have the existence of a model companion even for a theory T with the strict order property. Before proceeding to the main results, let us recall a few definitions from [24].

Deﬁnition 3.0.35. In all these deﬁnitions, we takex ¯ = (x1, . . . , xm),y ¯ = (y1, . . . , yn) and φ(¯x;y ¯) a formula in m + n-free variables.

• A theory T is said to have the Order Property if there exists a formula φ(¯x;y ¯), a model m ¯ n M |= T , and sequences ha¯i : i ∈ ωi ⊂ M , hbj : j ∈ ωi ⊂ M such that ¯ M |= φ(a ¯i; bj) ⇐⇒ i < j.

• A theory T is said to have the Strict Order Property if there exists a formula φ(¯x;y ¯), n a model M |= T , and a sequence ha¯i : i ∈ ωi ⊂ M such that

M |= ∃x¬φ(x, a¯i) ∧ φ(x, a¯j) ⇐⇒ i < j. CHAPTER 3. SALVAGING KIKYO-SHELAH 23

• A theory T is said to have the Independence Property if there exists a formula φ(¯x;y ¯), m a model M |= T , and a sequence ha¯i : i ∈ ωi ⊂ M such that for each I ⊆ ω, there is ¯b ∈ Mn for which ¯ I = {i < ω | M |= φ(a ¯i; b)}. • A theory T is said to be unstable if it has the order property; otherwise it is said to be stable. Theorem 3.0.36 (Shelah, [24]). A theory is unstable if and only if it has the Strict Order Property or the Independence Property. With these basic deﬁnitions in mind, we now move on to the main results we are interested in about the non-existence of TA for certain theories T . Kikyo showed that Theorem 3.0.37 (Kikyo, [15]). Let T be a model complete theory in a language L and σ a new unary function symbol. If T is unstable without the independence property, then Tσ has no model companion in Lσ. Kikyo and Shelah improved this result to the following: Theorem 3.0.38 (Kikyo-Shelah, [17]). Let T be a model complete theory in a language L and σ a new unary function symbol. If T has the strict order property, then Tσ has no model companion in Lσ.

3.1 Linear Order with Increasing Automorphism

Let LO be the theory of linear orders in the language L = {<}, where < is a binary relation symbol. Recall that this theory is axiomatized by the following L-sentences: ∀x¬(x < x), ∀x∀y∀z((x < y ∧ y < z) → x < z), ∀x∀y(x < y ∨ x = y ∨ y < x). As is well-known, the theory of linear orders has a model companion, namely the theory of non-trivial dense linear orders without endpoints (DLO), which is axiomatized by the above axioms together with ∀x∀y(x < y → ∃z((x < z) ∧ (z < y))) ∀x∃y∃z((y < x) ∧ (x < z)) ∃x∃y(x 6= y) Now we want to consider this structure with an automorphism σ and see if we can get a model companion. As noted earlier, σ cannot be a generic automorphism if we want model companion to exist. So we must put some restriction on σ. And the restriction we impose is the following. CHAPTER 3. SALVAGING KIKYO-SHELAH 24

Deﬁnition 3.1.1. An automorphism σ on LO is said to be increasing if

∀x(x < σ(x)).

+ We denote by LOσ the theory of linear orders together with the axioms denoting “σ is an increasing L-automorphism” in the language Lσ = {<, σ}. And we claim that this theory has a model companion in Lσ, namely, the theory of non-trivial dense linear orders with an + increasing automorphism, which we denote by DLOσ . It is worthwhile to note here that − there is an analogous theory LOσ with decreasing automorphisms too. + + + + Lemma 3.1.2. Every model of LOσ embeds in a model of DLOσ . Thus LOσ and DLOσ are co-theories.

+ Proof. Let M be a model of LOσ . We will build a chain of models M = M0 ⊂ M1 ⊂ M2 ⊂ Sω + ··· such that Mi is embedded in Mi+1 for all i, and i=0 Mi is a model of DLOσ . Suppose we have constructed Mi and now we will show how to construct Mi+1 in the induction step. Note that since σ is increasing, Mi does not have a largest element. Also, σ is increasing implies σ−1 is decreasing:

∀x(x < σ(x)) ⇐⇒ ∀x(σ−1(x) < σ−1(σ(x))) ⇐⇒ ∀x(σ−1(x) < x).

Hence, Mi does not have a smallest element either. We call a pair (a, b) of elements in Mi a bad pair if a < b and there is no x ∈ Mi such that a < x < b. For each such bad pair (a, b), we introduce a new element c(a,b) ∈ Mi+1 and extend the order by deﬁning a < c(a,b) < b. Thus, Mi+1 = Mi ∪ {c(a,b) :(a, b) is a bad pair}. −1 −1 We extend σ and σ on Mi+1 as follows: since σ and σ are L-automorphisms, ^ ¬∃x(a < x < b) =⇒ ¬∃x(σ(a) < x < σ(b)) ¬∃x(σ−1(a) < x < σ−1(b)).

Therefore, for any bad pair (a, b), (σ(a), σ(b)) and (σ−1(a), σ−1(b)) are also bad pairs, and hence we have introduced new elements c(σ(a),σ(b)) and c(σ−1(a),σ−1(b)) in Mi+1 such that σ(a) < −1 −1 c(σ(a),σ(b)) < σ(b) and σ (a) < c(σ−1(a),σ−1(b)) < σ (b). Deﬁne

−1 σ(c(a,b)) = c(σ(a),σ(b)) and σ (c(a,b)) = c(σ−1(a),σ−1(b)).

Now σ has a well-deﬁned inverse. So it is a bijective map on Mi+1. Moreover the way we deﬁned σ, it clearly preserves order. So σ is an L-automorphism of Mi+1. Finally, note that σ is increasing: Let (a, b) be a bad pair. Since σ is increasing, there are two choices

a < σ(a) < b < σ(b) or a < b < σ(a) < σ(b).

The ﬁrst case is ruled out as (a, b) is assumed to be a bad pair. Hence, we have

a < c(a,b) < b < σ(a) < c(σ(a),σ(b)) < σ(b). CHAPTER 3. SALVAGING KIKYO-SHELAH 25

−1 In particular c(a,b) < c(σ(a),σ(b)) = σ(c(a,b)). A similar argument shows that σ (c(a,b)) = + c(σ−1(a),σ−1(b)) < c(a,b). Thus, σ is increasing and Mi+1 is a model of LOσ . Also clearly Mi embeds in Mi+1. Sω + Finally let N = i=0 Mi. Clearly M embeds in N and moreover, N is a model of DLOσ : Let a, b ∈ N . Then there exists i such that a, b ∈ Mi. And so, there is c ∈ Mi+1 ⊂ N such that a < c < b. Also a < σ(a) in Mi, and hence a < σ(a) in N . Remark 3.1.3. It is also easy to see that, in the above proof, N is the smallest dense linear + order containing M, i.e., N embeds in any other model G of DLOσ containing M. This is because, by our construction, each Mi can be embedded in G and this embedding can be extended inductively to an embedding of N .

+ Now we show that DLOσ eliminates quantiﬁers. We use Test 2.1.24 (and Deﬁnition 2.1.23) for this purpose. To that end, we prove the following.

+ Lemma 3.1.4. DLOσ has algebraically prime models.

+ Proof. It is easy to see that (LOσ )∀ is the theory of linear orders with an increasing injective endomorphism. So we need to show that given a linear order with an increasing injective endomorphism (M, τ), we can extend it to a smallest dense linear order with an increasing automorphism. By Lemma 3.1.2 and Remark 3.1.3, it is enough to show that we can extend (M, τ) to a smallest linear order with an increasing automorphism (N , σ). Consider X = {(g, n): g ∈ M, n ∈ N, n ≥ 0}. We think of (g, n) as τ −n(g). We define an equivalence relation ∼ on X by (g, n) ∼ (h, m) if and only if τ m(g) = τ n(h). Let N = X/ ∼. For (g, n) ∈ X, let [(g, n)] denote the ∼-class of (g, n). We can order N by [(g, n)] < [(h, m)] if and only if τ m(g) < τ n(h). We define the map σ : N → N as follows: σ([(g, m)]) = [(τ(g), m)]. Clearly σ is an injective endomorphism of N . Moreover since τ is increasing, σ is increasing too. And finally, σ is an automorphism: for any [(x, m)] ∈ N , σ−1([(x, m)]) = [(x, m + 1)]. + Hence, (N , σ) is a model of LOσ . We can embed M into N by the map i(g) = [(g, 0)]. Clearly, i is injective and preserves order. Suppose that (N 0, σ˜) is a linear order with an increasing automorphism and j : M → N 0 is an embedding. Let h : N → N 0 be defined by h([(g, n)]) =σ ˜−n(j(g)). It is easy to verify that h is a well-defined embedding and j = h ◦ i.

+ Theorem 3.1.5. DLOσ eliminates quantiﬁers in the language Lσ.

+ Proof. Because of Lemma 3.1.4, all we are left to show is that whenever M, N |= DLOσ with M ⊆ N , then M is simply closed in N , i.e., for any quantifier-free L-formula φ(x;w ¯) and a¯ ∈ Mn, if N |= ∃xφ(x;a ¯), then M |= ∃xφ(x;a ¯). Recall that every quantifier-free formula is a finite boolean combination of atomic formulas, and can be re-written in a standard Disjunctive Normal Form (DNF). Since a formula in DNF is true if and only if one of the disjuncts is true, it suffices to consider the case where φ is a conjunction of atomic and CHAPTER 3. SALVAGING KIKYO-SHELAH 26

negated atomic formulas. If θ(x;w ¯) is atomic, then θ is equivalent to either σi(x) = g or g < σi(x) or σi(x) < g or x < σi(x) or x = σi(x) or σi(x) < x for some g ∈ M and i ∈ N. Since σ is increasing, the last three cases are ruled out because x < σi(x) and x = x are always true, and x < x and σi(x) ≤ x are never true, for i > 0. Thus we may assume that

^ i ^ i ^ i φ(x;w ¯) ↔ σ (x) = gi ∧ hij < σ (x) ∧ σ (x) < lij, i i,j i,j

i −i where gi, hij, lij ∈ M. If there is actually a conjunct σ (x) = gi, then x = σ (gi) ∈ M and we −i −i already have a solution in M. Otherwise, let h0 = maxi,j{σ (hij)} and l0 = mini,j{σ (lij)}. Then, c ∈ N satisﬁes φ(x;a ¯) if and only if h0 < c < l0. Since N |= ∃xφ(x;a ¯), we have in + particular h0 < l0. But h0, l0 ∈ M and M is a model of DLOσ . Hence there is b ∈ M such that h0 < b < l0. But then b satisﬁes φ(x;a ¯). So M |= ∃xφ(x;a ¯). As an immediate corollary, we get that

+ + Corollary 3.1.6. DLOσ is model complete and is the model completion of LOσ in Lσ.

3.2 Ordered Abelian Group with Automorphism

Now we study the theory of ordered abelian groups with a restricted automorphism and see if the new theory has a model companion. We work in the language of ordered groups with a symbol for the automorphism Lσ = {+, −, 0, <, σ}.

The Lσ-theory Tσ of ordered diﬀerence abelian groups is axiomatized by the following axioms: 1. Axioms of Abelian Groups in the language {+, -, 0}

• ∀x(x + 0 = 0 + x = x) • ∀x∀y∀z(x + (y + z) = (x + y) + z) • ∀x(x + (−x) = (−x) + x = 0) • ∀x∀y(x + y = y + x)

2. Axioms of Linear Order in the language {<} (see Section 3.1)

3. Axiom about interaction

• ∀x∀y∀z(x < y → x + z < y + z)

4. Axioms asserting σ is an L-automorphism, where L = {+, −, 0, <}.

• σ(0) = 0 CHAPTER 3. SALVAGING KIKYO-SHELAH 27

• ∀x∀y(σ(x + y) = σ(x) + σ(y)) • ∀x(σ(−x) = −σ(x)) • ∀x∀y(x < y → σ(x) < σ(y)) • ∀x∃y(σ(y) = x)

Remark 3.2.1. Note that Tσ is an ∀∃-theory in the language Lσ: the existence of inverse (∀x∃y(σ(y) = x)) is the only ∀∃-axiom. The universal theory of Tσ,(Tσ)∀, is the theory of ordered abelian groups with an injective endomorphism.

Unfortunately the theory of ordered abelian groups has the strict order property. As before, by Kikyo and Shelah’s theorem [17], we cannot hope to have a model companion of the theory of ordered difference abelian groups. However, if we restrict ourselves to a very specific kind of automorphisms, we do actually get a model companion. Each of the intended automorphisms√ is multiplication by an element of a real-closed field. For example, σ(x) = 2x, or σ(x) = 2x, or σ(x) = δx, where δ could be an infinite or infinitesimal element. The problem is that in general abelian groups such multiplication does not make sense. But since integers embed in any torsion-free abelian group, in particular any ordered abelian group, by imitating what we do for real numbers, we can make sense of such multiplication. m times z }| { For an abelian group G, multiplication by m ∈ N makes sense: mg := g + ··· + g . Taking additive inverses, multiplication by integers also makes sense: (−m)g := −(mg). m mg If G is torsion-free divisible, then multiplication by rational numbers makes sense: g = n n is defined to be the unique y ∈ G such that ny = mg.

Motivation. We carry this idea forward and deﬁne cuts in rational numbers to make sense of multiplication by irrationals. Let ρ be an element of a real closed ﬁeld K. Then, for any 0 < g ∈ G, we would like ρ · g to be an element of G such that, for all r ∈ Q,

rg ≶ ρ · g ⇐⇒ r ≶ ρ. Since we are typically interested in preserving the order on G, we also require that ρ > 0, because then g1 ≶ g2 ⇐⇒ ρ · g1 ≶ ρ · g2. Without loss of generality, we also require that ρ ≥ 1; otherwise, we can work with ρ−1 instead. Since ρ is an element of the real closed ﬁeld K, we can deﬁne the cut of ρ in the rationals by

cutQ(ρ) = {a ∈ K : for each q ∈ Q, q ≶ a ⇐⇒ q ≶ ρ}.

