QUANTIFIER ELIMINATION FOR THE RELATIVE FROBENIUS

THOMAS SCANLON

Abstract. Let (K, v) be a complete discretely valued field of characteristic zero with an algebraically closed residue field of positive characteristic. Let σ : K → K be a continuous automorphism of K inducing a Frobenius automorphism on the residue field. We prove quantifier-elimination for (K, v, σ) in a language with angular component maps and in a language with predicates on leading terms. The proof passes through a generalization of the main Ax-Kochen-Eršov and quantifier-elimination results of [5] to a wider class of D-henselian fields of pure characteristic zero.

1. Introduction

A valued difference field is a valued field (K, v) given together with an automorphism σ : K → K preserving the valuation in the sense that v(x) = v(σ(x)) holds universally on K^{×}. Examples of valued difference fields are legion though complete fields of positive residue characteristic given together with a relative Frobenius may be the mostly widely exploited.

Valued D-fields, a general framework for considering valued difference and differential fields, were introduced in [5] and quantifier elimination relative to the residue field and the value group for D-henselian fields with root-closed linearly differentially closed residue fields of characteristic zero was proved. However, quantifier elimination for valued difference fields eluded the methods of that paper for good reason: if the distinguished automorphism is nontrivial on the residue field, then the theory of (K, v, σ) cannot elimination quantifiers, even relative to the residue field and the value group. The culprit is the same obstruction to quantifier elimination for henselian (pure) fields with residue fields not closed under roots: the existential quantifier defining the ℓ-th powers cannot be eliminated simply by expanding the language for the residue field and the value group.

While the formula defining powers lies at the heart of the failure of quantifier elimination, it does not suffice to simply expand the language with power predicates. However, the solution to the problem for D-henselian fields is no more difficult than the solution for henselian (pure) fields.

Ax and Kochen [1] approached this problem by expanding the language to include a section of the valuation. If (K, v) is a valued field and χ : vK → K^{×} is a section of the valuation, then we may write any element x ∈ K^{×} as x = \frac{x}{χ(v(x))}χ(v(x)). If K is henselian of residue characteristic zero, then x is an n-th power if and only if n divides v(x) and the reduction of \frac{x}{χ(v(x))} is an n-th power in the residue field for any n ∈ ℤ_{+}.

By expanding the language with a section of the valuation one substantially alters the class of definable sets. Such a section is never definable in the pure valued field language and some of the new definable sets are quite pathological in comparison to those definable in the valued field language. For instance, the image of the section is an infinite definable subset of the field which has no interior. No set definable in the valued field language can have this property. For some valued fields, notably 𝔭-adic fields, it suffices to add power predicates [4], but in general the Basarab-Kuhlmann technology of additive multiplicative congruences (amc structures) is needed to obtain quantifier elimination in a definitional expansion of the valued field language [2].

Date: 29 August 1999.

While amc structures meet the condition of not expanding the class of definable sets, they may very well strike the reader as obscure (precise definitions are given in section 2). Angular component functions restore the transparency of the axioms in a language with a cross section while retaining the topological properties of the definable sets in the original language. An angular component function (of level zero for a pure valued field) is nothing more than a group homomorphism from the units of the field to the units of the residue field which is trivial on the 1-units and induces the identity on the residue field. If π : O_{K} → O_{K}/𝔪_{K} is the residue map and χ : vK → K^{×} is a section of the valuation, then the associated angular component map is x ↦ π(\frac{x}{χ(v(x))}) =: α(x). For K a henselian field of residue characteristic zero, x is an n-th power if and only if α(x) is an n-th power and n divides v(x). We use χ only to get α, the rest of the information supplied by χ is wasted.

In this work we modify the amc and angular component techniques to suit valued D- fields. We use angular component functions and amc structures of higher level to pass from mixed characteristic to pure characteristic zero.

The main theorem of this paper is an extension of the results of [5] to a complete axiomatization and quantifier elimination for a wider class of D-henselian fields. This class includes all D-henselian fields of characterization zero. Perhaps, the most important example of such a valued D-field is (W_{p^{∞}}(𝔽_{p}^{alg})[\frac{1}{p}], F_{q}), the field

of fractions of the Witt vectors of 𝔽_{p}^{alg} (also known as \begin{matrix} \\ ℚ_{p} & unr \end{matrix}, the completion of the maximal unramified extension of the p-adics) with the unique lifting of the q-power Frobenius. Independently from the current author, Luc Belair and Angus Macintyre have obtained a complete axiomatization for this structure [3].

This paper is organized as follows. We recall the formalism of amc structures and angular components and adapt them to valued D-fields. We then set out the languages to be used and state precisely the theorems to be proved. We present a standard valuation coarsening argument to reduce to the study of valued D-fields of pure characteristic zero. The remainder of the paper consists of a detailed guide to modifying the arguments of [5] to the expanded languages.

I originally envisioned the results of the current paper as a part of my thesis [6], but realized through conversations with my thesis advisor, Ehud Hrushovski, that predicates beyond those used in [6] were necessary. A number of people contributed to my understanding of this problem. I thank especially Luc Belair (for suggesting that angular components may be the most natural framework for quantifier elimination), Lou van den Dries (for supplying a preprint of [7]), Ehud Hrushovski (for his advice during the work on [6]), Franz-Viktor Kuhlmann (for bringing amc structures to my attention), and Angus Macintyre (for discussing his work on the relative Frobenius). During the writing of this paper I was partially supported by an NSF MSPRF.

2. Leading Terms

In this section (K, v) is a valued field with ring of integers O_{K,v} := {x ∈ K : v(x) ≥ 0} and value group vK. We write O_{K} for O_{K,v} if there is no risk of ambiguity. Ideals in O_{K,v} correspond to cuts in Γ_{K,v,≥0} := {v(x) : x ∈ O_{K}} via I ⇔ v(I) := {v(a) : a ∈ I}. For an element γ ∈ Γ_{K,v,≥0} write I_{γ} for {x ∈ O_{K} : v(x) > γ}. Note that I_{0} = 𝔪_{K}, the maximal ideal of O_{K}.

If I ⊂ O_{K} is a proper ideal, then because O_{K} is a local ring, 1 + I is a subgroup of K^{×} under multiplication. We denote by K_{I} the factor group K^{×}/(1 + I). The structure K_{I} has been defined merely as a group, but it inherits considerable structure from K.

The valuation v on K is defined by the exact sequence

1 ⟶ O_{K}^{×} ⟶ K^{×} ⟶^{v} vK ⟶ 1

Since 1 + I ≤ O_{K}^{×}, the valuation descends to K_{I} giving the following exact sequence.

1 ⟶ (O_{K}/I)^{×} ⟶ K_{I} ⟶^{v} vK ⟶ 1

If I ⊆ J ⊂ O_{K} are proper ideals, then we have a natural projection map π_{I,J} : K_{I} → K_{J}. As K_{(0)} ≅ K^{×}, we write π_{I} : K^{×} → K_{I} for the map π_{(0),I}.

The ultrametric triangle inequality on K implies that addition on K leaves a strong trace on K_{I}. When I is a prime ideal, addition (almost) descends to a binary operation on K_{I}. If x, x′, y, y′ ∈ K^{×} and π_{I}(x) = π_{I}(x′), π_{I}(y) = π_{I}(y′), and v(x + y) ≤ min{v(x), v(y)} + v(I), the π_{I}(x + y) = π_{I}(x′ + y′). We mostly use this construction when I = 𝔪_{K}.

There is also a trace of addition on K_{I} when I is not prime. However, the partial addition on K_{I} in general takes values in K_{J} for a different ideal J. Let I ⊆ J ⊂ O_{K} be proper ideals. For concreteness, we suppose that I = {x ∈ O_{K} : v(x) > δ} and that J = {x ∈ O_{K} : v(x) > ϵ}. We define a partial addition +_{I,J} : K_{I} × K_{I} → K_{J} defined by π_{I}(x)+_{I,J}π_{J}(y) := π_{J}(x+y) if min{v(x+y)−v(x), v(x+y)−v(y)} ≤ δ−ϵ. For general I ⊆ J ⊂ O_{K}, we define π_{I}(x) +_{I,J} π_{I}(y) := π_{J}(x + y) if for any choice of x′, y′ with π_{I}(x′) = π_{I}(x) and π_{I}(y′) = π_{I}(y) we have π_{J}(x + y) = π_{J}(x′ + y′).

