A MODEL COMPLETE THEORY OF VALUED D-FIELDS

THOMAS SCANLON

Abstract. The notion of a D-ring, generalizing that of a differential or a difference ring, is introduced. Quantifier elimination and a version of the Ax-Kochen-Ershov principle is proven for a theory of valued D-fields of residual characteristic zero.

The model theory of differential and difference fields has been extensively studied (see for example [7, 3]) and valued fields have proven to be amenable to model theoretic analysis (see for example [1, 2]). In this paper we subject a theory of valued fields possessing either a derivation or an automorphism interacting strongly with the valuation to such an analysis. Our theory differs from C. Michaux’s theory of henselian differential fields [8] on this last point: in his theory, the valuation and derivation have a very weak interaction.

In Section 1 we introduce the notion of a D-field and show that a differential ring may be regarded as a specialization of a difference ring. This formal connection supports the view that differential and difference algebra are instances of the same theory. We introduce our axioms in Section 5 and prove quantifier elimination in Section 7. This provides an example of a non-trivial difference ring admitting elimination of quantifiers in a natural language. Differential fields possessing a valuation compatible with the derivation in some way have appeared in many guises. In model theory, these fields have arisen as Hardy fields associated to O-minimal structures. However, in contrast to Hardy fields, the fields considered in this paper have the property that for each value in the value group there is some differential constant with that valuation. This restriction is intrinsic to the methods used here.

This paper derives from a chapter of my doctoral thesis [11] written under the direction of E. Hrushovski whom I now thank for his advice and careful reading of preliminary versions of this paper. I thank the referee for a very thorough reading of this paper, for detailed and constructive suggestions for improvements and for supplying a correct proof of Proposition 1.1.

1. Algebraic Preliminaries

In this paper, a valued field is a field K given together with a function v : K → Γ ∪ {∞} where Γ is an ordered abelian group and ∞ is a formal symbol defined by (∀γ ∈ Γ) (∞ > γ) ∧ (∞ + γ = ∞ = ∞ + ∞ = γ + ∞). v must satisfy v(x) = ∞ ⟺ x = 0, v(x · y) = v(x) + v(y), and v(x + y) ≥ min{v(x), v(y)}. The ring of integers of K is O_{K} := {x ∈ K : v(x) ≥ 0}. O_{K} is a local ring with maximal ideal 𝔪_{K} := {x ∈ K : v(x) > 0}. The residue field of K is k_{K} := O_{K}/𝔪_{K}. The quotient map π : O_{K} → k_{K} is called the residue map. If R is any (unital) ring then we denote the units of R by R^{×} := {x ∈ R : (∃y ∈ R) xy = 1}. We may recover Γ as K^{×}/O_{K}^{×}. We call an extension (L, w)/(K, v) of valued fields immediate if w(L^{×}) = v(K^{×}) and π(O_{L}) = π(O_{K}). As described in Section 4, we think of a valued field as a three-sorted structure: (K, k_{K}, Γ).

Date: 22 May 1997 revised 23 April 1999.

Let R be a commutative local ring with maximal ideal 𝔪. Let π : R → R/𝔪 denote the reduction map. R is said to be henselian if Hensel’s Lemma is valid in R. That is, whenever P (X) ∈ R[X] and a ∈ R such that π(P′(a)) ≠ 0 = π(P (a)), then there is some b ∈ R such that P (b) = 0 and π(a) = π(b). A henselization of R is a local homomorphism φ : R → R^{h} such that R^{h} is henselian and whenever R → S is a local homomorphism from R to a henselian ring S, then there is a unique local homomorphism R^{h} → S such that the following diagram commutes.

R ⟶ R^{h}
↓ ↙ ∃!
S

For a proof that henselizations exist, see [10].

By all rights the next proposition should be well known, but we could find no published proof. A proof of this proposition follows from Lemma 4.5 of the preprint [6]. The proof given below was supplied by the referee replacing my original erroneous proof.

Proposition 1.1. Let (K, v) be a valued field of equicharacteristic zero. Let (L, w) be a finite immediate extension of K. There is some α ∈ O_{L} integral over O_{K} such that L = K(α) and if P (X) ∈ O_{K}[X] is the minimal monic polynomial of α over O_{K}, then w(P′(α)) = 0.

■ Let M be a normal closure of L. Let w′ be an extension of w to M. Let R_{L} be the integral closure of O_{K} in L and R_{M} be the integral closure of O_{K} in M. Let 𝔭 := {x ∈ R_{M} : w′(x) > 0} be the prime ideal of w′ in R_{M}.

Claim 1.2. If σ ∈ Gal(M/K) \ Gal(M/L), then σ(𝔭) ∩ R_{L} ≠ 𝔭 ∩ R_{L}.