Clearly for all a ∈ cutQ(ρ), ρ · g and a · g are order-indistinguishable with respect to the rationals. This is a little bit of a problem because we would typically like to be able to CHAPTER 3. SALVAGING KIKYO-SHELAH 28

distinguish between b·g and (b+)·g, where b is an algebraic number, and is an infinitesimal. Pn i This is because if b is algebraic over Z, then b is a root of a polynomial L(x) = i=0 aix , with ai ∈ Z for all i = 0, . . . , n. Then for any 0 6= g ∈ G, we have L(b) · g = 0, but L(b + ) · g 6= 0. However, for any a ∈ K and any polynomial L(x) over Z, we also have L(a) ∈ K. In particular, L(a) > 0 or L(a) = 0 or L(a) < 0. So either L(a) · g > 0 for all 0 < g ∈ G, or L(a) · g = 0 for all g > 0, or L(a) · g < 0 for all g > 0. This is the property we take from this particular setting and apply to the general setting to make the “multiplication” work and define what we call multiplicative ordered difference abelian group (MODAG).

Coming back to the general situation, we have an ordered abelian group G and an automorphism σ : G → G. For i ∈ N, we denote

i times z }| { σi(x) := σ(σ(... (σ(x)) ...)).

Deﬁnition 3.2.2. There is a natural map Φ : Z[σ] → End(G), which maps any L := k k−1 mkσ + mk−1σ + ... + m1σ + m0 (thought of as an element of Z[σ] with the mi’s coming from Z), to an endomorphism L(·): G → G. Such an L is called a linear diﬀerence operator.

Due to this action of Z[σ], G has the structure of a Z[σ]-module, with the understanding that σ has an inverse. To turn it into an ordered Z[σ]-module, we further impose the following condition on σ (motivated from our earlier example with the real closed ﬁelds): for each L ∈ Z[σ], _ _ ∀x > 0 (L(x) > 0) ∀x > 0 (L(x) = 0) ∀x > 0 (L(x) < 0) (Axiom OM)

(OM stands for Ordered Module). This axiom also makes sense for σ an injective endomorphism. Axiom OM is consistent with Axioms 1-4 because any ordered abelian group is a model of these axioms with σ(x) = 2x for all x, say. Also, with this axiom, Z[σ] becomes a quasi- ordered ring with the order deﬁned as follows: L1 = L2 ⇐⇒ ∀x > 0 (L1 − L2)(x) = 0 , and L1 > L2 ⇐⇒ ∀x > 0 (L1 − L2)(x) > 0 . It is easy to see that the relation

L1 ≈ L2 ⇐⇒ L1 = L2 and L2 = L1 ⇐⇒ ∀x > 0 ((L1 − L2)(x) = 0) is an equivalence relation. Thus taking a quotient makes sense, and we deﬁne

Definition 3.2.3. Z[ρ] := Z[σ]/ ≈, where ρ is the image of σ under this quotient map. We also define Q(ρ) to be the fraction field of Z[ρ]. CHAPTER 3. SALVAGING KIKYO-SHELAH 29

Remark 3.2.4. Clearly then Z[ρ] is an (totally) ordered ring and admits an embedding into a real closed field. So ρ can also be simultaneously thought of as an element of a real closed field. It is also easy to see that Z[ρ] = Z[σ]/Ker(Φ), where Φ is as defined in Definition 3.2.2. Note that the kernel of Φ need not be trivial. For example, if σ(x) = 2x for all x, then σ − 2 ∈ Ker(Φ). Moreover G is an ordered module over the ordered ring Z[ρ] with the understanding that ρ has an inverse. So we can denote the automorphism on G equivalently by ρ·, i.e. σ(x) = ρ · x. Axiom OM then is equivalent to: for each L ∈ Z[ρ], _ _ ∀x > 0 (L · x > 0) ∀x > 0 (L · x = 0) ∀x > 0 (L · x < 0) .

Definition 3.2.5. For any ordered difference abelian group G satisfying Axiom OM, we define the set of Z[σ]-positivities of G as

ptpZ[σ](G) := {L ∈ Z[σ]: ∀x ∈ G (x > 0 =⇒ L(x) > 0)}. 0 0 We say that two G and G have the same ρ if ptpZ[σ](G) = ptpZ[σ](G ). We also say G is a MODAG with a given ρ if G satisfies a given consistent set of Z[σ]-positivities. Definition 3.2.6. An ordered difference abelian group is called multiplicative if it satisfies Axiom OM. The theory of such structures (also called MODAG) is axiomatized by Axioms 1-4 and Axiom OM. Note that this theory is also an ∀∃-theory. We also denote by MODAGρ the theory of the class of all MODAGs with a same ρ. Definition 3.2.7. If there is a non-zero L ∈ Z[σ] such that ∀x > 0(L(x) = 0), we say that ρ satisfies L and that ρ is algebraic (over the integers); otherwise we say ρ is transcendental. If ρ is algebraic, there is a minimal (degree) polynomial that it satisfies. Definition 3.2.8. A MODAG G is called divisible (or linear difference closed) if for any 0 6≈ L ∈ Z[σ] and b ∈ G, the equation L(x) = b has a solution in G. Definition 3.2.9 (Language for MODAG). We study MODAG in the language of ordered abelian groups together with a symbol for the automorphism, LOG,σ := {+, −, 0, <, σ}.

Definition 3.2.10. Let MODDAG be the LOG,σ-theory of non-trivial multiplicative ordered divisible difference abelian groups. This theory is axiomatized by the above axioms along with ∃x(x 6= 0) and the following additional infinite list of axioms: for each L ∈ Z[σ], ∀x(L(x) = 0) ∨ ∀x∃y(L(y) = x) ,

i.e., all non-zero linear diﬀerence operators are surjective. Thus, MODDAG is an ∀∃-theory. Similarly as above, we denote by MODDAGρ the theory of the class of all MODDAGs with a same ρ. CHAPTER 3. SALVAGING KIKYO-SHELAH 30

We now show that MODDAG is the model companion of MODAG. By abuse of terminology, we refer to both the theory and any model of the theory as MODAG (respectively MODDAG).

Remark 3.2.11. It might already be clear from the definitions above that for a given ρ, MODDAGρ is basically the theory of non-trivial ordered vector spaces over the ordered field Q(ρ) and then quantifier elimination actually follows from well-known results [25]. However, here we are doing things a little differently. Instead of proving the result for a particular ρ, we are proving it uniformly across all ρ using Axiom OM.

Lemma 3.2.12. MODAG and MODDAG are co-theories.

Proof. We will prove something stronger: for a fixed ρ, MODAGρ and MODDAGρ are co-theories. Any model of MODDAGρ is trivially a model of MODAGρ. So all we need to show is that we can embed any model of MODAGρ into a model of MODDAGρ. We will actually show something even stronger. Let (G, σ) be an ordered abelian group with an injective endomorphism that is multiplicative with a given ρ. Then we can extend it to a “smallest” model (H, σ) of MODDAGρ, which we call the (multiplicative) divisible hull of G. If G is trivial, we can embed it into Q(ρ). So without loss of generality we may assume, G is non-trivial. Let Z[σ]+ := {L ∈ Z[σ]: L > 0}. Define an equivalence relation ∼ on G × Z[σ]+ as follows: (g, L) ∼ (g0,L0) ⇐⇒ L0(g) = L(g0). Reflexivity and symmetry are obvious. For transitivity, suppose

(g, L) ∼ (g0,L0) and (g0,L0) ∼ (g00,L00).

Then, L0(g) = L(g0) and L0(g00) = L00(g0). Applying L00 to the ﬁrst equation and L to the second equation, we get L00 ◦L0(g) = L00 ◦L(g0) and L ◦ L0(g00) = L ◦ L00(g0). Since these operators commute with each other, we can rewrite this as:

L0 ◦ L00(g) = L00 ◦ L0(g) = L00 ◦ L(g0) = L ◦ L00(g0) = L ◦ L0(g00) = L0 ◦ L(g00),

0 00 00 0 0 00 00 i.e., L (L (g) − L(g )) = 0. Since L ∈ Z[ρ]+, i.e., L > 0, we have L (g) = L(g ), i.e., (g, L) ∼ (g00,L00). Let [(g, L)] denote the equivalence class of (g, L) and let H = G × Z[ρ]+/ ∼. We deﬁne + on H by [(g, L)] + [(h, P )] = [P (g) + L(h),L ◦ P ]. CHAPTER 3. SALVAGING KIKYO-SHELAH 31

To show that this is well-defined, let (g, L) ∼ (g0,L0). Want to show that [(g, L)] + [(h, P )] = [(g0,L0)] + [(h, P )], i.e., [(P (g) + L(h),L ◦ P )] = [(P (g0) + L0(h),L0 ◦ P )]. In other words, we want to show that (P (g) + L(h),L ◦ P ) ∼ (P (g0) + L0(h),L0 ◦ P ). But, L ◦ P (P (g0) + L0(h)) = L ◦ P ◦ P (g0) + L ◦ P ◦ L0(h) = P ◦ P ◦ L(g0) + L0 ◦ P ◦ L(h) = P ◦ P ◦ L0(g) + L0 ◦ P ◦ L(h) = L0 ◦ P ◦ P (g) + L0 ◦ P ◦ L(h) = L0 ◦ P (P (g) + L(h)). Hence, + is well-defined. Similarly, we define − by −[(g, L)] = [(−g, L)]. This is also well-defined. It follows easily that (H, +, −) is an abelian group, where [(0, 1)] is the identity and [(−g, L)] is the inverse of [(g, L)].

We define an endomorphism (which we still call) σ of H as follows: let σ([(g, L)]) = [(σ(g),L)]. It is easy to check that this is well-defined and defines an injective endomorphism of H.

For any 0 6≈ L ∈ Z[σ] and any [(h, P )] ∈ H, we have L([(h, P ◦ L)]) = [(L(h),P ◦ L)] = [(h, P )]. Hence, H is linear diﬀerence closed or divisible. In particular, σ is an automorphism of H.

We extend the order as follows: [(g, L)] < [(g0,L0)] ⇐⇒ L0(g) < L(g0). If g, h ∈ G with g < h, then [(g, 1)] < [(h, 1)]; so this extends the ordering on G. Moreover, for [(a1,L1)] < [(a2,L2)] and [(b1,P1)] ≤ [(b2,P2)], we have L2(a1) < L1(a2) and P2(b1) ≤ P1(b2). Then,

P1 ◦ P2 ◦ L2(a1) + L1 ◦ L2 ◦ P2(b1) < P1 ◦ P2 ◦ L1(a2) + L1 ◦ L2 ◦ P1(b2)

i.e., L2 ◦ P2(P1(a1) + L1(b1)) < L1 ◦ P1(P2(a2) + L2(b2))

i.e., [(P1(a1) + L1(b1),L1 ◦ P1)] < [(P2(a2) + L2(b2),L2 ◦ P2)]

i.e., [(a1,L1)] + [(b1,P1)] < [(a2,L2)] + [(b2,P2)] Also, [(a, L)] < [(b, P )] ⇐⇒ P (a) < L(b) ⇐⇒ σ(P (a)) < σ(L(b)) ⇐⇒ P (σ(a)) < L(σ(b)) ⇐⇒ [(σ(a),L)] < [(σ(b),P )]. CHAPTER 3. SALVAGING KIKYO-SHELAH 32

Finally, for any L ∈ Z[σ], and x > 0 and P ∈ Z[σ]+, we have L([(x, P )]) > [(0, 1)] ⇐⇒ [(L(x),P )] > [(0, 1)] ⇐⇒ L(x) > 0. Hence, H is a multiplicative ordered divisible diﬀerence abelian group.

Claim. G embeds into H. Proof. Deﬁne ι : G → H by ι(g) = [(g, 1)]. Then ι(0) = [(0, 1)]; ι(g + h) = [(g + h, 1)] = [(g, 1)] + [(h, 1)]; ι(−g) = [(−g, 1)] = −[(g, 1)]. Also, ι(σ(g)) = [(σ(g), 1)] = σ([(g, 1)]). And, g < h =⇒ ι(g) = [(g, 1)] < [(h, 1)] = ι(h).

0 0 0 Moreover, if H |= MODDAGρ and j : G → H is an embedding, then let h : H → H be given by h([(g, L)]) = [(j(g),L)]. It is routine to check that h is a well-deﬁned embedding, preserves order and j = h ◦ ι. We, thus, call H as the (multiplicative) divisible hull of G.

Remark 3.2.13. We have thus shown that for a ﬁxed ρ, MODAGρ and MODDAGρ are co- theories. In fact, since MODDAG∀ is the theory of ordered abelian group with multiplicative injective endomorphism, what we have actually shown is that MODDAGρ has algebraically prime models, namely the (multiplicative) dvisible hull.

Lemma 3.2.14. MODDAGρ has quantiﬁer elimination.

Proof. We have already shown that MODDAGρ has algebraically prime models. All we need to show now is that MODDAGρ is simply closed. So suppose G ⊆ H are two models of MODDAGρ. We want to show G ≺s H. Suppose ϕ(v, w¯) is a quantiﬁer-free formula,g ¯ ∈ G and for some h ∈ H, H |= ϕ(h, g¯). It suﬃces to consider the case where ϕ is a conjunction of atomic and negated atomic formulas. Pn Pn0 0 0 If θ(v, w¯) is atomic, then θ is equivalent to i=1 Li(wi)+L(v) = 0 or i=1 Li(wi)+L (v) > 0 0 0 for some L, L ,Li,Li ∈ Z[σ]. In particular, there is an element a ∈ G such that θ(v, g¯) is of the form L(v) = a or L(v) > a. Also note that L(v) 6= a is equivalent to L(v) > a or −L(v) > a. So we may assume that

^ ^ 0 ϕ(v, g¯) ↔ Li(v) = ai ∧ Li(v) > bi,

0 where ai, bi ∈ G and Li,Li ∈ Z[σ]. We may also assume that Li 6≈ 0 because either the corresponding ai is zero, in which case the equation is trivially true for all v, or the corresponding ai is non-zero, in which case the equation is inconsistent. If there is actually a conjunct Li(v) = ai with 0 6≈ Li, then we must have h ∈ G because G is divisible and a non-zero linear equation has a unique solution. So suppose ϕ(v, g¯) = V 0 0 0 0 0 Li(v) > bi. Let h1 = min{[(bi,Li)] : Li < 0} and h2 = max{[(bi,Li)] : Li > 0}. Since H |= ϕ(h, g¯), we have h2 < h < h1. In particular, h2 < h1. Now since G |= MODDAGρ, g + h G is densely ordered because if g < h, then g < < h. So there is d ∈ G such that 2 h2 < d < h1, and then G |= ϕ(d, g¯). Thus, G ≺s H. CHAPTER 3. SALVAGING KIKYO-SHELAH 33

Thus, MODDAGρ eliminates quatifiers. In particular, MODDAGρ is model complete. Moreover, for a fixed ρ, Q(ρ) with the induced ordering is a prime model of MODDAGρ. In particular, MODDAGρ is complete. Note that MODDAG is not complete; its completions are given by MODDAGρ by fixing a (consistent) set of Z[σ]-positivities. Finally we have, Theorem 3.2.15. MODDAG is the model companion of MODAG.