Other operations on K induce structure on K_{I}. For instance, if σ : K → K is a field endomorphism which maps I back to itself, then σ induces an endomorphism of K. In particular, if σ is an automorphism preserving the valuation (v(x) = v(σ(x)) for x ∈ K^{×}), then σ induces a valuation preserving automorphism of K_{I}. If D : K → K gives K the struture of a valued D-field, then D induces a partial function on K_{I}. As with the partial addition, take I ⊆ J ⊂ O_{K}. We define the partial function D_{I,J} : K_{I} → K_{J} by D_{I,J}π_{I}(x) := π_{J}(D(x)) if this assignment is well defined. If I = {x ∈ O_{K} : v(x) > δ} and J = {x ∈ O_{K} : v(x) > ϵ}, then D_{I,J}(π_{I}(x)) is defined if and only if v(Dx) ≤ v(x) + δ − ϵ.

Of course, other structure from K also descends to the leading terms, but we shall need nothing beyond what we have already lain out.

Let I be a set of ideals in O_{K}. A system of angular component functions for the pure valued field K relative to I is a family of group homomorphisms ac_{I} : K_{I} → (O_{K}/I)^{×} which are sections of the inclusions defining the valuation on K_{I} and which are compatible in the sense that if I ⊆ J are two elements of I, then π_{I,J} ∘ ac_{I} = ac_{J} ∘ π_{I,J}. The function ac_{I} is called an angular component function of level I. We abuse notation and write ac_{I} for ac_{I} ∘ π_{I} as well. Necessarily, an angular component function must preserve addition (as far as this makes sense), but it need not preserve any additional structure.

If (K, v, D, e) is a valued D-field, then recall that σ : K → K defined by σ(x) := eD(x) + x defines an endomorphism of K. If v(e) > 0, then σ is necessarily an automorphism and v(x) = v(σ(x)) holds for all x ∈ K^{×}. However, if v(e) = 0, the endomorphism σ need be neither surjective nor valuation preserving. From now on we include as part of the definition of a valued D-field that σ is a valuation preserving automorphism.

If K is a valued D-field and I is a family of ideals in O_{K}, then a system of D-field angular component functions for I is defined to be a system of angular component functions for I in the sense of pure valued fields which respect the Dstructure. That is, ac_{J}(D_{I,J}(x)) = π_{I,J}(Dac_{I}(x)) for x with v(Dx) = v(x) and ac_{I}(σ(x)) = σ(ac_{I}(x)) for all x.

From an angular component function ac_{I} : K_{I} → (O_{K}/I)^{×} we obtain a section of the valuation χ_{I} : Γ_{K} → K_{I} by the formula χ_{I}(γ) := \frac{x}{ac_{I}(x)} for any choice of x ∈ K_{I} with v(x) = γ. The condition that ac_{I} respects the D-structure is equivalent to χ_{I} taking values in the constants of K_{I}.

3. Languages

We treat valued D-fields as many-sorted structures having a basic sort K for the valued field itself and other sorts for the residue field, k; the value group, Γ; and, perhaps, also some other sorts for leading terms, K_{I}, and residue rings, O_{K}/I. All these sorts are interpretable in the basic sort; so nothing is lost by working in a one-sorted structure and treating the other sorts as merely interpreted.

The language of D-rings, L_{D−ring} := L(+, ·, −, D, 0, 1, e), is the expansion of the language of rings by a new unary function symbol D and a new constant symbol e. As mentioned in the previous section, if (R, D, e) is a D-ring, then the function σ : R → R defined by σ(x) := eD(x) + x is a ring endomorphism. We will insist that σ is actually an automorphism so that the L_{D−ring}-structures we consider will have a natural expansion to L_{D−ring}(σ, σ^{−1}).

We regard K and k as L_{D−ring}(σ, σ^{−1})-structures and Γ as an L(+, −, ≤, 0)- structure where the non-logical symbols have their usual meaning. These sorts are connected by the valuation v : K → Γ ∪ {∞} and residue map π : K → k ∪ {∞}, where we have adjoined a new symbol “∞” not belonging to any particular sort to ensure that π and v are total. We call the language described above the language of valued D-fields, L_{vdf}.

One could consider the language of valued D-fields augmented by arbitrarily complicated systems of angular component functions or leading term sorts. However, we need only multiplicative congruences modulo the 1-units in pure characteristic zero and with respect to the family of ideals {I_{n} := {x ∈ O_{K} : v(x) > nv(p)}}_{n∈ω} in mixed characteristic.

The language appropriate for considering a pure characteristic zero valued Dfield with angular components is L_{vdf}^{ac,0} := L_{vdf}(ac_{0}) where ac_{0} : K → k is a unary function symbol which will be interpreted as an angular component function of level 𝔪_{K} (and ac_{0}(0) := 0).

The language appropriate for mixed characteristic valued D-fields with angular components is more complicated to describe. First, we need to add new sorts, R_{n}, one for non-negative integer n. We identify R_{0} with k. Each R_{n} is treated as an L_{D−ring}(σ, σ^{−1})-structure. We add new functions connecting the sorts, π_{n} : K → R_{n} ∪ {∞}, π_{m,n} : R_{m} → R_{n} and ac_{n} : K → R_{n} for 0 ≤ n < m < ω. We will interpret R_{n} as O_{K}/p^{n+}O_{K} where p^{n+}O_{K} := {x ∈ O_{K} : v(x) > nv(p)}, the maps π_{n} and π_{m,n} as the natural projections, and {ac_{n}}_{n∈ω} as a system of angular component functions for {I_{n}}_{n∈ω}. We call this language L_{vdf}^{ac,ω}.

The languages for leading term structures are defined similarly. The language involving just leading terms of level 𝔪_{K} is obtained from L_{vdf} by adjoining a new sort K_{0} to be treated as an L_{D−ring}(σ, σ^{−1})-structure and a new function symbol π_{0} : K → K_{0}. Moreover, we extend the function v to be defined on K_{0} as well and treat k as a substructure of K_{0}. We will interpret K_{0} as K_{𝔪} ∪ {0}. The inclusion k ↪ K_{0} comes from the valuation exact sequence on K_{𝔪} together with 0 ↦ 0. We call this language L_{vdf}^{lt,0}.

As with the angular components, the mixed characteristic leading term language is more complicated than the pure characteristic zero leading term language. We add new sorts K_{n} to be treated as L_{D−ring}(σ, σ^{−1})-structures and function symbols π_{n} : K → K_{n}, π_{m,n} : K_{m} → K_{n}, D_{m,n} : K_{m} → K_{n}, and +_{m,n} : K_{m}^{2} → K_{n}. We identify k with K_{0} as in L_{vdf}^{lt,0} and we extend v to each K_{m}. Since we will interpret K_{m} as K_{I_{m}}, we no longer insist that + and D be total functions. Perhaps, one might like to think of the new sorts as simply L(·, σ, σ^{−1}, 1, e)-structures and the partial functions +_{n,m} and D_{n,m} as merely relations. Instead, we regard the value of any of our partial functions applied to a point where it is not defined to be ∞. This language is denoted by L_{vdf}^{lt,ω}.

In the following sections we prove quantifier elimination for certain valued Dfields relative to the residue field, value group and leading terms. Syntactically, this means that we allow expansions of the languages L_{vdf}^{ac,0}, L_{vdf}^{lt,0}, L_{vdf}^{ac,ω}, and L_{vdf}^{lt,ω} by predicates on the sorts k, Γ, R_{n}, or K_{n}. There are cases in which these additional predicates are not needed (K a D-henselian field with divisible value group and quantifier eliminable differential field of characteristic zero as residue field), but every field considered in this paper not already considered in [5] requires the new predicates.

4. Axioms

The class of valued D-fields considered in this paper is larger than that of [5] due to the relaxation of three conditions on the residue field. First, we no longer insist that the residue characteristic is zero. Secondly, we now allow for the possibility of v(e) = 0. Finally, we do not demand that the residue field be closed under roots. We require only that the residue field be linearly D-closed. As shown in [5], this is an intrinsic property of D-henselian fields. We also insist on generic characteristic zero. It ought to be possible to relax this condition as well, but given the current state of knowledge about the model theory of pure valued fields of positive characteristic, I expect this would take significantly new methods.