✠ If σ(𝔭) ∩ R_{L} = 𝔭 ∩ R_{L}, then the valuations w′ and w′ ∘ σ^{−1} agree on L. Since the extension M/L is Galois, Gal(M/L) acts transitively on the set of extensions of w to M (see 14.1 of [4]). Thus, w′ ∘ σ^{−1} = w′ ∘ τ for some τ ∈ Gal(M/L), but then w′ = w′∘(τσ). Since the extension w of v to L is immediate, the residual and ramification degrees of the conjugates of w′ over w are the same as they are over v. Call these common degrees f and e. As (K, v) and (L, w) have residual characteristic zero, they are each defectless in M (see Corollary 20.23 of [4]). Thus there are exactly \frac{[M:K]}{ef} conjugates of w′ under the action of Gal(M/K) and \frac{[M:L]}{ef} conjugates of w′ under the action of Gal(M/L). We conclude that the decomposition group of w′ over K, the isotropy group of w′ for the action of Gal(M/K), is a subgroup of Gal(M/L). So we must have τσ ∈ Gal(M/L) which implies that σ ∈ Gal(M/L) contrary to our hypothesis. ✠

By the Chinese remainder theorem we can find α ∈ R_{L} such that α ∈ σ(𝔭)∩R_{L} ⇔ σ ∈ Gal(M/L). Such an α works. Since σ(α) = α is only possible if α ∈ σ(𝔭) in which case σ ∈ Gal(M/L) we have the equality of fields L = K(α). The element α reduces to zero with respect to w′ but every other conjugate of α over K is a w′-unit. Thus, 0 is a simple root of the reduction of the minimal monic polynomial of α over O_{K}. ■

2. D-rings

To keep with the notation of the following sections, we use “e” rather than, say, “X”, to denote an indeterminate.

Let R be a commutative ℤ[e]-algebra. D_{e}(R) is the ring which as an abelian group is R^{2} with multiplication defined by (x_{1}, x_{2})∗(y_{1}, y_{2}) := (x_{1}y_{1}, x_{1}y_{2} +y_{1}x_{2} +ex_{2}y_{2}). D_{e} defines a functor from the category of commutative ℤ[e]-algebras to the category of commutative rings.

The projection onto the first co-ordinate defines a ring homomorphism π_{0} : D_{e}(R) → R. A D_{e}-structure on R is given by a section of this projection map. Concretely, such a structure is given by an additive function D : R → R satisfying the twisted Leibniz rule D(x · y) = xDy + yDx + e(Dx)(Dy) and D(1) = 0 defining a section φ : R → D_{e}(R) by φ(x) = (x, Dx).

Remark 2.1. From the standpoint of logic, the restriction to ℤ[e]-algebras corresponds to adding a constant symbol e to the language of rings.

Remark 2.2. Note that when e = 0 in R, a D_{e}-structure on R is simply given by a derivation.

Definition 2.3. A D-ring is a ℤ[e]-algebra R given with a function D : R → R defining a D_{e}-structure on R.

Proposition 2.4. Let (R, D) be a D-ring. The function σ : R → R defined by x ↦ eDx + x is a ring endomorphism of R.

■ Additivity is clear as is the fact that σ(1) = 1. For multiplication:

σ(xy) = eD(xy) + xy
= e(xDy + yDx + eDxDy) + xy
= e^{2}DxDy + exDy + eyDx + xy
= (eDx + x)(eDy + y)
■

Remark 2.5. If e is a non-zero divisor, then σ determines D. So when e is a unit, a D-ring is just a difference ring in disguise. That is, if e is a non-zero divisor, then D and σ are inter-definable. If e is a unit and one includes e^{−1} as a constant, then D is term definable from σ as Dx = \frac{σ(x)−x}{e}.

Of course, for fields e being zero or a unit exhaust the possibilities, but for more general rings there is an intermediate case.

Remark 2.6. The Leibniz rule may also be written as D(x · y) = xDy + σ(y)Dx.

Proposition 2.7. If x ∈ R^{×}, then D(\frac{1}{x}) = \frac{−Dx}{xσ(x)}.

■

0 = D(1)
= D(x^{−1}x)
= x^{−1}D(x) + σ(x)D(x^{−1})

Subtracting, we find that σ(x)D(x^{−1}) = −(x^{−1})Dx. As x ∈ R^{×}, we also have σ(x) ∈ R^{×}. Therefore, D(\frac{1}{x}) = \frac{−Dx}{σ(x)x}. ■

Proposition 2.8. If R is a D-ring and S ⊆ R is a multiplicative subset of R containing 1 and is closed under σ, then there is a unique structure of a D-ring on the localization S^{−1}R.

■ Proposition 2.7 shows how D must be defined. The original D_{e}-structure on R corresponds to a map R ⟶^{φ} D_{e}(R) given by x ↦ (x, Dx). By functoriality of D_{e}, there is a map D_{e}(R) ⟶^{D_{e}(i)} D_{e}(S^{−1}R). For any s ∈ S there is an inverse to i ∘ φ(s) = (s, Ds) in D_{e}(R), namely, (\frac{1}{s}, \frac{−Ds}{σ(s)s}). By the universal property of S^{−1}R, there is a unique ring homomorphism S^{−1}R ⟶ D_{e}(S^{−1}R) making the following diagram commute.