Proof. By Lemma 3.2.12, MODAG and MODDAG are co-theories. All we need to show now is that MODDAG is model complete. So let G ⊆ H be two models of MODDAG. Want to show that G ≺ H. Since G ⊆ H and both are non-trivial, in particular they have the same set of Z[σ]- positivities. Thus, for some ﬁxed ρ, G, H |= MODDAGρ. But MODDAGρ is model complete. Hence, G ≺ H. 34

Chapter 4

Multiplicative Valued Diﬀerence Field

4.1 Introduction

In this chapter, we are interested in studying the theory of valued difference fields in a more general context. As already noted in Section 2.3.2, to understand the theory of such structures, we need to specify how the automorphism interacts with the valuation. We also saw in the same section that two specific cases, namely those of the isometric automorphisms and the contractive automorphisms, have already been dealt with. However, these two cases are two ends of a spectrum, and the goal of this chapter is to bridge that gap to some extent. To that end, we make the following definition.

Definition 4.1.1. Let K = (K, v, Γ, σ) be a valued difference field. σ : K → K is called a multiplicative automorphism if the induced structure on Γ is that of a MODAG with ρ ≥ 1. Then K is called a multiplicative valued difference field.

All we need in the above definition is ρ > 0. However, if ρ ≤ 1, we can switch to σ−1 and then we will have ρ ≥ 1. It is clear then that this is a generalization over the isometric and the contractive cases. The case ρ = 1 is precisely the isometric case; and the case “ρ = ∞”, i.e., when all 0 < γ ∈ Γ satisfy for all b ∈ Z+, ρ · γ > bγ, is the contractive case. We can have other finite and infinitesimal values for ρ as well. One more thing needs mention here about the characteristics of the relevant fields. Any automorphism of a field is trivial on the integers. Thus for any n ∈ Z, we have σ(n) = n. In particular, this means that for any prime p, if v(p) > 0, then v(p) = v(σ(p)) = ρ·v(p), which implies ρ = 1. Thus the mixed characteristic case does not arise for ρ > 1, and the mixed characteristic case for ρ = 1 has already been dealt with in [5]. The equi-characteristic p case even without the automorphism is too non-trivial and is not known yet. In this thesis, we thus restrict ourselves only to the equi-characteristic zero case. CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 35

4.2 Pseudoconvergence and Pseudocontinuity

We are interested in proving an AKE-principle for (and hence, finding the model companion of) the theory of multiplicative valued difference fields. For the moment we view a multiplicative valued difference field as a 3-sorted structure K = (K, Γ, k; v, π), where K is the field sort in the language LR,σ, Γ is the value group sort in the language LOG,ρ·, k is the residue field sort in the language LR,σ¯, v is the valuation function, and π is the residue map. We denote the automorphism on the valued field K by σ, and the induced automorphisms on the residue field k and the value group Γ byσ ¯ and ρ· respectively. Our main axiom is Axiom 1. Γ is a MODAG with ρ ≥ 1, and v(σ(x)) = ρ · v(x) ∀x ∈ K. From now on, we assume that all our valued difference fields and valued difference field extensions satisfy Axiom 1. Our first goal is to prove pseudo-continuity of difference polynomials. It follows from [13] that if {aη} is a pc-sequence from K, and aη a with a ∈ K, then for any ordinary non-constant polynomial P (x) ∈ K[x], we have P (aη) P (a). Unfortunately, this is not true in general for non-constant σ-polynomials over valued difference fields. As it turns out, this is true when ρ is transcendental over the integers (which includes the contractive case “ρ = ∞”), but not true if ρ is algebraic (which includes the isometric case ρ = 1). Fortunately, in the algebraic case, we can remedy the situation by resorting to equivalent pc-sequences. We will follow the treatment of [5], [2] with appropriate modifications. We will, however, need the following basic lemma. But before that we need a definition. n+1 Definition 4.2.1. Let Γ be a MODAG with a given ρ, and let I = (i0, . . . , in) ∈ Z . We Pn j define |I|ρ := j=0 ijρ . Clearly |I|ρ ∈ Z[ρ].

Lemma 4.2.2. Let {γη} be an increasing sequence of elements in a MODAG Γ. Let A = n+1 {|Ii|ρ : Ii ∈ Z , i = 1, . . . , l} be a ﬁnite set with |A| = m, and for i = 1, . . . , m, let ci + ni · x, ci ∈ Γ, ni ∈ A, be linear functions of x with distinct ni. Then there is a µ, and an enumeration i1, i2, . . . , im of {1, . . . , m} such that for η > µ, ci1 + ni1 · γη < ci2 + ni2 · γη <

··· < cim + nim · γη.

Proof. Since Γ is a MODAG, there is a linear order amongst the ni’s. Suppose ni 6= nj ∈ A.

WMA ni < nj. Then either cj + nj · γη < ci + ni · γη for all η, or for some ηij, ci + ni · γηij ≤ cj + nj · γηij . But in the later case, for all η > ηij, we have ci + ni · γη < cj + nj · γη, as ni < nj and {γη} is increasing. Since A is a ﬁnite set, the set of all such ηij’s is also ﬁnite, and hence taking µ to be the maximum of those ηij’s, we have our result.

4.2.1 Basic Calculation

Suppose K is a multiplicative valued diﬀerence ﬁeld. Let {aη} be a pc-sequence from K with a pseudo-limit a in some extension. Let P (x) be a non-constant σ-polynomial over K of order ≤ n. CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 36

Case I. ρ is transcendental.

Let γη = v(aη − a). Then for each η we have,

X L X P (aη) − P (a) = P(L)(a) · σ(aη − a) =: QL(aη) n+1 n+1 L∈N L∈N 1≤|L|≤deg(P ) 1≤|L|≤deg(P )

To calculate v(P (aη)−P (a)), we need to calculate the valuation of each summand QL(aη). We claim that there is a unique L for which the valuation of QL(aη) is the minimum eventually. Suppose not. Note that the valuation of QL(aη)

L v(QL(aη)) = v(P(L)(a) · σ(aη − a) ) = v(P(L)(a)) + |L|ρ · γη is a linear function in γη. Thus, by Lemma 4.2.2, the only way there isn’t a unique L with 0 0 the valuation of QL(aη) minimum eventually is if there are L 6= L with |L|ρ = |L |ρ. But then,

0 0 |L|ρ = |L |ρ =⇒ |L − L |ρ = 0 0 0 0 1 0 n =⇒ (l0 − l0)ρ + (l1 − l1)ρ + ··· + (ln − ln)ρ = 0

which implies that ρ is algebraic over Z, a contradiction. Hence, the claim holds. In particular, there is a unique L0 such that eventually (in η),

v(P (aη) − P (a)) = v(P(L0)(a)) + |L0|ρ · γη,

which is strictly increasing. Hence, P (aη) P (a). Note that if ρ = ∞, then ρ is transcendental over Z. Hence, the contractive case is included in Case I.

Case II. ρ is algebraic. Since ρ satisfies some algebraic equation over the integers, there can be accidental cancela- 0 tions and we might have v(QL(aη)) = v(QL0 (aη)) for infinitely many η and L 6= L , and the above proof fails. To remedy this, we construct an equivalent pc-sequence {bη} such that P (bη) P (a). Put γη := v(aη − a); then {γη} is eventually strictly increasing. Since v is surjective, choose θη ∈ K such that v(θη) = γη. Set bη := aη + µηθη, where we demand that µη ∈ K and v(µη) = 0. Define dη by aη − a = θηdη. So v(dη) = 0 and dη depends on the choice of θη. Since a need not be in K, dη will not normally be in K either. Then,

bη − a = bη − aη + aη − a

= θη(µη + dη). CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 37

We impose v(µη + dη) = 0. This ensures bη a, and that {aη} and {bη} have the same n+1 width; so they are equivalent. Let A := {|L|ρ : L ∈ N and 1 ≤ |L| ≤ deg(P )}. Now,

X L P (bη) − P (a) = P(L)(a) · σ(bη − a) n+1 L∈N 1≤|L|≤deg(P ) X X L = P(L)(a) · σ(bη − a)

m∈A |L|ρ=m X X L = P(L)(a) · σ(θη(µη + dη))

m∈A |L|ρ=m X X L L = P(L)(a) · σ(θη) · σ(µη + dη)

m∈A |L|ρ=m X = Pm,η(µη + dη) m∈A

where Pm,η is the σ-polynomial over Khai given by

X L L Pm,η(x) = P(L)(a) · σ(θη) · σ(x) .

|L|ρ=m

Since P 6∈ K, there is an m ∈ A such that Pm,η 6= 0. For such m, pick L = L(m) with L |L|ρ = m for which v(P(L)(a) · σ(θη) ) is minimal, so L Pm,η(x) = P(L)(a) · σ(θη) · pm,η(σ(x)),

where pm,η(x0, . . . , xn) has its coeﬃcients in the valuation ring of Khai, with one of its coeﬃcients equal to 1. Then

v(Pm,η(µη + dη)) = v(P(L)(a)) + m · γη + v(pm,η(σ(µη + dη))).

This calculation suggests a new constraint on {µη}, namely that for each m ∈ A with Pm,η 6= 0, v(pm,η(σ(µη + dη))) = 0 (eventually in η).

Assume this constraint is met. Then Lemma 4.2.2 yields a ﬁxed m0 ∈ A such that if m ∈ A and m 6= m0, then eventually in η,

v(Pm0,η(µη + dη)) < v(Pm,η(µη + dη))

For this m0 we have, eventually in η,

v(P (bη) − P (a)) = v(P(L)(a)) + m0 · γη, L = L(m0),

which is increasing. So P (bη) P (a), as desired. To have {µη} satisfy all constraints, we introduce an axiom (scheme) about K which involves only the residue ﬁeld k of K : CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 38

Axiom 2. For each integer d > 0 there is y ∈ k such thatσ ¯d(y) 6= y.

By [9], p. 201, this axiom implies that there are no residualσ ¯-identities at all, that is, for every non-zero f ∈ k[x0, . . . , xn], there is a y ∈ k with f(σ¯(y)) 6= 0 (and thus the set {y ∈ k : f(σ¯(y)) 6= 0} is inﬁnite). Now note that the pm’s are over Khai, and we need µ¯η ∈ k. The following lemma will take care of this.

0 Lemma 4.2.3 ([2, Lemma 3.3]). Let k ⊆ k be a ﬁeld extension, and p(x0, . . . , xn) a non- 0 zero polynomial over k . Then there is a non-zero polynomial f(x0, . . . , xn) over k such that whenever y0, . . . , yn ∈ k and f(y0, . . . , yn) 6= 0, then p(y0, . . . , yn) 6= 0.

Consider an m ∈ A with non-zero Pm,η, and deﬁne

n qm,η(x0, . . . , xn) := pm,η(x0 + dη, . . . , xn + σ (dη)).

Then the reduced polynomial

¯ n ¯ q¯m,η(x0, . . . , xn) :=p ¯m(x0 + dη, . . . , xn +σ ¯ (dη)) is also non-zero for each η. By Lemma 4.2.3, we can pick a non-zero polynomial fη(x0, . . . , xn) ∈ k[x0, . . . , xn] such that if y ∈ OK and fη(σ¯(¯y)) 6= 0, thenq ¯m,η(σ¯(¯y)) 6= 0 for each m ∈ A with Pm,η 6= 0.

¯ Conclusion: if for each η the element µη ∈ OK satisﬁesµ ¯η 6= 0, µ¯η + dη 6= 0, and fη(σ¯(¯µη)) 6= 0, then all constraints on {µη} are met. Axiom 2 allows us to meet these constraints, even if instead of a single P (x) of order ≤ n we have ﬁnitely many non-constant σ-polynomials Q(x) of order ≤ n and we have to meet simultaneously the constraints coming from each of those Q’s. This leads to:

Theorem 4.2.4. Suppose K satisﬁes Axiom 2. Suppose {aη} in K is a pc-sequence and aη a in an extension with γη := v(a − aη). Let Σ be a ﬁnite set of σ-polynomials P (x) over K.

• If ρ is transcendental, then P (aη) P (a), for all non-constant P ∈ K[x]; more speciﬁcally there is a unique L0 = L0(P ) such that for all I 6= L0, eventually

v(P (aη) − P (a)) = v(P(L0)(a)) + |L0|ρ · γη < v(P(I)(a)) + |I|ρ · γη.

• If ρ is algebraic, then there is a pc-sequence {bη} from K, equivalent to {aη}, such that P (bη) P (a) for each non-constant P ∈ Σ; more speciﬁcally there is a unique m0 = m0(P ) such that for all I with |I|ρ 6= m0, eventually

v(P (bη) − P (a)) = min v(P(L0)(a)) + |L0|ρ · γη < v(P(I)(a)) + |I|ρ · γη. |L0|ρ=m0 CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 39

Corollary 4.2.5. The same result, where a is removed and one only asks that {P (bη)} is a pc-sequence.