The axioms for valued D-fields given in [5] do not suffice for the more general valued D-fields we consider here. Since the language has been expanded and the conditions on the residue field relaxed, we need to modify the axioms for valued D-fields. In stating the axioms, we give the axioms for valued D-fields in the basic language of valued D-fields, L_{vdf}.

The first three axioms describe valued D-fields.

Axiom 1. K is D-field, k is a D-field and Γ is an ordered abelian group. [Note: we require a D-field to be an L_{D−ring}(σ, σ^{−1})-structure in which σ is an automorphism with inverse σ^{−1} satisfying the equation σ = eD + id.]

Axiom 2. The inequality v(Dx) ≥ v(x) holds universally as does the equality v(x) = v(σ(x)).

Axiom 3. K is a valued field with value group a subgroup of Γ via v and residue field a subfield of k via π. The map π restricted to O_{K} is a map of L_{D−ring}(σ, σ^{−1})- structures.

The next there axioms finish the description of D-henselian fields.

Axiom 4. The maps π and v are surjective.

Axiom 5. K has enough constants: v((K^{D})^{×}) = Γ. That is, for any γ ∈ Γ there is some x ∈ K with Dx = 0 and v(x) = γ.

Axiom 6. D-Hensel’s Lemma: If P (X) ∈ O_{K}⟨X⟩_{D} and a ∈ O_{K} with v(P (a)) > 0 = v(\frac{∂}{∂X_{i}} P (a)) for some non-negative integer i, then there is some b ∈ O_{K} with P (b) = 0 and v(a − b) > 0.

A valued D-field with an angular component function (of level zero) is a valued D-field given together with an angular component function ac_{0} : K → k of level 𝔪. Likewise, a valued D-field with a system of angular component functions is an expansion of a valued D-field to the language L_{vdf}^{ac,ω} in which the angular component function symbols are interpreted as a system of angular components for {I_{n} := {x ∈ O_{K} : v(x) > nv(p)}}_{n∈ω}.

Remark 4.1. In [5] the axioms for a valued D-field are given relative to a fixed theory of the residue field and value group. We present a semantic version of that completeness result below. The syntactic form is somewhat more complicated for mixed characteristic valued D- fields. In particular, it does not suffice to specify the theory of the residue field and of the value group in order to give a completion of the theory of D-henselian fields.

Remark 4.2. Proposition 5.3 of [5] applies to the modified axioms for D-henselian fields also. Thus, if K is a D-henselian field, its residue field, k, is linearly D-closed.

Our main theorem is, roughly, that the completions of theory of D-henselian fields depend on the theories of the auxiliary sorts (and their inter-relations) and that relative to the auxiliary theories we have quantifier elimination. The precise formulations of this theorem is slightly different for each of the languages introduced in the previous section.

Theorem 4.3. Let K be a D-henselian field of characteristic zero. The theory of K in L_{vdf}^{ac,ω} is determined by the theory of D-henselian fields of characteristic zero, the theory of ⋃_{n∈ω} R_{n}(K), the theory of Γ_{K} and the induced structure on ℤ[e]. The theory of K in L_{vdf}^{lt,ω} is determined by the theory of D-henselian fields of characteristic zero, the theory of ⋃_{n∈ω} K_{n}, and the induced structure on ℤ[e]. Moreover, if L ⊇ L_{vdf}^{ac,ω} is an expansion of the language L_{vdf}^{ac,ω} by predicates on ⋃_{n∈ω} R_{n}(K) so that ⋃_{n∈ω} R_{n}(K) eliminates quantifiers or L ⊇ L_{vdf}^{lt,ω} is an expansion of the language L_{vdf}^{lt,ω} by predicates on ⋃_{n∈ω} K_{n} so that ⋃_{n∈ω} K_{n} eliminates quantifiers, then K eliminates quantifiers in L.

Remark 4.4. Note that if K is a D-henselian field of pure characteristic zero, then the L_{vdf}^{ac,ω} and L_{vdf}^{ac,0} structures (and the L_{vdf}^{lt,ω} and L_{vdf}^{lt,0} structures) on K are essentially the same so that we can work with the simpler languages.

Theorem 4.3 is stated syntactically, but we prove it semantically. Theorem 4.3 is equivalent to the following condition (we use the same notation of the theorem):

If L is a |K|^{+}-saturated D-henselian field for which ⋃_{n∈ω} R_{n}(L) ∪ Γ(L) ≡ ⋃_{n∈ω} R_{n}(K) ∪ Γ(K) or ⋃_{n∈ω} L_{n} ≡ ⋃_{n∈ω} K_{n}, A ⊆ K is a countable substructure, b ∈ K, and f : K → L is a partial L-isomorphism with dom(f) = A, then f extends to a partial isomorphism \overline{f} : K → L with b ∈ dom(\overline{f}).

There are other versions of this test and we will use at least one of these. For example, we may assume GCH and that K and L are saturated of the same cardinality, and then conclude that f extends to an isomorphism between K and L.

5. Reduction to Pure Characteristic Zero

We have stated on many occasions that the theory of the relative Frobenius motivates our study of more general D-henselian fields. In this section we show that the relative Frobenius fits into the framework of D-henselian fields and that standard coarsening arguments permit us to reduce our general problem to the case of pure characteristic zero D-henselian fields.

We start with some observations about the relation between linear D-closedness, linear difference closedness, and D-henselianness.

Lemma 5.1. Let (K, σ) be a difference field. Let e ∈ K^{×} be a unit. Give K the structure of a D-field by setting D(x) := \frac{σ(x)−x}{e}. Then, K is linearly Dclosed if and only if K is linearly difference closed (ie for any nonzero polynomial

∑_{j=0}^{m} a_{j}X^{j} ∈ K[X] the linear difference operator ∑_{j=0}^{m} a_{j}σ^{j} is surjective on K).

Proof:

An easy induction shows for any natural number n that D^{n} = (∏_{j=0}^{n−1} σ^{j}(e))σ^{n} + {lower order terms} and σ^{n} = (∏_{j=0}^{n−1} σ^{j}(e)^{−1})D^{n} + {lower order terms}. Thus, for any sequence (a_{0}, …, a_{m}) ∈ K^{m+1} we have

∑_{j=0}^{m} a_{j}D^{j} = a_{m}( ∏_{j=0}^{m−1} σ^{j}(e))σ^{m} + lower order terms

∑_{j=0}^{m} a_{j}σ^{j} = a_{m}( ∏_{j=0}^{m−1} σ^{j}(e)^{−1})D^{m} + lower order terms

So, if K is linearly difference closed and a_{m} ≠ 0, then the operator ∑_{j=0}^{m} a_{j}D^{j}is

equal to a difference operator having a non-zero leading co-efficient and is therefore surjective. Likewise, linear D-closedness implies linear difference closedness. ⊣

Corollary 5.2. If K is an algebraically closed field of characteristic p > 0, e ∈ K^{×}, n ∈ ℤ \ \{0\} and D : K → K is defined by D(x) := \frac{x^{p^{n}} −x}{e}, then K is linearly Dclosed.

Lemma 5.3. If (K, v, D) is D-henselian, then K is linearly D-closed.

Proof: Let L = ∑_{j=0}^{n} a_{j}D^{j} ∈ K[D] be a non-zero linear D-operator. Let α ∈ K.

If α = 0, then L(0) = 0 = α so that we may assume that α ≠ 0. Let ϵ ∈ K

with Dϵ = 0 and v(ϵ) = max\{v(α) − v(a_{j})\}. Let Q(Y) := ∑_{j=0}^{n} \frac{a_{j}ϵ}{α}D^{j}(Y) − 1. By

our choice of ϵ, Q(Y) ∈ O_{K}⟨X⟩_{D} and the reduction of Q, π(Q), is a non-constant affine D- polynomial over the residue field of K. Since the residue field of K is linearly D-closed, we can find some b ∈ O_{K} such that π(Q(b)) = 0. As π(Q) is residually affine, v(\frac{∂}{∂X_{i}} Q(b)) = 0 for some i. Thus, DHL applies to Q at b and we can find some a ∈ O_{K} with v(a − b) > 0 and Q(a) = 0. Let ã := aϵ. Then L(ã) = α(Q(a) + 1) = α. Thus, L is surjective on K. ⊣

As with pure fields, D-Hensel’s lemma takes many forms. In the next lemma we show the equivalence between the version of D-Hensel’s lemma already given and an ostensibly stronger version in which the approximate solution is not assumed to be a simple solution in the residue field.