Proof. By an observation of Macintyre, any pc-sequence in any expansion of valued fields (for example, a valued difference field) has a pseudo-limit in an elementary extension of that expansion. In particular, {aη} has a pseudo-limit a in an elementary extension of K. Use this a and Theorem 4.2.4.

4.2.2 Reﬁnement of the Basic Calculation. The following improvement of the basic calculation will be needed later on.

Theorem 4.2.6. Suppose K satisﬁes Axiom 2 and ρ is algebraic. Let {aη} be a pc-sequence from K and let aη a in some extension. Let P (x) be a σ-polynomial over K such that

(i) P (aη) 0,

(ii) P(L)(bη) 6 0, whenever |L| ≥ 1 and {bη} is a pc-sequence in K equivalent to {aη}.

Let Σ be a ﬁnite set of σ-polynomials Q(x) over K. Then there is a pc-sequence {bη} in K, equivalent to {aη}, such that P (bη) 0, and Q(bη) Q(a) for all non-constant Q in Σ.

Proof. By augmenting Σ, we can assume P(L) ∈ Σ for all L. Let n be such that all Q ∈ Σ have order ≤ n. Let {θη} and {dη} be as before. By following the proof in the basic calculation and using Axiom 2, we get non-zero polynomials fη ∈ k[x0, . . . , xn] and a sequence {µη} satisfying the constraints ¯ µη ∈ O, µ¯η 6= 0, µ¯η + dη 6= 0, fη(σ¯(¯µη)) 6= 0, such that, by setting bη := aη + θηµη, we have

Q(bη) Q(a) for each non-constant Q ∈ Σ.

We would like to constrain {µη} further so that we also have P (bη) 0. Letting A := {|L|ρ : L ∈ Nn+1 and 1 ≤ |L| ≤ deg(P )}, we have

P (bη) = P (aη + θηµη) X X L = P (aη) + P(L)(aη) · σ(θηµη)

m∈A |L|ρ=m X X L L = P (aη) + P(L)(aη) · σ(θη) · σ(µη)

m∈A |L|ρ=m X = P (aη) + Qm,η(µη) m∈A CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 40

where Qm,η is the σ-polynomial over K given by

X L L Qm,η(x) = P(L)(aη) · σ(θη) · σ(x) .

|L|ρ=m

Since P(L)(aη) P(L)(a) and P(L)(aη) 6 0, v(P(L)(aη)) settles down eventually. Let γL be this eventual value. For each m ∈ A such that Qm,η 6= 0, let L = L(m) be such that L P(L)(aη) · σ(θη) has minimal valuation. Then, for such Qm,η, we can write (eventually in η), Qm,η(x) = cm,η · qm,η(σ(x)),

where v(cm,η) = γL + |L|ρ · γη and qm,η is a polynomial over O with at least one coeﬃcient 1. This suggests another constraint on {µη}, namely, for each m ∈ A such that Qm,η 6= 0, v(qm,η(σ(µη))) = 0 (eventually in η); equivalently,q ¯m,η(σ¯(¯µη)) 6= 0. As usual, this constraint can be met by Axiom 2. And then, by Lemma 4.2.2, we have a unique m0 such that eventually in η, X v Qm,η(µη) = v(Qm0,η(µη)) = γL + m0 · γη, L = L(m0), m∈A

which is increasing. Now, if v(P (aη)) 6= v(Qm0,η(µη)), we do nothing. However, if v(P (aη)) = γL + m0 · γη, then replacing µη by a variable x, consider P (a ) + P P P (a ) · σ(θ )L · σ(x)L η m∈A |L|ρ=m (L) η η = P (a ) 1 + P P P (a )−1P (a ) · σ(θ )L · σ(x)L η m∈A |L|ρ=m η (L) η η

= P (aη)Hη(σ(x))

where Hη(y0, . . . , yn) is a polynomial over O with at least one coeﬃcient 1. So if we add the ¯ extra requirement that Hη(σ¯(¯µη)) 6= 0, easily fulﬁlled as before, we get that eventually

v(P (bη)) = min{v(P (aη)), v(Qm0,η(µη))},

and since both of these are increasing, we have P (bη) 0.

4.3 Around Newton-Hensel Lemma

For the moment we consider the basic problem of how to start with a ∈ K and P (a) 6= 0, and ﬁnd b ∈ K with v(P (b)) > v(P (a)). n+1 For any multi-index I = (i0, i1, . . . , in) ∈ N , we have (see Deﬁnitions 2.2.5, 2.2.6 and 4.2.1)

n n I i0 i1 n in X j X j v(σ(a) ) = v(a (σ(a)) ··· (σ (a)) ) = ijv(σ (a)) = ijρ · v(a) = |I|ρ · v(a). j=0 j=0 CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 41

Recall that, if I = ei := (0,..., 0, 1, 0,... 0) with 1 at the i-th place, we denote P(ei) by P(i). i And then, for γ ∈ Γ, we have |ei|ρ · γ = ρ · γ. By abuse of notation, we will often identify ei with i. For example, we will write J 6= i (for some multi-index J) to actually mean J 6= ei. Hopefully, this should be clear from the context. Let K be a multiplicative valued difference field. Let P (x) be a σ-polynomial over K of order ≤ n, and a ∈ K. Let I, J, L ∈ Nn+1. Definition 4.3.1. We say (P, a) is in σ-hensel configuration if P is not constant and there is 0 ≤ i ≤ n and γ ∈ Γ such that

i j (i) v(P (a)) = v(P(i)(a)) + ρ · γ ≤ v(P(j)(a)) + ρ · γ whenever 0 ≤ j ≤ n,

(ii) v(P(J)(a)) + |J|ρ · γ < v(P(L)(a)) + |L|ρ · γ whenever 0 6= J < L and P(J) 6= 0. We say (P, a) is in strict σ-hensel conﬁguration if the inequality in (i) is strict for j 6= i.

Remark 4.3.2. Note that if (P, a) is in (strict) σ-hensel conﬁguration, then P(J)(a) 6= 0 whenever J 6= 0 and P(J) 6= 0, so P (a) 6= 0, and therefore γ as above satisﬁes

j v(P (a)) = min v(P(j)(a)) + ρ · γ, 0≤j≤n so is unique, and we set γ(P, a) := γ. If (P, a) is not in σ-hensel conﬁguration, then we set γ(P, a) := ∞. If (P, a) is in strict σ-hensel conﬁguration, then i is unique and we set i(P, a) := i.

Remark 4.3.3. Suppose P is non-constant, P (a) 6= 0, v(P (a)) > 0 and v(P(J)(a)) = 0 for all J 6= 0 with P(J) 6= 0. Then (P, a) is in σ-hensel configuration with γ(P, a) = v(P (a)) > 0 and any i with 0 ≤ i ≤ n for ρ = 1; and for ρ > 1, (P, a) is in strict σ-hensel configuration with γ(P, a) = v(P (a)) > 0 and i(P, a) = 0. Now given (P, a) in (strict) σ-hensel configuration, we aim to find b ∈ K such that v(P (b)) > v(P (a)) and (P, b) is in (strict) σ-hensel configuration. This, however, requires an additional assumption on the residue field k, namely that k should be linear difference- closed. We will justify later on why this assumption is necessary.

Axiom 3n. If α0, . . . , αn ∈ k are not all 0, then the equation

n 1 + α0x + α1σ¯(x) + ··· + αnσ¯ (x) = 0 has a solution in k.

Lemma 4.3.4. Suppose K satisﬁes Axiom 3n, and (P, a) is in σ-hensel conﬁguration. Then there is b ∈ K such that 1. v(b − a) ≥ γ(P, a), v(P (b)) > v(P (a)), CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 42

2. either P (b) = 0, or (P, b) is in σ-hensel conﬁguration. For any such b, we have v(b − a) = γ(P, a) and γ(P, b) > γ(P, a). Proof. This is the same proof as [1, Lemma 4.4]. But we include it here for the sake of completeness, and also to set the ground for the next lemma.

Step 1. Let γ = γ(P, a). Pick ∈ K with v() = γ. Let b = a + u, where u ∈ K is to be determined later; we only impose v(u) ≥ 0 for now. Consider

X J P (b) = P (a) + P(J)(a) · σ(b − a) . |J|≥1

P J Therefore, P (b) = P (a) · (1 + |J|≥1 cJ · σ(u) ), where P (a) · σ()J c = (J) . J P (a)

From v() = γ and the fact that (P, a) is in σ-hensel conﬁguration, we obtain min0≤j≤n v(cj) = 0 and v(cL) > 0 for |L| > 1. Then imposing v(P (b)) > v(P (a)) forcesu ¯ to be a solution of the equation X j 1 + c¯j · σ¯ (x) = 0. 0≤j≤n

By Axiom 3n, we can take u with this property, and then v(u) = 0, so v(b − a) = γ(P, a), and v(P (b)) > v(P (a)).

Step 2. Assume that P (b) 6= 0. It remains to show that then (P, b) is in σ-hensel conﬁguration with γ(P, b) > γ. Let J 6= 0,P(J) 6= 0 and consider

X L P(J)(b) = P(J)(a) + P(J)(L)(a) · σ(b − a) . L6=0

Note that P(J)(a) 6= 0. Since K is of equi-characteristic zero, v(P(J)(L)(a)) = v(P(J+L)(a)). Therefore, for all L 6= 0,

L v(P(J)(L)(a) · σ(b − a) ) > v(P(J)(a)),

hence v(P(J)(b)) = v(P(J)(a)). Since P (b) 6= 0, we can pick γ1 ∈ Γ such that

j P (b) = min v(P(j)(a)) + ρ · γ1. 0≤j≤n

i Then γ < γ1 : Pick 0 ≤ i ≤ n such that v(P (a)) = v(P(i)(a)) + ρ · γ. So

i i ρ · γ = v(P (a)) − v(P(i)(a)) < v(P (b)) − v(P(i)(a)) ≤ ρ · γ1. CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 43

Also for I, J 6= 0 and θ ∈ Γ with θ > 0, we have |J|ρ · θ < |L|ρ · θ for J < L (here we are using the fact that ρ > 0). Thus the inequality

v(P(J)(a)) + |J|ρ · γ < v(P(L)(a)) + |L|ρ · γ together with γ1 > γ yields

v(P(J)(a)) + |J|ρ · γ1 < v(P(L)(a)) + |L|ρ · γ1.

Hence, (P, b) is in σ-hensel conﬁguration with γ(P, b) = γ1.

Lemma 4.3.5. Suppose K satisﬁes Axiom 3n and ρ > 1, and (P, a) is in σ-hensel conﬁgu- ration. Then there is c ∈ K such that 1. v(c − a) ≥ γ(P, a), v(P (c)) > v(P (a)),

2. either P (c) = 0, or (P, c) is in strict σ-hensel configuration. For any such c, we have v(c − a) = γ(P, a), γ(P, c) > γ(P, a); and if (P, a) was already in strict σ-hensel configuration, then i(P, c) ≤ i(P, a). Proof. Let γ = γ(P, a) and i = i(P, a) (in case (P, a) is in strict σ-hensel configuration). Since (P, a) is in σ-hensel configuration, by Lemma 4.3.4, there is b ∈ K such that v(b−a) = γ(P, a), v(P (b)) > v(P (a)), γ(P, b) > γ(P, a) = γ and either P (b) = 0 or (P, b) is in σ-hensel configuration. If P (b) = 0, let c := b and we are done. So suppose P (b) 6= 0. Then, letting γ1 = γ(P, b), we have for some 0 ≤ j0 ≤ n,

j0 j v(P (b)) = v(P(j0)(a)) + ρ · γ1 ≤ v(P(j)(a)) + ρ · γ1 for all 0 ≤ j ≤ n. If the above inequality is strict for j 6= j0, we are done: Then (P, b) is in strict σ-hensel i conﬁguration with i(P, b) = j0 and γ(P, b) = γ1. Moreover, if i < j0, then ρ · (γ1 − γ) ≤ j0 ρ · (γ1 − γ) as γ1 − γ > 0 and ρ ≥ 1, and we have

i j0 v(P(i)(a)) + ρ · γ < v(P(j0)(a)) + ρ · γ i i j0 j0 =⇒ v(P(i)(a)) + ρ · γ + ρ · (γ1 − γ) < v(P(j0)(a)) + ρ · γ + ρ · (γ1 − γ) i j0 =⇒ v(P(i)(a)) + ρ · γ1 < v(P(j0)(a)) + ρ · γ1, which is a contradiction. So j0 ≤ i. Thus, we have, γ(P, b) > γ(P, a) and i(P, b) ≤ i(P, a). Let c := b. However, if there is no such unique j0, then it means there are 0 ≤ j0 < j1 < ··· < jm ≤ n such that

j0 j1 jm v(P (b)) = v(P(j0)(a)) + ρ · γ1 = v(P(j1)(a)) + ρ · γ1 = ··· = v(P(jm)(a)) + ρ · γ1. CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 44

Since (P, b) is in σ-hensel configuration, we can find b0 ∈ K such that v(P (b0)) > v(P (b)) > v(P (a)), γ(P, b0) > γ(P, b) > γ(P, a), v(b0 − b) = γ(P, b) and either P (b0) = 0 or (P, b0) is in σ-hensel configuration. It follows that

v(b0 − a) = v(b0 − b + b − a) ≥ min{v(b0 − b), v(b − a)} = min{γ(P, b), γ(P, a)} =⇒ v(b0 − a) = γ(P, a) since, γ(P, a) < γ(P, b).

0 0 0 If P (b ) = 0, we are done. So suppose P (b ) 6= 0. Let γ2 = γ(P, b ). Since γ2 − γ1 > 0 and ρ > 1 (this is where we crucially use this hypothesis), we have

j0 j1 jm ρ · (γ2 − γ1) < ρ · (γ2 − γ1) < ··· < ρ · (γ2 − γ1).

But then by doing the same trick as in the previous paragraph, we obtain

j0 j1 jm v(P(j0)(a)) + ρ · γ2 < v(P(j1)(a)) + ρ · γ2 < ··· < v(P(jm)(a)) + ρ · γ2.

Thus, we have succeeded in finding a better approximation b0 than b in the sense that (P, b0) is in σ-hensel configuration with its minimal valuation occurring at a possibly lower index than that of (P, b). Since i(P, a) is finite, there are only finitely many possibilities for this index to go down. So by repeating this step finitely many times, we end up at our required c with v(c − a) = γ(P, a) such that either P (c) = 0 or (P, c) is in strict σ-hensel configuration with γ(P, c) > γ(P, a) and i(P, c) ≤ i(P, a).

Lemma 4.3.6. Suppose K satisﬁes Axiom 3n, and (P, a) is in σ-hensel conﬁguration. Sup- pose also there is no b ∈ K such that P (b) = 0 and v(b − a) = γ(P, a). Then there is a pc-sequence {aη} in K with the following properties:

1. a0 = a and {aη} has no pseudolimit in K;

2. {v(P (aη))} is strictly increasing, and thus P (aη) 0;

0 3. v(aη0 − aη) = γ(P, aη) whenever η < η ;

0 4. (P, aη) is in σ-hensel conﬁguration with γ(P, aη) < γ(P, aη0 ) for η < η ;

0 0 5. for any extension K of K and b, c ∈ K such that aη b and v(c − b) ≥ γ(P, b), we have aη c.

Proof. We will build the sequence by transﬁnite recursion. Start with a0 := a. Suppose for some ordinal λ > 0, we have built the sequence {aη}η<λ such that

(i) (P, aη) is in σ-hensel conﬁguration, for all η < λ, CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 45

0 (ii) v(aη0 − aη) = γ(P, aη) whenever η < η < λ,

0 (iii) v(P (aη0 )) > v(P (aη)) and γ(P, aη0 ) > γ(P, aη) whenever η < η < λ. Now we will have to deal with the inductive case. If λ is a successor ordinal, say λ = µ + 1, then by Lemma 4.3.4, there is aλ ∈ K such that v(aλ − aµ) = γ(P, aµ), v(P (aλ)) > v(P (aµ)) and γ(P, aλ) > γ(P, aµ). Then the extended sequence {aη}η<λ+1 has the above properties with λ + 1 instead of λ. Suppose λ is a limit ordinal. Then {aη} is a pc-sequence and P (aη) 0. If {aη} has no pseudolimit in K, we are done. Otherwise, let aλ ∈ K be a pseudolimit of {aη}. Then v(aλ − aη) = v(aη+1 − aη) = γ(P, aη); also, for any η < λ,

X I P (aλ) = P (aη) + P(I)(aη) · σ(aλ − aη) ; |I|≥1 since P (aη) has the minimal valuation of all the summands, we have v(P (aλ)) ≥ v(P (aη)) for all η < λ. Since {v(P (aη))}η<λ is increasing by inductive hypothesis, we get v(P (aλ)) > v(P (aη)) for all η < λ. And then by Step 2 of Lemma 4.3.4, it follows that (P, aλ) is in σ-hensel configuration with γ(P, aλ) > γ(P, aη) for all η < λ. Thus the extended sequence {aη}η<λ+1 satisfies all the above properties with λ + 1 instead of λ. Eventually we will have a sequence cofinal in K, and hence the building process must come to a stop, yielding a pc-sequence satisfying (1), (2), (3) and (4). Now aη b. Thus v(b − aη) = v(aη+1 − aη) = γ(P, aη) for all η, and (P, b) is in σ-hensel configuration with γ(P, b) > γ(P, aη) for all η. In particular,

v(c − aη) = v(c − b + b − aη)

≥ min{v(c − b), v(b − aη)}

≥ min{γ(P, b), γ(P, aη)}

=⇒ v(c − aη) = γ(P, aη)

Since {γ(P, aη)} is increasing, we have aη c. It follows similarly (with ideas from the proof of Lemma 4.3.5) that

Lemma 4.3.7. Suppose K satisﬁes Axiom 3n, ρ > 1 and (P, a) is in strict σ-hensel conﬁg- uration. Suppose also there is no b ∈ K such that P (b) = 0 and v(b − a) = γ(P, a). Then there is a pc-sequence {aη} in K with the following properties:

1. a0 = a and {aη} has no pseudolimit in K;

2. {v(P (aη))} is strictly increasing, and thus P (aη) 0;

0 3. v(aη0 − aη) = γ(P, aη) whenever η < η ; CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 46

4. (P, aη) is in strict σ-hensel conﬁguration with γ(P, aη) < γ(P, aη0 ) and i(P, aη0 ) ≤ 0 i(P, aη) for η < η ;

0 0 5. for any extension K of K and b, c ∈ K such that aη b and v(c − b) ≥ γ(P, b), we have aη c. Definition 4.3.8. A multiplicative valued difference field K is called (strict) σ-henselian if for all (P, a) in (strict) σ-hensel configuration there is b ∈ K such that v(b − a) = γ(P, a) and P (b) = 0.

By Axiom 3 we mean the set {Axiom 3n : n = 0, 1, 2,...}. So Axiom 3 is really an axiom scheme and K satisfies Axiom 3 if and only if k is linear difference closed. Corollary 4.3.9. If K is maximally complete as a valued field and satisfies Axiom 3, then K is σ-henselian (strict σ-henselian if ρ > 1). In particular, if K is complete with discrete valuation and satisfies Axiom 3, then K is σ-henselian (strict σ-henselian if ρ > 1). Lemma 4.3.10. 1. If K is σ-henselian, then K satisfies Axiom 3. 2. If K satisfies Axiom 3, then K satisfies Axiom 2.

n Proof. (1) Assume that K is σ-henselian and let Q(x) = 1 + α0x + α1σ¯(x) + ··· + αnσ¯ (x) ∈ khxi such that not all αi’s are zero. We want to find b ∈ k such that Q(b) = 0. n Let P (a) = 1 + a0x + a1σ(x) + ··· + anσ (x), where for all i, ai ∈ K, ai = 0 if αi = 0, and v(ai) = 0 witha ¯i = αi if αi 6= 0. It is easy to see that (P, 0) is in σ-hensel configuration with γ(P, 0) = 0. By σ-henselianity, there is a ∈ K such that v(a) = 0 and P (a) = 0. Set b :=a ¯. (2) For K to satisfy Axiom 2, we need for each d ∈ Z+, an element a ∈ k such that d d σ¯ (a) 6= a. Consider the linear difference polynomial Pd(x) =σ ¯ (x) − x + 1 over k. Since d K satisfies Axiom 3, there is a ∈ k such that Pd(a) = 0, i.e.,σ ¯ (a) = a − 1. In particular, σ¯d(a) 6= a. Remark 4.3.11. 1. If Γ = {0}, then K is σ-henselian. 2. If Γ 6= {0} and K is σ-henselian, then K satisfies Axiom 3 by Lemma 4.3.10(1). And then by Lemma 4.3.10(2), K satisfies Axiom 2 as well. 3. If ρ > 1 and K satisfies Axiom 3, then K is σ-henselian iff K is strict σ-henselian: the “only-if” direction is trivial, and the “if” direction follows from Lemma 4.3.5. However, the “if” direction is not true for ρ = 1.

Deﬁnition 4.3.12. {aη} is said to be of σ-algebraic type over K if P (bη) 0 for some σ-polynomial P (x) over K and an equivalent pc-sequence {bη} in K. Otherwise, {aη} is said to be of σ-transcendental type. If {aη} is of σ-algebraic type over K, then a minimal σ-polynomial of {aη} over K is a σ-polynomial P (x) over K with the following properties: CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 47

(i) P (bη) 0 for some pc-sequence {bη} in K equivalent to {aη};

(ii) Q(bη) 6 0 whenever Q(x) is σ-polynomial over K of lower complexity than P (x) and {bη} is a pc-sequence in K equivalent to {aη}.

Lemma 4.3.13. Suppose K satisﬁes Axiom 2. Let {aη} from K be a pc-sequence of σ- algebraic type over K with minimal σ-polynomial P (x) over K, and with pseudolimit a in some extension. Let Σ be a ﬁnite set of σ-polynomials Q(x) over K. Then there is a pc- sequence {bη} in K, equivalent to {aη}, such that, with γη := v(a − aη):

(I) v(a − bη) = γη, eventually, and P (bη) 0;

(II) if Q ∈ Σ and Q 6∈ K, then Q(bη) Q(a);

(III) (P, bη) is in σ-hensel conﬁguration with γ(P, bη) = γη, eventually;

(IV) (P, a) is in σ-hensel conﬁguration with γ(P, a) > γη eventually.

0 If ρ > 1, then (P, bη) is actually in strict σ-hensel conﬁguration. Also there is some a , 0 0 pseudolimit of {aη}, such that (I), (II) and (IV ) hold with a replaced by a , and (P, a ) is in 0 strict σ-hensel conﬁguration with γ(P, a ) > γη eventually.

Proof. Let P have order n. Let us augment Σ with all P(I) for 1 ≤ |I| ≤ deg(P ). In the rest of the proof, all multi-indices range over Nn+1. Also since P is a minimal polynomial of {aη}, there is an equivalent sequence {cη} such that P (cη) 0. Now if ρ is transcendental, then by Theorem 4.2.4, Q(cη) Q(a) for all Q ∈ Σ and Q 6∈ K. Let bη := cη. Thus, {bη} satisﬁes (I) and (II). And if ρ is algebraic, then by Theorem 4.2.6, there is a pc-sequence {bη}, equivalent to {cη} (and hence to {aη}), such that (I) and (II) hold. Theorem 4.2.4 also shows that in the transcendental case, there is a unique L0 such that eventually for all I 6= L0,

v(P (bη) − P (a)) = v(P(L0)(a)) + |L0|ρ · γη < v(P(I)(a)) + |I|ρ · γη,

and in the algebraic case there is a unique m0 such that eventually for all I with |I|ρ 6= m0,

v(P (bη) − P (a)) = min v(P(L0)(a)) + m0 · γη < v(P(I)(a)) + |I|ρ · γη. (4.1) |L0|ρ=m0

We will show that in either case |L0| = 1. Since for ρ > 1, there is a unique L0 such that |L0|ρ = m0 and |L0| = 1, this gives us that for ρ > 1 (both algebraic and transcendental), there is a unique L0 such that eventually for all I 6= L0,

v(P (bη) − P (a)) = v(P(L0)(a)) + |L0|ρ · γη < v(P(I)(a)) + |I|ρ · γη. (4.2)

This actually gives the strict σ-hensel conﬁguration of (P, bη) for ρ > 1. CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 48

For any I such that P(I) 6= 0, we claim that if I < J, then

v(P(I)(a)) + |I|ρ · γη < v(P(J)(a)) + |J|ρ · γη

eventually: Theorem 4.2.4 with Σ = {P,P(I)} shows that we can arrange that our sequence {bη} also satisﬁes v(P(I)(bη) − P(I)(a)) ≤ v(P(I)(L)(a)) + |L|ρ · γη,

eventually for all L with |L| ≥ 1. Since v(P(I)(bη)) = v(P(I)(a)) eventually (as P is a minimal polynomial for {aη}), this yields

v(P(I)(a)) ≤ v(P(I)(L)(a)) + |L|ρ · γη = v(P(I+L)(a)) + |L|ρ · γη. For L with I + L = J, this yields

v(P(I)(a)) ≤ v(P(J)(a)) + |J − I|ρ · γη.

As {γη} is increasing, we have eventually in η,

v(P(I)(a)) < v(P(J)(a)) + |J − I|ρ · γη.

Since eventually v(P(I)(bη)) = v(P(I)(a)), we have

v(P(I)(bη)) + |I|ρ · γη < v(P(J)(bη)) + |J|ρ · γη, and

v(P(I)(a)) + |I|ρ · γη < v(P(J)(a)) + |J|ρ · γη

It follows that |L0| = 1 (for ρ = 1, this means m0 = 1). In particular, we have established (4.1) with m0 = 1 for ρ = 1, and (4.2) for ρ > 1. Since P (bη) 0, this yields v(P (a)) > v(P (bη)) eventually, i.e, v(P (bη) − P (a)) = v(P (bη)). It follows from this and (4.1) that (P, bη) is in σ-hensel configuration eventually with γ(P, bη) = γη; and it follows from (4.2) that for ρ > 1, (P, bη) is in strict σ-hensel configuration. Finally by Step 2 of Lemma 4.3.4, it follows that (P, a) is also in σ-hensel configuration with γ(P, a) > γη eventually; and for ρ > 1, if (P, a) is already in strict σ-hensel configuration, we are done. Otherwise follow the proof of Lemma 4.3.5 to find the required a0.

4.4 Immediate Extensions

Throughout this section, K = (K, Γ, k; v, π) is a multiplicative valued difference field satisfying Axiom 2. Note that then any immediate extension of K also satisfies Axiom 2. We state here a few basic facts on immediate extensions.

Lemma 4.4.1. Let {aη} from K be a pc-sequence of σ-transcendental type over K. Then K has a proper immediate extension (Khai, Γ, k; va, πa) such that: CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 49

1. a is σ-transcendental over K and aη a;

2. for any extension (K1, Γ1, k1; v1, π1) of K with ptpZ[ρ](Γ1) = ptpZ[ρ](Γ), and any b ∈ K1 with aη b, there is a unique embedding

(Khai, Γ, k; va, πa) −→ (K1, Γ1, k1; v1, π1)

over K that sends a to b.

Proof. See [2, Lemma 6.2]. All that is needed in the proof is the pseudo-continuity of the σ-polynomials (upto equivalent sequences). So the same proof works here. As a consequence of both (1) and (2) of Lemma 4.4.1, we have:

Corollary 4.4.2. Let a from some extension of K be σ-algebraic over K and let {aη} be a pc-sequence in K such that aη a. Then {aη} is of σ-algebraic type over K.

Lemma 4.4.3. Let {aη} from K be a pc-sequence of σ-algebraic type over K, with no pseudolimit in K. Let P (x) be a minimal σ-polynomial of {aη} over K. Then K has a proper immediate extension (Khai, Γ, k; va, πa) such that:

1. P (a) = 0 and aη a;

2. for any extension (K1, Γ1, k1; v1, π1) of K with ptpZ[ρ](Γ1) = ptpZ[ρ](Γ), and any b ∈ K1 with P (b) = 0 and aη b, there is a unique embedding

(Khai, Γ, k; va, πa) −→ (K1, Γ1, k1; v1, π1)

over K that sends a to b.