Lemma 5.4. Let (K, v, D) be a valued D-field with enough constants and linearly D-closed residue field. The following are equivalent.

1. For all D-polynomials P (X) ∈ O_{K}⟨X⟩ and elements a ∈ O_{K} such that v(P (a)) > 0 = min\{v(\frac{∂}{∂X_{i}} P (a)) : i ∈ ω\}, there is some b ∈ O_{K} with P (b) = 0 and v(a − b) = v(P (a)).

2. Given P (X) ∈ O_{K}⟨X⟩ and a ∈ O_{K} define γ := min\{v(\frac{∂}{∂X_{i}} P (a)) : i ∈ ω\}. If v(P (a)) > 2γ, then there is some b ∈ O_{K} with P (b) = 0 and v(a − b) = v(P (a)) − γ.

Proof: The implication from (2) to (1) is immediate so we concentrate on proving (1) to (2).

Let P (X) ∈ O_{K} and a ∈ O_{K} be given with γ = min\{v(\frac{∂}{∂X_{i}} P (a)) : i ∈ ω\} having v(P (a)) > 2γ.

If P (a) = 0, then there is nothing to prove so we assume now that P (a) ≠ 0. Let ϵ ∈ K^{D} with v(ϵ) = v(P (a)) − γ. Let Q(Y) := \frac{1}{P (a)}P (a + ϵY). I claim that

Q(Y) ∈ O_{K}⟨Y ⟩_{D}. To see this, expand P (a + ϵY) ≡ P (a) + ∑ \frac{∂}{∂X_{i}} P (a)ϵD^{i}Y + (ϵ^{2}). Of course, P (a) divides P (a), so the constant term of Q(Y) is integral. By our choice of ϵ and the definition of γ, v(\frac{∂}{∂X_{i}} P (a)ϵ) = v(\frac{∂}{∂X_{i}} P (a)) + v(ϵ) ≥ v(P (a)). Thus, the linear term of Q(Y) is integral. Finally, we have v(ϵ^{2}) = 2v(P (a))−2γ = v(P (a)) + (v(P (a)) − 2γ) > v(P (a))

which implies that the higher terms in Q(Y) have co-efficients in the maximal ideal of O_{K}.

So, not only is Q(Y) integral, but its reduction is a nonzero affine D-polynomial. As the residue field is linearly D-closed, there is some b ∈ K such that v(Q(b)) > 0 and v(b) = 0. By DHL in the original form, there is some c ∈ O_{K} such that Q(c) = 0 and v(c − b) = v(Q(b)). Set d := a + ϵc. Then P (d) = 0 and v(a − d) = v(ϵ) + v(c) = v(ϵ) = v(P (a)) − γ. ⊣

Lemma 5.5. If (K, v, D) is a D-henselian field and w is a coarsening of the valuation v, then (K, w, D) is also a D-henselian field.

Proof: We check the axioms.

Let w̃ : vK → wK be the homorphism for which w = w̃ ∘v. Note that w̃ is order preserving.

Axiom 1 makes no mention of the valuation so it remains true.

For Axiom 2 let x ∈ K, then by hypothesis v(Dx) ≥ v(x) and v(σ(x)) = v(x). Applying w̃, we obtain w(Dx) = w̃ ∘ v(Dx) ≥ w̃ ∘ v(x) = w(x) and w(σ(x)) = w̃ ∘ v(σ(x)) = w̃ ∘ v(x) = w(x).

Axioms 3 and 4 state merely that the extra sorts are interpreted as the value group and residue field. This does not change upon passage from v to w.

For Axiom 5 let γ ∈ wK. Let γ̃ ∈ vK with w̃(γ̃) = γ. By Axiom 5 for v we can find ϵ ∈ K with v(ϵ) = γ̃ and Dϵ = 0. Apply w̃ and we see that w(ϵ) = w̃ ∘ v(ϵ) = w̃(γ̃) = γ.

The only axiom requiring real proof is Axiom 6, D-Hensel’s lemma. For this we use the strengthened version of D-Hensel’s lemma. If P (X) ∈ O_{K,w}⟨X⟩ and a ∈ O_{K,w} with w(P (a)) > 0 = w(\frac{∂}{∂X_{i}} P (a)) for some i, we can scale so that P (X) ∈ O_{K,v} and a ∈ O_{K,v}. (Note: this move uses the fact that (K, v, D) has enough constants.) The hypothesis that w(\frac{∂}{∂X_{i}} P (a)) = 0, does not mean that v(\frac{∂}{∂X_{i}} P (a)) = 0. Rather, because w is a refinement of v, we can conclude from this and w(P (a)) > 0 that v(P (a)) > 2v(\frac{∂}{∂X_{i}} P (a)). By the strengthened version of DHL, there is some b such that P (b) = 0 and v(a − b) = v(P (a)) − v(\frac{∂}{∂X_{i}} P (a)). We have w(a − b) = w̃ ∘ v(a − b) = w̃ ∘ v(P (a)) − w̃ ∘ v(\frac{∂}{∂X_{i}} P (a)) = w(P (a)) as required. ⊣

With the above lemmata in place, we can reduce the our main problem to the case of pure characteristic zero valued D-fields.

Proposition 5.6. If Theorem 4.3 is true for D-henselian fields of pure characteristic zero, then it is true generally.

Proof: We deal with valued D-fields in the leading term language first, but much of the argument applies equally well to the situation of angular component functions.

We start with two mixed characteristic valued D-field K and L satisfying the hypotheses of Theorem 4.3. We may (and do) assume that K is ℵ_{1}-saturated and that the map f is actually an isomorphism between ⋃_{n∈ω} K_{n} and ⋃_{n∈ω} L_{n}. Let w_{K} be the valuation on K having valuation ring O_{K}[\frac{1}{p}] and w_{L} the valuation on L with valuation ring O_{L}[\frac{1}{p}]. The structures (K, w_{K}, D) and (L, w_{L}, D) are valued D-fields of pure characteristic zero. Our task is to show that the hypotheses of Theorem 4.3 are true of these structures and that an embedding with respect to an expansion of L_{vdf}^{lt,0} by predicates on w-zero leading terms induces an embedding of K into L for L_{vdf}^{lt,ω}.

The saturation hypotheses on L and K imply that we may recover the zero leading term structure on K with respect to w_{K} (respective, on L with respect to w_{L}) from the p^{n}-leading term structures. That is, ℵ_{1}-saturation implies that the natural maps

K^{×}/1 + 𝔪_{K,w_{K}} ⟶^{ψ_{K}} lim_{n→∞^{⟵}} K^{×}/1 + p^{n+}O_{K}

and

L^{×}/1 + 𝔪_{L,w_{L}} ⟶^{ψ_{L}} lim_{n→∞^{⟵}} L^{×}/1 + p^{n+}O_{L}

are isomorphisms. These maps preserve more than just the multiplicative structure. They are isomorphisms of L_{D−ring}(σ, σ^{−1})-structures as well: if x, y ∈ K and w_{K}(x + y) = w_{K}(x) = w_{K}(y), then for some n ∈ ω we have v(x + y) < min{v(x) + nv(p), v(y) + nv(p)} so that for m > n the expression π_{2m}(x) +_{2m,m} π_{2m}(y) = π_{m}(x + y) is well-defined. Likewise, the D and σ-structure on K_{𝔪_{K,w_{K}}} is determined by the structure on {K_{n}}_{n∈ω}. Thus, the map f induces an isomorphism between the zero-leading term structure of K (with respect to w_{K}) and the zero-leading term structure of L (with respect to w_{L}). In order to meet the hypotheses of Theorem 4.3, we replace L by a |K|^{+}-saturated elementary extension M of L. (Note that (L, w_{L}) is not even ℵ_{1}-saturated.) By Lemma 5.5, (L, w_{L}, D) is D-henselian. So, assuming the theorem in pure characteristic zero we obtain an embedding g : (K, w_{K}, D) → (M, w, D).

We use now the flexibility in the choice of L. The leading term structure inherits a trace of the valuation v on K. That is, we may (and do) continue to consider K_{0} has having the valuation v as well as w_{K}. Thus the embedding K → M can be taken to preserve v as well!