Proof. See [1, Lemma 5.3].

Deﬁnition 4.4.4. K is said to be σ-algebraically maximal if it has no proper immediate σ-algebraic extension; and K is said to be maximal if it has no proper immediate extension.

Corollary 4.4.5. 1. K is σ-algebraically maximal if and only if each pc-sequence in K of σ-algebraic type over K has a pseudolimit in K;

2. if K satisﬁes Axiom 3 and is σ-algebraically maximal, then K is σ-henselian.

Proof. (1) The “only if” direction follows from Lemma 4.4.3. For the “if” direction, suppose for a contradiction that K1 := (K1, Γ, k; v1, π1) is a proper immediate σ-algebraic extension of K. Since the extension is proper, there is a ∈ K1 \ K. Since the extension is immediate, we can ﬁnd a pc-sequence {aη} from K such that aη a. Since the extension is σ-algebraic, CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 50

a is σ-algebraic over K. Then by Corollary 4.4.2, {aη} is of σ-algebraic type over K. So by assumption, there is b ∈ K such that aη b. But then by Lemma 4.4.3(2), we have ∼ ∼ (Khai, Γ, k; va, πa) = (Khbi, Γ, k; vb, πb) = (K, Γ, k; v, π), i.e., a ∈ K, a contradiction. (2) Let P (x) be a σ-polynomial over K of order ≤ n, and a ∈ K be such that (P, a) is in σ-hensel conﬁguration. If there is no b ∈ K such that v(b − a) = γ(P, a) and P (b) = 0, then by Lemma 4.3.6, there is a σ-algebraic pc-sequence {aη} in K such that {aη} has no pseudolimit in K. But then by part (1) of this corollary, K is not σ-algebraically maximal, a contradiction. It is clear that K has σ-algebraically maximal immediate σ-algebraic extensions, and also maximal immediate extensions. Provided that K satisﬁes Axiom 3, both kinds of extensions are unique up to isomorphism, but for this we need one more lemma: Lemma 4.4.6. Let K0 be a σ-algebraically maximal extension of K satisfying Axiom 3. Let {aη} from K be a pc-sequence of σ-algebraic type over K, with no pseudolimit in K, and 0 with minimal σ-polynomial P (x) over K. Then there exists b ∈ K such that aη b and P (b) = 0.

0 Proof. By Corollary 4.4.5(1), there exist a ∈ K such that aη a. If P (a) = 0, we are done. So let us assume P (a) 6= 0. But then by Lemma 4.3.13(IV), (P, a) is in σ- 0 hensel conﬁguration with γ(P, a) > v(a − aη) eventually. Since K satisﬁes Axiom 3, by Corollary 4.4.5(2), there is b ∈ K0 such that v(b − a) = γ(P, a) and P (b) = 0. Finally v(b − aη) = v(b − a + a − aη) = v(a − aη), since v(b − a) = γ(P, a) > v(a − aη). Thus, aη b. Together with Lemmas 4.4.1 and 4.4.3, this yields: Theorem 4.4.7. 1. Suppose K0 is a proper immediate σ-henselian extension of K, and 0 let a ∈ K \ K. Let K1 be a σ-henselian extension of K satisfying Axiom 2, such that every pc-sequence from K1 of length at most card(Γ) has a pseudolimit in K1. Then Khai embeds in K1 over K. 2. Suppose K0 is a proper immediate σ-henselian σ-algebraic extension of K, and let a ∈ 0 K \ K. Let K1 be a σ-henselian extension of K satisfying Axiom 2, such that every pc-sequence of σ-algebraic type over K1 and of length at most card(Γ) has a pseudolimit in K1. Then Khai embeds in K1 over K.

Proof. (1) By the classical theory, there is a pc-sequence {aη} from K such that aη a and {aη} has no pseudolimit in K. By assumption, there is b ∈ K1 such that aη b. If {aη} is of σ-transcendental type, then by Corollary 4.4.2, both a and b must be σ- transcendental over K. Now apply Lemma 4.4.1. CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 51

If {aη} is of σ-algebraic type, let P (x) be a minimal polynomial for {aη}. By Theorem 4.2.6, we get an equivalent pc-sequence {bη} from K with bη a, such that P (bη) 0, P (bη) P (a), P(L)(bη) P(L)(a) (but not to 0) for all |L| ≥ 1, (P, bη) is in σ-hensel configuration eventually, and either P (a) = 0, or (P, a) is also in σ-hensel configuration with 0 γ(P, a) > γ(P, bη) eventually. If P (a) = 0, we are done. Otherwise, since K is σ-henselian, 0 0 0 0 we have a ∈ K such that P (a ) = 0 and v(a − a) = γ(P, a). Since γ(P, a) > γ(P, bη) 0 0 0 0 eventually, we have bη a . Thus, in either case, we have a ∈ K such that P (a ) = 0 and 0 0 0 0 bη a . Similarly, we get b ∈ K1 such that bη b and P (b ) = 0. By Lemma 4.4.3, Kha0i is isomorphic to Khb0i as multiplicative valued difference fields over K, with a0 mapped to b0. Now, a is immediate over Kha0i. If it is not already in Kha0i, then we may repeat the argument and conclude by a standard maximality argument.

(2) By Corollary 4.4.2, there is a pc-sequence {aη} from K of σ-algebraic type pseudo- converging to a, but with no pseudolimit in K. Then the calculation in (1) works noting that every extension or pc-sequence considered will be of σ-algebraic type.

Corollary 4.4.8. Suppose K satisﬁes Axiom 3. Then all its maximal immediate extensions are isomorphic over K, and all its σ-algebraically maximal immediate σ-algebraic extensions are isomorphic over K.

Proof. We have already noticed the existence of both kinds of maximal immediate extensions. By Corollary 4.4.5(2), they are also σ-henselian. The desired uniqueness then follows by a standard maximality argument using Theorem 4.4.7 (1) and (2). We now state minor variants of the last two results using the notion of saturation from model theory.

0 + Lemma 4.4.9. Let K be a |Γ| -saturated σ-henselian extension of K. Let {aη} from K be a pc-sequence of σ-algebraic type over K, with no pseudolimit in K, and with minimal 0 σ-polynomial P (x) over K. Then there exists b ∈ K such that aη b and P (b) = 0. 0 Proof. By the saturation assumption, there is a pseudolimit a ∈ K of {aη}. If P (a) = 0, we are done. So let’s assume P (a) 6= 0. But then by Lemma 4.3.13(IV), (P, a) is in σ-hensel 0 0 conﬁguration with γ(P, a) > v(a−aη) eventually. Since K is σ-henselian, there is b ∈ K such that v(b−a) = γ(P, a) and P (b) = 0. Finally, aη b, since v(b−a) = γ(P, a) > v(a−aη). In combination with Lemmas 4.4.1 and 4.4.3, this yields:

Corollary 4.4.10. Suppose that K satisﬁes Axiom 3 and K0 is a |Γ|+-saturated σ-henselian extension of K. Let K∗ be a maximal immediate extension of K. Then K∗ can be embedded in K0 over K. CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 52

4.5 Example and Counter-example

We will now show the consistency of our axioms by building models for our theory. The canonical models we have in mind are the generalized power series fields k((tΓ)), also known as the Hahn series. Given any difference field k of characteristic zero with automorphismσ ¯, and any MODAG Γ with automorphism σ(γ) = ρ · γ, we form the multiplicative difference valued field k((tΓ)) as follows. As a set, k((tΓ)) := {f :Γ → k | supp(f) := {x ∈ Γ: f(x) 6= 0} is well-ordered in the ordering induced by Γ}. An element f ∈ k((tΓ)) is thought of as a formal power series X f ↔ f(γ)tγ γ∈Γ (f + h)(γ) := f(γ) + h(γ) X (fh)(γ) := f(α)h(β) α+β=γ v(f) := min supp(f) X σ(f) := σ¯(f(γ))tρ·γ γ∈Γ If we choose ρ ≥ 1, k((tΓ)) satisfies Axiom 1. Also if we impose thatσ ¯ is a linear difference closed automorphism on k, then k((tΓ)) satisfies Axiom 2 and Axiom 3 as well. Moreover, using the fact that k((tΓ)) is maximally complete [20], it follows from Corollary 4.3.9 that k((tΓ)) is σ-henselian for ρ ≥ 1, and strict σ-henselian for ρ > 1. Also note that the residue field of k((tΓ)) is k, and the value group is Γ.

Now we will justify why we need Axiom 3 (at least for the case ρ > 1). We will provide an example that shows why Axiom 3 cannot be dropped. This example is adapted from [1], Example 5.11. Example 4.5.1. Let ρ be any element of a real-closed field and ρ > 1. Let Γ be a MODAG with the given ρ. Let k be any field of characteristic zero, construed as a difference field equipped with its identity automorphism. And let K be the multiplicative valued difference field (k((tΓ)), Γ, k; v, π). −n Γn For each n, let Γn := ρ Z[ρ] and let Kn := (k((t )), Γn, k; v, π). Let [ Γn K∞ := k((t )), Γ, k; v, π . n

Then K∞ equipped with the restriction of σ, is a valued difference subfield of K and σ is multiplicative. Let us define a sequence {an} as follows: n X −ρ−i an = t . i=1 CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 53

We claim that {an} is a pc-sequence: Note that since ρ > 1, we have for i < j ∈ N, −ρ−i −i −j −ρ−j −ρ−(n+1) −(n+1) v(t ) = −ρ < −ρ = v(t ). Hence, v(an+1 − an) = v(t ) = −ρ , which is increasing as n → ∞. −ρ−i −ρ−i+1 Also it is clear that {an} has no pseudolimit in K∞. Moreover, since σ(t ) = t , we have for P (x) = σ(x) − x − t−1, −ρ−n P (an) = t 0, and hence {an} is of σ-algebraic type over K∞. Now K∞ is a union of henselian valued ﬁelds, and hence is henselian. Moreover it is of characteristic zero. Hence K∞ is algebraically maximal. Therefore, P (x) is a minimal σ-polynomial of {an} over K∞. Also, P (an) + 1 0,

and so P (x) + 1 is a minimal σ-polynomial of {an} over K∞ as well. By Lemma 4.4.3, there 0 0 are immediate extensions K∞hai, K∞ha i of K∞ such that an a, P (a) = 0, and an a , 0 P (a ) + 1 = 0. Let L1, L2 be σ-algebraically maximal, immediate, σ-algebraic extensions of 0 K∞hai, K∞ha i respectively. Now we claim that L1 and L2 are not isomorphic over K∞. Suppose for a contradiction that they are isomorphic. Then there is b ∈ L1 such that P (b) + 1 = 0. Since P (a) = 0 we have Q(a, b) := σ(b − a) − (b − a) + 1 = 0. We claim that this is only possible when v(b − a) = 0 : if v(b − a) > 0, then since ρ > 1, we have v(σ(b − a)) > v(b − a) > 0 = v(1). Hence, v(Q(a, b)) = v(1) = 0, and thus Q(a, b) 6= 0, a contradiction; similarly, if v(b − a) < 0, then v(σ(b − a)) < v(b − a) < 0 = v(1), in which case again v(Q(a, b)) = 0, a contradiction. Thus, v(b − a) = 0. But then, b − a ∈ k and b − a is a solution of σ¯(x) − x + 1 = 0, which is impossible sinceσ ¯ = id, contradiction. Here we considered a particular instance of failure of Axiom 3; namely, whenσ ¯ is the identity, the aboveσ ¯-linear equation does not have a solution in k. However, one can produce a similar construction for any non-degenerate inhomogeneousσ ¯-linear equation which does not have a solution in k.

4.6 Extending the Residue Field and the Value Group

Let K = (K, Γ, k; v, π) and K0 = (K0, Γ0, k0; v0, π0) be two multiplicative valued difference 0 0 fields with ptpZ[ρ](Γ) = ptpZ[ρ](Γ ). Let O and O be their respective ring of integers, and let σ denote both their difference operators. Let E = (E, ΓE, kE; v, π) be a common multiplicative valued difference subfield of both K and K0, that is, E ≤ K, E ≤ K0. Then we have: CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 54

Lemma 4.6.1. Let a ∈ O and assume α = π(a) is σ¯-transcendental over kE. Then

P L • v(P (a)) = minL{v(bL)} for each σ-polynomial P (x) = bLσ(x) over E;

× × • v(Ehai ) = v(E ) = ΓE, and Ehai has residue ﬁeld kEhαi;

0 • if b ∈ O is such that β = π(b) is σ¯-transcendental over kE, then there is a valued diﬀerence ﬁeld isomorphism Ehai → E 0hbi over E sending a to b.

Proof. See [2, Lemma 2.5].

Lemma 4.6.2. Let P (x) be a non-constant σ-polynomial over the valuation ring of E whose reduction P¯(x) has the same complexity as P (x). Let a ∈ O, b ∈ O0, and assume that P (a) = 0,P (b) = 0, and that P¯(x) is a minimal σ¯-polynomial of α := π(a) and of β := π0(b) over kE. Then

× • Ehai has value group v(E ) = ΓE and residue ﬁeld kEhαi;

• if there is a difference field isomorphism kEhαi → kEhβi over kE sending α to β, then there is a valued difference field isomorphism Ehai → Ehbi over E sending a to b.

Proof. See [2, Lemma 2.6]. Now we will show how to extend the value group. Recall that Γ is a model of MODAG. Before stating the results, we need a couple of deﬁnitions.

P L Deﬁnition 4.6.3. For a given σ-polynomial P (x) = bLσ(x) over K and a ∈ K, we say a is generic for P if v(P (a)) = min{v(bL) + |L|ρ · v(a)}. Deﬁnition 4.6.4. An element a ∈ K (or K0) is said to be generic over E if a is generic for P L all σ-polynomials P (x) = bLσ(x) over E.