In the language with the angular component functions, we need to work to produce an angular component function for the coarsened valuation. In this case, the L_{vdf}^{ac,0} structure is not canonically determined by the L_{vdf}^{ac,ω} structure. However, the indeterminacy may be traced to the choice of a section of K^{×}/O_{K,v}^{×} → K^{×}/O_{K,v}[\frac{1}{p}]^{×} so that we can keep it under control.

The ℵ_{1}-saturation of K ensures that the natural map

O_{K,v}/p^{∞}O_{K,v} ⟶ lim_{n→∞^{⟵}} O_{K,v}/p^{n+}O_{K,v}  lim_{n→∞^{⟵}} R_{n}(K)

is an isomorphism. The angular component functions

K^{×} ⟶^{ac_{n}} (O_{K}/p^{n+}O_{K})^{×}

patch together to give a section

K^{×}/1 + p^{∞}O_{K} ⟶^{α} (O_{K}/p^{∞}O_{K})^{×}

of the inclusion (O_{K}/p^{∞}O_{K})^{×} ↪ K^{×}/1 + p^{∞}O_{K}. However, α is not an angular component function. For α to be an angular component function we would need α restricted to O_{K}[\frac{1}{p}]^{×}/1 + p^{∞}O_{K} to be the identity, but this is not the case as, for instance, α(p) = 1.

The coarsened valuation w_{K} corresponds to the exact sequence

1 → O_{K}[\frac{1}{p}]^{×}/O_{K}^{×} → K^{×}/O_{K}^{×} → K^{×}/O_{K}[\frac{1}{p}]^{×} → 1
| |  | |  | |
1 →  Γ_{w|v}  →  Γ_{v}  →  Γ_{w}  → 1

Let ψ : Γ_{v} → Γ_{w|v} be a section of the inclusion Γ_{w|v} ↪ Γ_{v}.

From the splittings

K^{×}/1 + p^{n}O_{K} ⟶^{ac_{n}} (O_{K}/p^{n}O_{K})^{×}

we obtain splittings

Γ_{v} ⟶^{χ_{n}} K^{×}/1 + p^{n}O_{K}

of the sequences

1 ⟶ (O_{K}/p^{n}O_{K})^{×} ⟶ K^{×}/1 + p^{n}O_{K} ⟶^{v} Γ_{v} ⟶ 1

. Let χ := lim_{n→∞^{⟵}} χ_{n} : Γ_{v} → lim_{n→∞^{⟵}} K^{×}/1 + p^{n}O_{K} = K_{𝔪_{K,w}}. Let β : K^{×} →

(O_{K}[\frac{1}{p}]/p^{∞}O_{K}[\frac{1}{p}])^{×} be defined by β(x) := α(x)χ(ψ(v(x))). This function will serve as ac_{0} for the coarsened valuation. We check now that it has the requisite properties.

First, we remark that β does take values in (O_{K}[\frac{1}{p}]/p^{∞}O_{K}[\frac{1}{p}])^{×} as claimed. For any x ∈ K we have ψ(v(x)) ∈ Γ_{w|v} so that χ_{n}(ψ(v(x)) ∈ O_{K}[\frac{1}{p}]^{×}/1+p^{n}O_{K}. Thus, χ(ψ(v(x)) ∈ O_{K}[\frac{1}{p}]^{×}/1 + p^{∞}O_{K} = O_{K}[\frac{1}{p}]^{×}/1 + p^{∞}O_{K}[\frac{1}{p}] = (O_{K}/p^{∞}O_{K}[\frac{1}{p}])^{×}. By construction, α(x) ∈ O_{K}^{×}/1 + p^{∞}O_{K} ↪ (O_{K}[\frac{1}{p}]/p^{∞}O_{K}[\frac{1}{p}])^{×}. Thus, β(x) ∈ (O_{K,w}/𝔪_{K,w})^{×}.

Secondly, β is a homorphism. This is clear from the construction.

Thirdly, β is a section. Let x ∈ (O_{K,w}/𝔪_{K,w})^{×}. Write x = x̃χ(v(x)). Note that v(x̃) = 0 so that x̃ ∈ O_{K,v}^{×}/1 + 𝔪_{K,w}. We compute

β(x) = α(x̃ · χ(v(x))) · χ(v(x̃ · χ(v(x))))
= α(x̃) · α(χ(v(x))) · χ(v(x̃)) · χ(v(χ(v(x))))
= x̃ · 1 · 1 · χ(v(x))
= x

As noted in the section on leading terms, the fact that β is a section of (O_{K,w}/𝔪_{K,w})^{×} ↪ K_{𝔪_{K,w}} implies already that β preserves addition as far as this makes sense.

Fourthly, β preserves the difference structure.

β(σ(x)) = α(σ(x)) · χ(ψ(v(σ(x))))
= lim_{n→∞^{⟵}} ac_{n}(σ(x)) · lim_{n→∞^{⟵}} χ_{n}(ψ(v(x)))
= lim_{n→∞^{⟵}} σ(ac_{n}(x)) · lim_{n→∞^{⟵}} χ_{n}(ψ(v(x)))
= σ(α(x)) · χ(ψ(v(x)))
= σ(α(x) · χ(ψ(v(x)))
= σ(β(x))

The penultimate equality uses the fact that the image of χ is contained in the fixed field of σ. Why is this? An element x is equal to χ(γ) if and only if v(x) = γ and α(x) = 1. The element σ(x) satisfies the same defining conditions.

Finally, β preserves the D-structure as far as this makes sense. The section χ also maps to D-constants as we have the functional equality α∘χ ≡ 1. Thus, for any γ we have w(D(χ(γ))) > w(χ(γ)) = γ for otherwise α(D(χ(γ))) = D(α(χ(γ))) = D(1) = 0 which is impossible.

So, let x ∈ K^{×}/1 + 𝔪_{K,w}. Let x̃ ∈ K^{×} lift x. We suppose that w(Dx̃) = w(x̃). This means that v(Dx̃) = v(x̃) + γ < v(x̃) + nv(p) for some n ∈ ℤ_{+}. Write x = x̄χ(v(x)). Let ˜ x̄ ∈ K^{×} lift x̄. Then as w(Dy) > w(y) for any y lifting χ(v(x)) we have Dx̃ = (D˜x̄) · χ(v(x))(1 + 𝔪_{K,w}). We compute

β(Dx̃) = β((D˜x̄)χ(v(x)))
= β(D˜x̄)χ(ψ(v(χ(v(x))))
= α(D˜x̄)χ(ψ(v(D˜x̄)))χ(ψ(v(x)))
= Dα(x̄)χ(ψ(γ))χ(ψ(v(x)))
= Dα(x̄)χ(ψ(γ + v(x))
= D(α(x̄) · χ(ψ(v(Dx̃))))
= D(β(x))

We now have a way of regarding (K, w) and an L_{vdf}^{ac,0}-structure and can complete the argument as we did for L_{vdf}^{lt,0}. That is, we assume GCH and that K and L are saturated of the same cardinality (which, it follows, must be at least ℵ_{1}). From the hypotheses that Γ_{K} ≡ Γ_{L} and that the auxiliary structure of K admits quantifier elimination in L, we conclude that f is compatible with an isomorphism of the value groups vK = Γ_{K} ≅ Γ_{L} = vL. Relative to this isomorphism we choose the same splitting ψ for both K and L. The L-embedding f : A → L is then also an embedding relative to the coarsened valuation. As before, we elementarily embed L into a |K|^{+}-saturated model M and extend f to \overline{f} : A(⟨b⟩) → M using the hypothesis that the theorem is true for pure characteristic zero D-henselian fields. As above, we conclude that \overline{f} is also an L-embedding. ⊣

6. The proofs in equicharacteristic zero

In the previous section we reduced the proof of the main theorem to the case of pure characteristic zero D-henselian fields. The proof we present here is a reworking of [5]. The reader should have a copy of that paper in hand as we constantly refer to it. As in that paper, we assume that L and K are ℵ_{1}-saturated and that we have an isomorphism given between a countable substructure of K and of L and that we aim to extend the isomorphism to some specified new element. The strategy for this extension is precisely that of [5]. We need to strengthen the conclusions of most of the lemmata proven there, but the proofs do not change substantially. Since most of this proof follows the proof of [5], we indicate the corresponding lemmata and propositions in parentheses. For the remainder of this section, “valued D-field” means “valued D-field of equicharacteristic zero.”