P L Lemma 4.6.5. Assume K satisﬁes Axiom 2. Then, for each γ ∈ Γ and P (x) = bLσ(x) over K, there is a ∈ K such that v(a) = γ and a is generic for P .

Proof. Let c ∈ K be such that v(c) = γ. If c is already generic for P , set a := c and we are done. Otherwise, pick ∈ K such that v() = 0 (we will decide later how to choose ) and P L L × set a := c. Note that v(a) = v(c) = γ. Then, P (a) = bLσ(c) σ() . Choosing d ∈ K L such that v(d) = min{v(bLσ(c) )} = min{v(bL) + |L|ρ · γ}, we can write P (a) = dQP (), where QP () is over the ring of integers of K. Consider QP (¯), aσ ¯-polynomial over k; choose ¯ ∈ k such that QP (¯) 6= 0, which is possible since K satisﬁes Axiom 2. Let ∈ K be such that π() = ¯. Then v(QP ()) = 0, and hence v(P (a)) = v(d) = min{v(bL) + |L|ρ · γ}. Thus, a is generic for P and v(a) = γ. CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 55

Remark 4.6.6. It is clear from the proof of Lemma 4.6.5 that if we replace P (x) by a ﬁnite set of σ-polynomials {P1(x),...,Pm(x)} of possibly diﬀerent orders, then by choosing

¯ ∈ k such that it does not solve any of the related m equations QPi (x) = 0 over k (which is again possible to do as K satisﬁes Axiom 2), we can ﬁnd a ∈ K such that a is generic for {P1,...,Pm}.

P L Deﬁnition 4.6.7. Let P (x) = bLσ(x) be a σ-polynomial over K and a ∈ K. Write P (ax) = dQP (x), where d ∈ K is such that v(d) = min{v(bL) + |L|ρ · v(a)}. Then QP (x) is a σ-polynomial over OK , and thus QP (x) is aσ ¯-polynomial over k. We say QP (x) is a k-σ¯-polynomial corresponding to (P, a).

Lemma 4.6.8. Let γ ∈ Γ \ ΓE. Let κ be an inﬁnite cardinal such that |kE| ≤ κ. Assume K, K0 satisfy Axiom 2 and are κ+-saturated. Then

(i) there is a ∈ K, generic over E, with v(a) = γ;

(ii) Ehai has value group ΓEhγi, and residue ﬁeld kEhai of size ≤ κ;

0 0 0 (iii) if γ ∈ Γ is such that there is an isomorphism ΓEhγi → ΓEhγ i over ΓE of MODAGs, and a0 ∈ K0 is such that a0 is generic over E with v(a0) = γ0, then there is a valued diﬀerence ﬁeld isomorphism Ehai → Eha0i over E sending a to a0.

Proof. (i) Fix c ∈ K such that v(c) = γ. For each σ-polynomial P (x) over E, let QP (x) be a k-¯σ-polynomial corresponding to (P, c), and deﬁne

ϕP (y) := QP (y) 6= 0

i.e. ϕP (y) is the ﬁrst-order formula with parameters from k saying “y is not a root of QP ”. Let

p(y) := {ϕP (y) | P is a σ-polynomial over E}.

By Axiom 2, p(y) is finitely consistent, and hence consistent. So it is a type over E. Moreover by cardinality considerations, |p(y)| ≤ κ<ω = κ (since κ is infinite). Since K is κ+-saturated, p(y) is realized in K. In particular, there is y ∈ k such that y is not a root of any QP . Choosing ∈ O with π() = y and setting a := c, we then have that v(a) = γ and a is generic for all σ-polynomials P (x) over E, i.e., a is generic over E. × (ii) Since a is generic over E, it is clear that v(Ehai ) = ΓEhγi, which clearly has size at most κ. Moreover, since each element of the residue field comes from an element of valuation zero, the size of kEhai is at most the size of the set

{P (x) | P is a σ-polynomial over E and v(P (a)) = 0},

which, again by cardinality considerations, is at the most κ. Thus |kEhai| ≤ κ. CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 56

(iii) Finally, if b is generic over E, then v(P (b)) 6= ∞ for any σ-polynomial P (x) over E; i.e., P (b) 6= 0. So b is σ-transcendental over E. In particular, a and a0 are σ-transcendental over E. Thus there is a difference field isomorphism ψ : Ehai → Eha0i over E sending a 0 0 0 ∼ 0 0 to a . But, since v(a) = γ, v(a ) = γ ,ΓEhγi = ΓEhγ i over ΓE, and a and a are both generic over E, the valuations are already determined and matched up on both sides, i.e., ψ is actually a valued difference field isomorphism.

4.7 Embedding Theorem

To prove completeness and relative quantifier elimination of the theory of multiplicative valued difference fields, we use Test 2.1.22. To that end we would like to know when can we extend isomorphism between “small” substructures. The main theorem of this section, Theorem 4.7.1, gives an answer to that question. For the moment we will work in the 4-sorted language L4vdf , where we have our usual 3 sorts for the valued field K, the value group Γ and the residue field k, and we add to it a fourth sort for the RV . (For details on RV , refer to Section 2.3.) This represents the language of the leading terms introduced in [4], and explained further in [23] and [11]. As in the previous sections, we replace the symbol for the induced automorphismσ ˜ on the value group Γ by ρ·. We could have just worked with a 2-sorted language with K and RV (known as the leading term language). But the 2-sorted language is interpretable in and also interprets the 4-sorted language. So to make things more transparent we stick to the 4-sorted language. Recall that we are always dealing with the equi-characteristic zero case. 0 0 0 Now we describe the embeddings. Let K = (K, Γ, k, RV ; v, π, vrv, ι, rv) and K = (K , Γ , 0 0 0 0 0 0 0 k ,RV ; v , π , vrv, ι , rv ) be two σ-henselian multiplicative valued difference fields satisfy- 0 0 ing Axiom 1 with ptpZ[ρ](Γ) = ptpZ[ρ](Γ ). By Lemma 4.3.10, K and K satisfy Axiom 2 and Axiom 3. We denote the difference operator of both K and K0 by σ, and their 0 rings of integers by O and O respectively. Let E = (E, ΓE, kE,RVE; v, π, vrv, ι, rv) and 0 0 0 0 0 0 0 0 E = (E , ΓE0 , kE0 ,RVE0 ; v , π , vrv, ι , rv ) be valued difference subfields of K and K respectively. We say a bijection ψ : E → E0 is an admissible isomorphism if it has the following properties:

1. ψ is an isomorphism of multiplicative valued diﬀerence ﬁelds in the language L4vdf ;

0 2. the induced isomorphism ψrv : RVE → RVE in the language Lrv,σrv is elementary, i.e.,

for all formulas ϕ(x1, . . . , xn) in Lrv,σrv , and ξ1, . . . , ξn ∈ RVE, 0 RV |= ϕ(ξ1, . . . , ξn) ⇐⇒ RV |= ϕ(ψrv(ξ1), . . . , ψrv(ξn));

3. the induced isomorphism ψr : kE → kE0 of diﬀerence ﬁelds is elementary, i.e., for all formulas ϕ(x1, . . . , xn) in LR,σ¯, and α1, . . . , αn ∈ kE, 0 k |= ϕ(α1, . . . , αn) ⇐⇒ k |= ϕ(ψr(α1), . . . , ψr(αn)); CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 57

4. the induced isomorphism ψv :ΓE → ΓE0 is elementary, i.e, for all formulas ϕ(x1, . . . , xn) in LOG,ρ·, and γ1, . . . , γn ∈ ΓE,

0 Γ |= ϕ(γ1, . . . , γn) ⇐⇒ Γ |= ϕ(ψv(γ1), . . . , ψv(γn)).

(Note that it is enough to maintain (1) and (2) above, since (3) and (4) are consequences of (2) because of Proposition 2.3.26). Our main goal is to be able to extend such admissible isomorphisms. For this we need certain degree of saturation (see Definition 2.1.16) on K and K0. Fix an infinite cardinal κ and let us assume that K and K0 are κ+-saturated. Since the language is countable, such models exist assuming GCH (see Corollary 2.1.18). We then say a substructure E = 0 (E, ΓE, kE,RVE; v, π, vrv, ι, rv) of K (respectively of K ) is small if |ΓE|, |kE| ≤ κ. While extending the isomorphism, we do it in steps and at each step we typically extend the isomorphism from some E to Ehai, which is obviously small if E is; and then reiterate the process κ many times, which again preserves smallness. Eventually we reiterate this process countably many times and take union of an increasing sequence of countably many small fields, which also preserves smallness. Having said all that, we now state the Embedding Theorem. Theorem 4.7.1 (Embedding Theorem). Suppose K, K0, E, E 0 are as above with K, K0 κ+- saturated and E, E 0 small. Assume ψ : E → E0 is an admissible isomorphism and let a ∈ K. Then there exist b ∈ K0 and an admissible isomorphism ψ0 : Ehai ∼= E0hbi extending ψ with ψ(a) = b. Proof. The theorem is obvious if Γ = {0}. So let us assume that Γ 6= {0}. Also without loss of generality, we may assume a ∈ OK . We will extend the isomorphism in steps. There are three cases to consider:

I. There exists c ∈ Ehai such that π(c) ∈ k \ kE;

II. There exists c ∈ Ehai such that v(c) ∈ Γ \ ΓE;

III. For all c ∈ Ehai, we have π(c) ∈ kE and v(c) ∈ ΓE.

Step I: Extending the residue ﬁeld

× Let c ∈ Ehai be such that α := π(c) 6∈ kE. Since k ,→ RV , α ∈ RV . By saturation of 0 0 0 ∼ 0 K , we can ﬁnd α ∈ RV and an Lrv-isomorphism RVEhαi = RVE0 hα i extending ψrv and sending r 7→ r0 that is elementary as a partial map between RV and RV 0. Note that then α0 ∈ k0. Now we have two cases to consider. 0 Subcase I. α (respectively, α ) isσ ¯-transcendental over kE (respectively, kE0 ). In that case, pick d ∈ O and d0 ∈ O0 such that π(d) = α and π(d0) = α0. Then by Lemma 4.6.1, there is ∼ 0 0 an admissible isomorphism Ehdi = E hd i extending ψ with small domain (Ehdi, ΓE, kEhαi) and sending d to d0. CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 58

¯ Subcase II. α isσ ¯-algebraic over kE. Let P (x) be a σ-polynomial over OE such that P (x) is a minimalσ ¯-polynomial of α. Pick d ∈ O such that π(d) = α. If P (d) 6= 0 already, then (P, d) is in σ-hensel conﬁguration with γ(P, d) > 0. Since K is σ-henselian, there is e ∈ O such that P (e) = 0 and π(e) = π(d) = α. Likewise, there is e0 ∈ O0 such that P ψ(e0) = 0 and π(e0) = α0, where P ψ is the diﬀerence polynomial over E0 corresponding to P under ψ. Then by Lemma 4.6.2, there is an admissible isomorphism Ehei ∼= E 0he0i extending ψ with 0 small domain (Ehei, ΓE, kEhαi) and sending e to e . Note that in either case, we have been able to extend the admissible isomorphism to a small domain that includes α. Since E is small, so is Ehai, i.e., |kEhai| ≤ κ. Thus, by repeating Step I κ many times, we are able to extend the admissible isomorphism to a small domain E1 such that for all c ∈ Ehai with π(c) 6∈ kE, we have π(c) ∈ kE1 . Continuing this countably many times, we are able to build an increasing sequence of small domains

E = E0 ⊂ E1 ⊂ · · · ⊂ Ei ⊂ · · · such that for each c ∈ Eihai with π(c) 6∈ kEi , we have

π(c) ∈ kEi+1 . Taking the union of these countably many small domains, we get a small domain, which we still call E, such that ψ extends to an admissible isomorphism with domain E and for all c ∈ Ehai, we have π(c) ∈ kE, i.e., we are not in Case I anymore.

Step II: Extending the value group

Let c ∈ Ehai be such that γ := v(c) 6∈ ΓE. Let b ∈ K be generic over E with v(b) = γ. Let 0 0 0 ∼ 0 0 r := rv(b). By saturation of K , ﬁnd r ∈ RV and an Lrv-isomorphism RVEhri = RV hr i 0 0 extending ψrv, sending r 7→ r , that is elementary as a partial map between RV and RV . Let b0 ∈ K0 be such that rv0(b0) = r0. 0 0 P L 0 We claim that b is generic over E : for any P (x) = bLσ(x) with bL ∈ E , let ψ−1 P L P (x) = aLσ(x) be the corresponding σ-polynomial over E with aL ∈ E and ψ(aL) = ψ−1 bL. Since b is generic over E, we have v(P (b)) = min{v(aL) + |L|ρ · γ}, and hence by Lemma 2.3.21, we have

ψ−1 X L X L X L rv(P (b)) = rv(aLσ(b) ) = rv(aL)σ(rv(b)) = rv(aL)σ(r) .

Then

0 0 ψ−1 rv (P (b )) = ψrv(rv(P (b))) X L X 0 0 L X 0 0 L = ψrv rv(aL)σ(r) = rv (bL)σ(r ) = rv (bLσ(b ) ),

0 0 0 and hence by Lemma 2.3.20 again, we have v(P (b )) = min{v(bL) + |L|ρ · v(b )}, i.e., b is generic over E 0. Then by Lemma 4.6.8, since K satisﬁes Axiom 2, there is an admissible =∼ isomorphism from Ehbi / E 0hb0i extending ψ and sending b 7→ b0. Thus we have been able to extend the admissible isomorphism to a small domain that includes γ. Since E is small, so is Ehai, i.e., |ΓEhai| ≤ κ. Thus, by repeating Step II κ many times, we are able to extend the admissible isomorphism to a small domain E1 such that for CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 59

all c ∈ Ehai with v(c) 6∈ ΓE, we have v(c) ∈ ΓE1 . Continuing this countably many times, we are able to build an increasing sequence of small domains E = E0 ⊂ E1 ⊂ · · · ⊂ Ei ⊂ · · ·

such that for each c ∈ Eihai with v(c) 6∈ ΓEi , we have v(c) ∈ ΓEi+1 . Taking the union of these countably many small domains, we get a small domain, which we still call E, such that ψ extends to an admissible isomorphism with domain E and for all c ∈ Ehai, we have v(c) ∈ ΓE, i.e., we are not in Case II anymore.