Lemma 6.1 (Lemma 7.4 for L_{vdf}^{ac,0}). If K is a valued D-field with enough constants, Q_{1}, …, Q_{n} ∈ K[X] are polynomials over K with Q_{i} = ∑ q_{i,j}X^{j}, and γ ∈ Γ_{K}, then there is some ϵ ∈ K with Dϵ = 0, v(ϵ) = γ, and v(Q_{i}(ϵ)) =

min{v(q_{i,j}) + jγ}, and ac_{0}(Q_{i}(ϵ)) = ∑_{{j:v(Q_{i}(ϵ))=v(q_{i,j})+jγ}} ac_{0}(q_{i,j})ac_{0}(ϵ)^{j} for each i.

The version of this lemma for L_{vdf}^{lt,0} is almost the same.

Lemma 6.2 (Lemma 7.4 for L_{vdf}^{lt,0}). If K is a valued D-field with enough constants, Q_{1}, …, Q_{n} ∈ K[X] are polynomials over K with Q_{i} = ∑ q_{i,j}X^{j}, and γ ∈ Γ_{K}, then there is some ϵ ∈ K with Dϵ = 0, v(ϵ) = γ, and v(Q_{i}(ϵ)) = min{v(q_{i,j}) + jγ}, and

π_{0}(Q_{i}(ϵ)) = ∑_{{j:v(Q_{i}(ϵ))=v(q_{i,j})+jγ}} π_{0}(q_{i,j})π_{0}(ϵ)^{j} for each i.

Proof: Let ϵ be given by the original Lemma 7.4. The stronger conclusions follow from the facts that ac_{0} and π_{0} respect addition as long as the valuation of the sum is equal to the minimal valuation of a summand. ⊣

Remark 6.3. To pin down the structure on K(ϵ) one needs to know more about ϵ than just that it is a D-constant with valuation γ. At the very least, one needs to know ac_{0}(ϵ) or π_{0}(ϵ).

Lemma 6.4 (Lemma 7.8 for L_{vdf}^{ac,0}). Let p ∈ S_{1,Γ}(Γ_{K}) be a one-type in the valuation group sort extending {nx ≠ γ : n ∈ ℤ_{+}, γ ∈ v(K^{×})}. There is a unique (up to L_{vdf}^{ac,0}-isomorphism)structure of a valued D-field on K(x) such that v(x) ⊧ p, Dx = 0, ac_{0}(x) = 1, and for any polynomial P (X) = ∑ p_{j}X^{j} ∈ K[X] we have v(P (x)) = min{v(p_{j}) + jv(x)} and ac_{0}(P (x)) = ac_{0}(p_{j}) where v(P (x)) = v(p_{j}) + jv(x).

Lemma 6.5 (Lemma 7.8 for L_{vdf}^{lt,0}). Let K be a valued D-field. Let p ∈ S_{1,K_{0}}(K_{0}) be a one-type in the leading term sort extending {nv(x) ≠ γ : γ ∈ v(K^{×}), n ∈ ℤ_{+}} and Dx = 0. There is a unique structure of a valued D-field on K(x) such that π_{0}(x) ⊧ p, Dx = 0 , and for any polynomial P (X) = ∑ p_{j}X^{j} ∈ K[X] we have v(P (x)) = min{v(p_{j}) + jv(x)} and π_{0}(P (x)) = π_{0}(p_{j})π_{0}(x)^{j} where v(P (x)) = v(p_{j}) + jv(x).

Proof: As in the original proof, the valued D-field structure (in the languages L_{vdf}^{ac,0} or L_{vdf}^{lt,0}) is determined by our hypotheses. The verification that these prescriptions give valued D-fields is routine. ⊣

The next lemma is the version of Lemma 7.11 of [5] in the strengthened languages. The statement is (superficially) precisely the same as the original statement.

Lemma 6.6 (Lemma 7.11). If K is a valued D-field, R is the henselization of O_{K}, and L is the field of fractions of R, then L has a unique structure of a valued D-field extending K.

The above statement means in particular that if K is considered as an L_{vdf}^{ac,0}- structure (L_{vdf}^{lt,0}-structure, respectively), then the L_{vdf}^{ac,0}_{K}-structure (L_{vdf}^{lt,0}_{K}-structure, respectively) on L is determined.

Proof: There are two changes to the proof.

First, Proposition 2.8 of [5] does not apply when v(e) = 0. However, in that case we work interchangeably with σ and σ^{−1}. The map σ : R → R ↪ R^{h} induces a map σ : R^{h} → R^{h} as does σ^{−1} : R → R ↪ R^{h}.

Secondly, the henselization R^{h} of R has the property that for any x ∈ R^{h} there is some y ∈ R with π_{0}(x) = π_{0}(y). Hence, the henselization does not change K_{0} and any equation involving ac_{0}(x) may be understood in terms of ac_{0}(y). ⊣

Lemma 6.7 (Lemma 7.12). Let K be a valued D-field. Given a type p ∈ S_{1,k}(k_{K}) and a D-polynomial P ∈ O_{K}⟨X⟩ such that if x ⊧ p, then

• π(P)(x) = 0,

• P and π(P) have the same total degree, and

• if Q(X) ∈ k_{K}⟨X⟩ with Q(x) = 0, then T.degP ≤ T.degQ; then there is a unique valued D-field L = K(⟨a⟩) such that P (a) = 0 and π(a) ⊧ p. When K is considered as an L_{vdf}^{ac,0}-structure (L_{vdf}^{lt,0}-structure, respectively; or even L-structure for L the language mentioned in the statement of Theorem 4.3), then L is unique up to L_{vdf}^{ac,0}_{K}-isomorphism (L_{vdf}^{lt,0}_{K}-isomorphism, L_{K}-isomorphism, respectively).

The proof of Lemma 7.12 in [5] superficially uses the hypothesis v(e) > 0, but this condition may be dropped by replacing the henselization used there with the strict henselization.

Proof: A few changes are needed. We retain the notation of [5].

The proof of uniqueness falls into two cases. When considering n ≤ m, we know that v(Q(a)) is the Gauss valuation of Q. That is, if α ∈ K with αQ ∈ O_{K}⟨X⟩ and π(αQ) ≠ 0, then v(Q(a)) = −v(α). It follows that π_{0}(Q(a)) = Q(π_{0}(a)). As ac_{0} factors through K_{0}, this pins down ac_{0}(Q(a)) as well.

The proof of Lemma 7.12 in [5], the case of n > m seems to fail for v(e) = 0. However, this is not the case. We rewrite the proof here.

As the case of v(e) > 0 was treated in [5], we assume that v(e) = 0. In this case we may work interchangeably with σ and D. So, we treat Q as a σ-polynomial instead of as a D-polynomial. As before, let G(X) := F (a, …, σ^{m−1}(a), X). We are considering the extension K(a, …, σ^{n}(a))/K(a, …, σ^{n−1}(a)).

The element σ^{n}(a) = σ^{n−m}(σ^{m}(a)) satisfies σ^{n−m}(G). Thus, the minimal polynomial Q_{n} of σ^{n}(a) over K_{n−1} divides σ^{n−m}(G). Now, σ^{m}(b) = π_{0}(σ^{m}(a)) is a simple root of π(G) and σ induces an automorphism of the zero leading terms so σ^{n}(b) is a simple root of π(σ^{n−m}(G)). [The remark in [5] that σ induces the identity on the residue field is irrelevant.] This implies that σ^{n−m}(b) is a simple root of Q_{n} as well so that the extension K_{n}/K_{n−1} is determined as an extension of valued fields by the conditions π(σ^{n−m}(a)) = σ^{n−m}(b) and Q_{n}(σ^{n−m}(a)) = 0. Moreover, the above argument shows that K_{n} is contained in the strict henselization of K_{n−1}.

When v(e) = 0, it need not be the case that K(⟨a⟩) = K_{+} := ⋃_{n∈ω} K(a, …, D^{n}a) since σ need not be surjective on K_{+}. To correct this, we need to consider the extesions of the form K_{+}(σ^{−1}(a), …, σ^{−n}(a))/K_{+}. These extensions are analyzed just as the extensions defined with positive powers of σ: work with factors of the polynomial σ^{−n}(H) where H(Y) := F (Y, σ(a), …, σ^{m}(a)).

Finally, we need to modify the existence proof.

As before, let b ⊧ p. We will produce an extension of σ to a valued difference field L and an embedding of k_{K}(⟨b⟩) into k_{L} so that we can find some a ∈ L with P (a) = 0 and π(a) = b. We then restrict to K(⟨a⟩).

Let K′ := K(X_{0} …, X_{m−1}). Give K′ the structure of a valued field (with angular components if need be) by setting v(∑ q_{α}X^{α}) := min{v(q_{α})}. Identify the residue field of K′ with k_{K}(b, …, σ^{m−1}(b)) via π(X_{j}) := σ^{j}(b). We extend ac_{0} by the equation ac_{0}(∑ q_{α}X^{α}) := ∑_{{α:v(q_{α})is minimal}} ac_{0}(q_{α}) ∏_{j=0}^{m−1} σ^{j}(b)^{α_{j}}.

Let L be the strict henselization of K. Continue the embedding k_{K}(⟨b⟩) ↪ k_{L}. Let y ∈ with F (X_{0}, …, X_{m−1}, y) = 0 and π(y) = σ^{m}(b). Note that y is uniquely determined by these conditions. Define σ : K′ → L by σ|_{K} is the original σ, σ(X_{j}) := X_{j+1} for j < m − 1 and σ(X_{m−1}) = y.

We observe that σ preserves the valuation. For this it suffices to consider el-

ements of K[X_{0}, …, X_{m−1}]. Let z = ∑_{j=0}^{ℓ} q_{j}(X_{0}, …, X_{m−2})X_{m−2}^{j}. Scaling, we

may assume that v(z) = 0. That is, q_{j} ∈ O_{K}[X_{0}, …, X_{m−1}] for each j and

π(q_{j}) ≠ 0 for some j. By definition, σ(z) = ∑_{j=0}^{ℓ} q_{j}^{σ}(X_{1}, …, X_{m−1})y^{j}. As σ

preserve the valuation on K and by the way that v is defined on K′, we see that v(q_{j}^{σ}(X_{1}, …, X_{m−1})) = v(q_{j}(X_{0}, …, X_{m−2})) for each j. As y ∈ O_{L}, we

see that v(σ(z)) ≥ 0. However, π(σ(z)) = ∑_{j=0}^{ℓ} π(q_{j}^{σ})(σ(b), …, σ^{m−1}(b))σ^{m}(b)^{j} =

σ(∑_{j=0}^{ℓ} π(q_{j})(b, …, σ^{m−2}(b))σ^{m−1}(b)^{j}) ≠ 0. Hence, v(σ(z)) = 0.

By the universal property of the strict henselization there is an extension of σ to L. The extension to L is not unique, but if we insist that it lift σ on k(⟨b⟩), then it is uniquely determined on the difference field generated by K and X_{0}.

If we let z ∈ L be the unique root to F^{σ^{−1}}(z, X_{0}, …, X_{m−1}) = 0 and π(z) = σ^{−1}(b) and define σ^{−1} : K′ → L by σ^{−1}|_{K} is the original σ^{−1}, σ^{−1}(X_{0}) := z, and σ^{−1}(X_{j}) := X_{j−1} for j > 0, then exactly as above we see that σ^{−1} extends to a valuation preserving endomorphism of L and that it is uniquely determined on the difference field generated by K and X_{0}. Consequently, σ is an automorphism of K(⟨X_{0}⟩). ⊣

Proposition 7.16 of [5] remains true in the new languages with the same proof as long as the reference to Lemma 7.12 is understood to mean the revised version, our Lemma 6.7.

The statement and proof of Proposition 7.17 requires a number of changes.

Proposition 6.8 (Proposition 7.17). Let K be a valued D-field. Assume that v(K^{×}) = v((K^{D})^{×}). Let η ∈ K^{D}. We suppose that tp(v(η)) ⊢ n|v(η) and that for no m dividing n do we have v(η) ∈ mv(K^{×}).

If K is regarded as an L_{vdf}^{ac,0}-structure, then we fix a type p(x) ∈ S_{1,k}(k_{K}) with p(x) ⊢ {Dx = 0} and p(x^{n}) = tp(ac(η)/k_{K}). We then conclude that there is a unique extension of valued D-fields of the form K(ϵ) in which ac_{0}(ϵ) ⊧ p, ϵ^{n} = η, and D(ϵ) = 0.

If K is regarded as an L_{vdf}^{lt,0}-structure, then we assume that tp(π_{0}(η)) ⊢ (∃y) y^{n} = π_{0}(η) ∧ σ(y) = y and fix some type p(x) ∈ S_{1,K_{0}}(K_{0}) with p(x^{n}) = tp(π_{0}(η)/K_{0}) and p(x) ⊢ Dx = 0. We conclude that there is a unique extension of valued D-fields of the form K(ϵ) in which ϵ ⊧ p, Dϵ = 0, and ϵ^{n} = η.

Proof: The uniqueness part of the proof is easier than before (as we have pinned down the structure).

For the existence proof, we remark merely that, writing ϵ, the prescription

ac_{0}( ∑_{j=0}^{n−1} a_{j}ϵ^{j}) := ac_{0}(a_{j}) where j is chosen so as to minimize v(a_{j}) + \frac{j}{n}v(η) cor-

rectly defines an angular component function. ⊣

Remark 6.9. This is the unique place in the proof where angular components or leading terms make any difference at all!

We need a stronger version of Proposition 7.18. The following proposition is a combined version of Propositions 7.18 and 7.19.

Proposition 6.10 (Propositions 7.18 and 7.19). The isomorphism extends so that, writing A for the domain of the isomorphism, for any a ∈ A^{×} one may find some b ∈ A with Db = 0, v(a) = v(b), and for any positive integer n if n divides the valuation of b, then there is some c_{n} ∈ A with c_{n}^{n} = b and Dc_{n} = 0.

Proof: Let a be in A, the current domain of f, so that there is no b ∈ A already satisfying the stated conclusions.

By Axiom 5 and compactness we can find a system of types p_{ℓ}(x) ∈ S_{1,K_{0}}(A_{0}) indexed by those positive integers ℓ with tp(v(a)) ⊢ ℓ|v(a) such that

• p_{1}(x) ⊢ v(x) = v(a),

• p_{1}(x) ⊢ v(∑ b_{j}x^{j}) = min{v(b_{j})+jv(a)} for each polynomial ∑ b_{j}X^{j} ∈ A[X],

• p_{ℓ}(x) ⊢ Dx = 0, and

• x ⊧ p_{ℓm}(x) ⇒ x^{m} ⊧ p_{ℓ}.

Let \overline{ϵ_{ℓ}} ∈ p_{ℓ}(K) and \overline{η_{ℓ}} ∈ f(p_{ℓ})(L).

By Axiom 4, ℵ_{1}-saturation, and DHL, we may find ϵ_{1} ∈ K^{D} and η_{1} ∈ L^{D} such that π_{0}(ϵ_{1}) = \overline{ϵ_{1}} and π_{0}(ϵ_{2}) = \overline{(}ϵ_{2}). By Lemma 6.7, the partial isomorphism extends to ϵ_{1} ↦ ϵ_{2}.

By Axiom 4, ℵ_{1}-saturation, residue characteristic zero, and DHL we find ϵ_{ℓ} ∈ K^{D} and η_{ℓ} ∈ L^{D} so that ϵ_{ℓm}^{m} = ϵ_{ℓ}, η_{ℓm}^{m} = η_{ℓ}, π_{0}(ϵ_{ℓ}) = \overline{ϵ_{ℓ}}, and π_{0}(η_{ℓ}) = \overline{η_{ℓ}}. By Proposition 6.8 the isomorphism extends by setting \overline{f}(ϵ_{ℓ}) = η_{ℓ}. ⊣

Propositions 7.20 and 7.21 have essentially the same form and proof as in the original version.

Subsection 7.2 of [5] on immediate extensions is the most involved portion of that paper and the reader may very well fear repeating those arguments here. Certainly, you say, there must be some new obstruction buried among the pseudo-convergent sequences. Fortunately, this is not the case. The arguments in [5] (almost) suffice for what is needed here. Why? The basic reason is that immediate extensions do not affect the zero leading terms.