Step III: Immediate Extension After doing Steps I and II, we are reduced to the case when Ehai is an immediate extension of E where both fields are equipped with the valuation induced by K. Let Ehai be the valued difference subfield of K that has Ehai as the underlying field. In this situation, we would like to extend the admissible isomorphism, not just to Ehai, but to a maximal immediate extension of Ehai and use Corollary 4.4.10. However, for that we need E to satisfy Axiom 2 and Axiom 3. Since Axiom 3 implies Axiom 2 by Lemma 4.3.10, it is enough to extend E such that it satisfies Axiom 3. Recall that K satisfies Axiom 3. Now to make E satisfy Axiom 3, for each linearσ ¯-polynomial P (x) over kE, if there is already no solution to P (x) in kE, find a solution α ∈ k and follow Step I. Since there are at most κ many such polynomials, we end up in a small domain. Thus, after doing all these, we can assume E satisfies Axiom 2 and Axiom 3. Let E ∗ be a maximal immediate valued difference field extension of Ehai. Then E ∗ is a maximal immediate extension of E as well. Similarly let E 0∗ be a maximal immediate extension of E 0. Since such extensions are unique by Corollary 4.4.8, and by Corollary 4.4.10 they can be embedded in K (respectively K0) over E (respectively E 0) by saturatedness of K (respectively K0), we have that ψ extends to a valued field isomorphism ∗ ∼ 0∗ E = E . Since kE∗ = kE and ΓE∗ = ΓE, it follows by the 5-Lemma on the following diagram that RVE∗ = RVE: × 1 / kE / RVE / ΓE / 0 .

id id

× 1 / kE∗ / RVE∗ / ΓE∗ / 0 Thus, the isomorphism is actually admissible. It remains to note that a is in the underlying diﬀerence ﬁeld of E ∗.

4.8 Completeness and Quantiﬁer Elimination Relative to RV

We now state some model-theoretic consequences of Theorem 4.7.1. We use ‘≡’ to denote the relation of elementary equivalence, and ‘4’ to denote the relation of elementary submodel. CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 60

Recall that we are working in the 4-sorted language L4vdf with sorts K for the valued ﬁeld in the language LR,σ, Γ for the value group in the language LOG,ρ·, k for the residue

field in the language LR,σ¯, and RV for the RV in the language Lrv,σrv ; and the sorts are connected by the valuation maps v : K → Γ ∪ {∞} and vrv : RV → Γ, the residue map π : O → k, the inclusion map ι : k → RV ∪ {∞}, and the rv map rv : K → RV ∪ {∞}. Let 0 0 0 0 0 0 0 0 0 0 K = (K, Γ, k, RV ; v, π, vrv, ι, rv) and K = (K , Γ , k ,RV ; v , π , vrv, ι , rv ) be two σ-henselian multiplicative valued difference fields (in the 4-sorted language) of equi-characteristic zero 0 satisfying Axiom 1 with ptpZ[ρ](Γ) = ptpZ[σ](Γ ). 0 0 Theorem 4.8.1. K ≡L4vdf K if and only if RV ≡Lrv,σrv RV .

Proof. The “only if” direction is obvious. For the converse, note that (Q, {0}, Q, Q; v, π, vrv, ι, rv), with v(q) = 0, π(q) = q, rv(q) = q, vrv(q) = 0 and ι(q) = q for all 0 6= q ∈ Q, is a substructure of both K and K0, and thus the identity map between these two substructures is an admissible isomorphism. Now apply Theorem 4.7.1.

Theorem 4.8.2. Let E = (E, ΓE, kE,RVE; v, π, vrv, ι, rv) be a σ-henselian multiplicative

valued diﬀerence subﬁeld of K, satisfying Axiom 1, such that RVE 4Lrv,σrv RV . Then E 4L4vdf K.

Proof. Take an elementary extension K0 of E. Then K0 satisﬁes Axiom 1, and is also σ- 0 henselian. Moreover (E, ΓE, kE,RVE; ··· ) is a substructure of both K and K , and hence the 0 identity map is an admissible isomorphism. Hence, by Theorem 4.7.1, we have K ≡L4vdf K . 0 Since E 4L4vdf K , this gives E 4L4vdf K.

Corollary 4.8.3. Theory of K in the language L4vdf is decidable if and only if theory of RV is decidable.

The proofs of these results use only weak forms of the Embedding Theorem, but now we turn to a result that uses its full strength: a relative elimination of quantifiers for the theory of σ-henselian multiplicative valued difference fields of equi-characteristic zero and satisfying Axiom 1.

Theorem 4.8.4. Let T be the L4vdf -theory of σ-henselian multiplicative valued difference fields of equi-characteristic zero satisfying Axiom 1, and φ(x) be an L4vdf -formula. Then there is an L4vdf -formula ϕ(x) in which all occurrences of field variables are free, such that

T ` φ(x) ↔ ϕ(x).

Proof. Let ϕ range over L4vdf -formulas in which all occurrences of ﬁeld variables are free. l m n s For a model K = (K, Γ, k, RV ; v, π, vrv, ι, rv) of T and a ∈ K , γ ∈ Γ , α ∈ k and r ∈ RV , let fqftpK(a, γ, α, r) := {ϕ : K |= ϕ(a, γ, α, r)}. CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 61

Let K, K0 be models of T and suppose

(a, γ, α, r) ∈ Kl × Γm × kn × RV s, (a0, γ0, α0, r0) ∈ K0l × Γ0m × k0n × RV 0s

0 0 0 0 are such that fqftpK(a, γ, α, r) = fqftpK0 (a , γ , α , r ). It suﬃces to show that

0 0 0 0 tpK(a, γ, α, r) = tpK0 (a , γ , α , r ).

Let E (respectively E 0) be the multiplicative valued diﬀerence subﬁeld of K (respectively K0) generated by a, γ, α and r (respectively a0, γ0, α0 and r0). Then there is an admissible isomorphism E → E 0 that maps a 7→ a0, γ 7→ γ0, α 7→ α0 and r 7→ r0. Now apply Theorem 4.7.1.

4.9 Completeness and Quantiﬁer Elimination Relative to (k, Γ)

Although the leading term language is already interpretable in the language of pure valued fields and is therefore closer to the basic language, we would now like to move to the 3-sorted language L3vdf , with a sort for the valued field K, a sort for the value group Γ, and a sort for the residue field k. It is well-known that in the presence of a “cross-section”, the two-sorted structure (K,RV ) is interpretable in the three-sorted structure (K, Γ, k). As a result any admissible isomorphism, as defined in the section on Embedding Theorem, boils down to one that satisfies properties (1), (3) and (4) only, because in the presence of a cross-section, (2) follows from (3) and (4). What that effectively means is that now we have completeness relative to the value group and the residue field. Let us now make all these explicit. Let K = (K, Γ, k; v, π) be a multiplicative valued difference field. Recall that we construe K× as a left Z[σ]-module (w.r.t. multiplication) under the action

n X j I ijσ a = σ(a) , j=0

where I = (i0, . . . , in) (we will freely switch between these two notations and the corresponding I or the ij’s will be clear from the context); similarly we construe Γ also as a left Z[σ]-module (w.r.t. addition) under the action

n n X j X j ijσ γ = ijρ · γ. j=0 j=0

With these actions in place, we make the following CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 62

Deﬁnition 4.9.1. A cross-section s :Γ → K× on K is a group homomorphism such that Pn j for all γ ∈ Γ and τ = j=0 ijσ ∈ Z[σ], we have v(s(γ)) = γ, and s((τ)γ) = (τ)s(γ).

Example 4.9.2. For Hahn diﬀerence ﬁelds k((tΓ)), the map given by s(γ) = tγ is a cross- section.

I Pn j Since v(σ(a) ) = j=0 ijρ · v(a), we have an exact sequence of Z[σ]-modules

ι v 1 / O× / K× / Γ / 0 , where O× is multiplicative group of units of the valuation ring O. Clearly then, existence of a cross-section on K corresponds to this exact sequence being a split sequence. Before we proceed further, we need a few preliminaries from algebra.

Preliminaries from algebra

Let R be a commutative ring with identity (for our case R = Z[σ]). Deﬁnition 4.9.3. For two left R-modules N ⊆ M, N is said to be pure in M (notation: p N / M ) if for any m×n matrix (rij) with entries in R, and any set y1, . . . , ym of elements of N, if there exist elements x1, . . . , xn ∈ M such that

n X rijxj = yi for i = 1, . . . , m j=1

0 0 then there also exist elements x1, . . . , xn ∈ N such that

n X 0 rijxj = yi for i = 1, . . . , m. j=1 Deﬁnition 4.9.4. If M,N are left R-modules and f : N → M is an injective homomorphism p f of left R-modules, then f is called pure injective if f(N) is pure in M (notation: N / M ). Deﬁnition 4.9.5. A left R-module E is called pure-injective if for any pure injective module homomorphism f : X → Y , and an arbitrary module homomorphism g : X → E, there exists a module homomorphism h : Y → E such that hf = g, i.e. the following diagram commutes:

p f X / Y p p g p p p h p E xp CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 63

Theorem 4.9.6. Every |R|+-saturated left R-module E is pure-injective.

Proof. See [8, page 171].

Since Z[σ] is countable, any ℵ1-saturated Z[σ]-module is pure injective by Theorem 4.9.6. × So let us assume K is ℵ1-saturated. Then O is also ℵ1-saturated (as it is interpretable in K) and hence pure-injective. Now since we have the exact sequence

ι v 1 / O× / K× / Γ / 0 ,

if we can show that O× is pure in K×, then we will be able to complete the following diagram

p ι O× / K× o o o id o oh wo o O× which will give us a splitting of the above exact sequence as Z[σ]-modules. Unfortunately, O× is not pure in K× in general, not even in the transcendental case. To make this happen, we need to impose additional conditions on the residue field or on the value group or on the valued field itself. For example, one could demand that Γ be a flat Z[σ]-module. Since Γ = K×/O×, it then implies that O× will be pure in K×. Or one could also directly impose that O× be pure in K×, which is a first-order condition. Restrictions on the residue field, such as closing it under taking roots with respect to Q(σ)-monomials, are also worth considering. This is a future research direction. In any case, once we have the cross-section in place, we have the following result.

Proposition 4.9.7. Suppose K has a cross-section s :Γ → K×. Then RV is interpretable in the two-sorted structure (Γ, k) with the first sort in the language of MODAG and the second in the language of difference fields.

Proof. Let S = (Γ × k×) ∪ {(0, 0)}. Note that S is a definable subset of Γ × k (in particular, the second co-ordinate is zero only when the first is too). Define f : S → RV ∪ {∞} by

s(γ)a if a 6= 0 f((γ, a)) = ∞ if (γ, a) = (0, 0)

Now it follows from [11, Proposition 3.1.6], that f is a bijection, and that the inverse images of multiplication and ⊕ on RV are deﬁnable in S. Moreover, if a 6= 0, then v(s(γ)a) = v(s(γ)) + v(a) = γ + 0 = γ, and if a = 0, then v(∞) = ∞. Thus the inverse image of the valuation map is {h(γ, a), γi} ∪ {h(0, 0), ∞i}. Finally, since σ(s(γ)a) = s(σ(γ))¯σ(a), the inverse image of the diﬀerence operator on RV is given by {h(γ, a), (σ(γ), σ¯(a))i}. Hence the result follows. CHAPTER 4. MULTIPLICATIVE VALUED DIFFERENCE FIELD 64

As an immediate corollary of Proposition 4.9.7, we have

0 0 0 0 0 0 0 0 Corollary 4.9.8. If K = (K, Γ, k, RV ; v, π, vrv, ι, rv) and K = (K , Γ , k ,RV ; v , π , vrv, 0 0 ι , rv ) are two multiplicative valued difference fields satisfying Axiom 1 with ptpZ[ρ](Γ) = 0 0 0 ptpZ[ρ](Γ ), have a cross-section, and Γ ≡LOG,ρ· Γ (as MODAGs) and k ≡LR,σ¯ k (as differ- 0 ence fields), then RV ≡Lrv,σrv RV .

This allows us to work in the 3-sorted language L3vdfs (eliminating the need for the RV sort), where we have a symbol s for the cross-section. Combining Corollary 4.9.8 with Theorems 4.8.1, 4.8.2 and 4.8.4 and Corollary 4.8.3, we then have the following nice results. Let K = (K, Γ, k; v, π, s) and K0 = (K0, Γ0, k0; v0, π0, s0) be two σ-henselian multiplicative 0 valued diﬀerence ﬁelds, satisfying Axiom 1 with ptpZ[ρ](Γ) = ptpZ[ρ](Γ ), of equi-characteristic zero, and having a cross-section. Then,

0 Theorem 4.9.9. K ≡L3vdfs K (as multiplicative valued difference fields with a cross-section) 0 0 if and only if Γ ≡LOG,ρ· Γ (as MODAGs) and k ≡LR,σ¯ k (as difference fields).

Theorem 4.9.10. Let E = (E, ΓE, kE; v, π) be a σ-henselian multiplicative valued diﬀerence

subﬁeld of K, satisfying Axiom 1 and having a cross-section, such that ΓE 4LOG,ρ· Γ (as

MODAGs) and kE 4LR,σ¯ k (as difference fields). Then E 4L3vdfs K (as a multiplicative valued difference field with a cross-section).

Theorem 4.9.11. Theory of K in the language L3vdfs is decidable if and only if theories of Γ and k are decidable.

Since we have expanded the language to contain a symbol for the cross-section, we also have the following relative quantiﬁer elimination result.

Theorem 4.9.12. Let T be the L3vdfs-theory of σ-henselian multiplicative valued difference fields of equi-characteristic zero satisfying Axiom 1 and having a cross-section, and φ(x) be an L3vdfs-formula. Then there is an L3vdfs-formula ϕ(x) in which all occurrences of field variables are free, such that T ` φ(x) ↔ ϕ(x). 65

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