Lemma 6.11. If x_{α} ⇒ a, then either x_{α} ⇒ 0 or π_{0}(x_{α}) = π_{0}(a) for α ≫ 0.

Proof: If x_{α} ⇏ 0, then v(x_{α}) = v(x_{β}) for 0 ≪ α < β. As x_{α} ⇒ a, we have v(x_{α} − a) < v(x_{β} − a) = v(x_{β} − x_{α} + x_{α} − a) for α < β. By the ultrametric inequality, we conclude that v(x_{α} − a) = v(x_{β} − x_{α}) which is greater than v(x_{α}) as \{x_{α}\} is pseudo-convergent. Thus, we can write a = x_{α}(\frac{a}{x_{α}}) = x_{α}(\frac{x_{α}+a−x_{α}}{x_{α}}) = x_{α}(1 + \frac{a−x_{α}}{x_{α}}) ∈ x_{α}(1 + 𝔪_{K}). That is, π_{0}(a) = π_{0}(x_{α}). ⊣

Note that since ac_{0} factors through π_{0}, Lemma 6.11 implies that if x_{α} ⇒ a and x_{α} ⇏ 0, then ac_{0}(x_{α}) = ac_{0}(a) eventually.

The lemmata and propositions of section 7.2 of [5] remain true. There is one proof, that of Lemma 7.47, which mildly uses the hypothesis that v(e) > 0. However, this use is cosmetic. In the penultimate paragraph of that proof, the justification given for the observation that K′ embeds into the fraction field of the henselization of O_{K(a,... ,D^{m−1}a)} is that K′ and K have the same value group. However, the correct observation is that K′ is an immediate extension of the pure characteristic zero valued field K and hence an immediate finite extension of K(a, …, D^{m−1}a) so that Proposition 1.1 of [5] applies. In the case that v(e) = 0, the universal extension of the henselization should be applied to σ : O_{K(a,... ,D^{m−1}(a)} → O_{K(a,... ,D^{m−1}a)}^{h} rather than to the map φ considered in [5].

With Lemma 6.11 in place we see that the conclusions to all the lemmata and propositions of Section 7.2 also pin down the leading term and angular component structure. More precisely, Proposition 7.32 of [5] asserts (among other things) the existence and uniqueness of certain immediate extensions of valued D-fields based on the data of a minimal D-polynomial and a pseudo-convergent approximate solution. Note that prima facie the conclusion of Proposition 7.32 only applies to the language without leading terms and angular components. However, Lemma 7.41 of [5] strengthens this conclusion slightly by showing that, under a fullness hypothesis, if Q is a ≺-small D-polynomial and x_{α} ⇒ c, then Q(x_{α}) ⇒ Q(c). Thus, Lemma 6.11 shows that the leading terms (and, hence, angular components) of every element of K⟨a⟩ are determined as well.

With these observations we finish the proof as in subsection 7.3 of [5]. For the reader’s convenience, we repeat this argument.

Proof of Main Theorem: We are given K an ℵ_{1}-saturated D-henselian field of equicharacteristic zero, L another ℵ_{1}-saturated D-henselian field for which we assume either that L and K are treated as L_{vdf}^{lt,0}-structures and L_{0} ≡ K_{0} or that K and L are given as L_{vdf}^{ac,0}-structures and that k(K) ≡ k(L) and Γ(K) ≡ Γ(L), A ⊆ K a countable substructure, b ∈ K, and f : A → L is a partial isomorphism for a language obtained from L_{vdf}^{lt,0} or L_{vdf}^{ac,0} by adding predicates to the auxiliary sorts so that they eliminate quantifiers. We are charged with extending f to a partial isomorphism \overline{f} : A(⟨b⟩) → L. We actually overshoot a bit, extending f so that its domain contains a full immediate extension of some countable elementary submodel of K containing A and b. That is, we fix some B ≺ K, an elementary submodel of K containing A(⟨b⟩), and aim to extend f beyond B.

Using the modified version of Propositions 7.20 and 7.21 we extend f so that its domain has value group containing Γ(B) and residue field containing k(B) (and, hence, dom(f)_{0} ⊇ B_{0}). We then use the modified version of Proposition 7.51 to extend f to an immediate full extension of B. This finishes the proof. ✠

7. Eliminating the leading terms

The angular component functions and leading term structures used in this paper clutter our results. In this section we note that the results of the previous section suffice for a complete axiomatization of the theory of a D-henselian field of characteristic zero.

To begin, we note Theorem 4.3 implies a complete axiomatization in L_{vdf}.

Theorem 7.1. Let K be a D-henselian field of characteristic zero. Then the theory of K in L_{vdf} is determined by the theory of D-henselian fields of characteristic zero, the atomic theory of K, and the theory of the value group of K, and the theory of ⋃_{n∈ω} R_{n}(K) considered as a multi-sorted structure.

Proof: As the statement of this theorem is absolute, we may and do assume GCH. Let K′ ≻ K be a saturated elementary extension of K′. Let χ : vK′ → ((K′)^{D})^{×} be a section of the valuation on K′. If the residue characteristic of K is p, define ac_{n}(x) := \frac{x}{χ(v(x))} + p^{n+}O_{K′} for x ∈ (K′)^{×} and n ∈ ω. If the residue characteristic is zero, set ac_{0}(x) := \frac{x}{χ(v(x)} + 𝔪_{K′} for x ∈ (K′)^{×}. With these functions K′ is a Dhenselian field of characteristic zero with a system of angular component functions. By Theorem 4.3, the theory of K′ in L_{vdf}^{ac,ω} is determined by this fact and the theory of its value group and of ⋃ R_{n}(K′). Hence, the theory of its reduct to L_{vdf} is determined by the same things and thus, the same is true of K. ⊣

Remark 7.2. Theorem 7.1 is true with L_{vdf} replaced by an expansion of L_{vdf} by new predicates on Γ_{K} and on ⋃_{n∈ω} R_{n}(K).

If we specialize K, then the statement of Theorem 7.1 becomes cleaner.

Corollary 7.3. The completions of the theory of D-henselian fields of pure characteristic zero are determined by specifying the atomic theory, the theory of the residue field, and the theory of the value group.

Remark 7.4. The generalized power series construction of [5] shows that we may take any D-linearly closed field of characteristic zero and choose for e any element of non-negative valuation in the generalized power series ring.

Corollary 7.5. Let p be a rational prime. Let K := W_{p^{∞}}(𝔽_{p}^{alg})[\frac{1}{p}] be the field of fractions of the Witt vectors over 𝔽_{p}^{alg}. Let σ : K → K be the relative Frobenius. Let v : K → ℤ ∪ {∞} be the p-adic valuation on K. Then the theory of (K, v, σ) is axiomatized by saying that the function D(x) := σ(x) − x makes K into a Dhenselian field of characteristic zero with e = 1, residue field algebraically closed of characteristic p > 0, value group a Z-group with least positive element v(p), and D(x) ≡ x^{p} − x (mod p) for x ∈ O_{K}.

The reader may substitute finitely ramified extensions of W_{p^{∞}}(k)[\frac{1}{p}] where k is an algebraically closed field of characteristic p and powers of the relative Frobenius in the above corollary to obtain other axiomatizations of concrete theories of standard valued difference fields.

References

[1] J. Ax and S. Kochen, Diophantine problems over local fields. III. Decidable fields, Ann. of Math. (2) 83 (1966), 437–456.

[2] Ş. Basarab and F.-V. Kuhlmann, An isomorphism theorem for Henselian algebraic extensions of valued fields, Manuscripta Math.77 (1992), no. 2-3, 113–126.

[3] L. Belair and A. Macintyre, manuscript on the Frobenius, 1999.

[4] A. Macintyre, On definable subsets of p-adic fields, J. Symbolic Logic 41 (1976), no. 3, 605–610.

[5] T. Scanlon, A model complete theory of valued D-fields, J. Symbolic Logic (to appear).

[6] T. Scanlon, Model theory of valued D-fields, Ph. D. thesis, Harvard University, May 1997.

[7] L. van den Dries, On the elementary theory of rings of Witt vectors with a multiplicative set of representatives for the residue field, Manuscripta Math. 98 (1999), no. 2, 133–137.

University of California at Berkeley, Department of Mathematics, Evans Hall, Berkeley, California 94720, USA

   E-mail address: scanlon@math.berkeley.edu