R ⟶^{φ} D_{e}(R)
i↓  ↓^{D_{e}(i)}
S^{−1}R ⟶^{∃!} D_{e}(S^{−1}R)
■

Let us also calculate Dx^{n}.

Proposition 2.9. If R is a D-ring, x ∈ R, and n is a positive integer, then

Dx^{n} = ∑_{i=1}^{n} \binom{n}{i}e^{i−1}x^{n−i}(Dx)^{i}

■ We check this by induction on n. For n = 1 the assertion is obvious. Let us now try the case of n + 1.

Dx^{n+1} = D(x^{n}x)
= x^{n}Dx + x(Dx^{n}) + eDx(Dx^{n})
= x^{n}Dx + (x + eDx) ∑_{i=1}^{n} \binom{n}{i}e^{i−1}x^{n−i}(Dx)^{i}

= x^{n}Dx + ∑_{j=1}^{n} \binom{n}{j}e^{j−1}x^{(n+1)−j}(Dx)^{j} + ∑_{ℓ=1}^{n} \binom{n}{ℓ}e^{ℓ}x^{n−ℓ}(Dx)^{ℓ+1}

= x^{n}Dx + ∑_{j=1}^{n} \binom{n}{j}e^{j−1}x^{(n+1)−j}(Dx)^{j} + ∑_{t=2}^{n+1} \binom{n}{t − 1}e^{t−1}x^{(n+1)−t}(Dx)^{t}

= (n + 1)x^{n}Dx + ∑_{j=2}^{n+1}[\binom{n}{j} + \binom{n}{j − 1}]e^{j−1}x^{(n+1)−j}(Dx)^{j}

= ∑_{j=1}^{n+1} \binom{n + 1}{j}e^{j−1}x^{(n+1)−j}(Dx)^{j}
■

Lemma 2.10. Let R be a local ring with maximal ideal 𝔪. Assume that e ∈ 𝔪. Then D_{e}(R) is also a local ring with maximal ideal π_{0}^{−1}𝔪.

■ Let (x, y) ∈ D_{e}(R) \ π_{0}^{−1}𝔪. That is, x ∈ R^{×}. Since e ∈ 𝔪, x + ey ∈ R^{×} as well. The inverse to (x, y) is then (\frac{1}{x}, \frac{−y}{x(x+ey)}). ■

Proposition 2.11. If R is a henselian local ring with maximal ideal 𝔪 and e ∈ 𝔪, then D_{e}(R) is also henselian.

■ Denote the reduction map R → R/𝔪 by x ↦ \overline{x}. Denote the induced map R[X] → (R/𝔪)[X] by P (X) ↦ \overline{P} (X) as well. Consider R as a subring of D_{e}(R) via r ↦ (r, 0). Let P (X) ∈ D_{e}(R)[X] and let (x, y) ∈ D_{e}(R) such that \overline{x} is a simple root of \overline{π_{0}(P)}(X). Since R is henselian, there is a unique a ∈ R such that π_{0}(P)(a) = 0 and \overline{a} = \overline{x}. Let ϵ := (0, 1). Since π_{0}(P)(a) = 0, there is some Q(Y) ∈ R[Y] with

P (a + ϵY) = ϵQ(Y)

Taylor expand P (a + ϵY) to compute that the linear term of Q(Y) is π_{0}(P′(a))Y and that all the higher order terms involve e as a factor. Hence, \overline{Q} is a linear polynomial and therefore has a unique solution in R/𝔪. As R is henselian, there is a unique lifting of this solution to some b ∈ R. The pair (a, b) is then the unique solution to P (X) = 0 with \overline{a} = \overline{x}. ■

The free algebra in the variety of D-rings has a particularly simple description.

Proposition 2.12. Let R be a D-ring. There is an extension of D-rings R → R⟨X⟩ universal with respect to simple extensions of R. As a ring, R⟨X⟩ = R[{D^{n}X}_{n=0}^{∞}], the polynomial ring in countably many indeterminates.

■ Let R′ := R[{D^{n}X}_{n=0}^{∞}]. Let φ : R → D_{e}(R) be the ring homomorphism making R into a D-ring. The inclusion R → R′ induces a map D_{e}(R) → D_{e}(R′). Let φ′ : R → D_{e}(R′) be the composite of this inclusion with φ. By the universality property of the polynomial ring, there is a unique map of rings φ̃ : R′ → D_{e}(R′) which agrees with φ′ on R and which sends D^{n}X ↦ (D^{n}X, D^{n+1}X). Thus, R′ is a D-ring extending R.

Let us check now universality. Let ψ : R → S be any map of D-rings. Let s ∈ S. By the universality property of the polynomial ring, there is a unique map of rings ψ̃ : R⟨X⟩ → S which agrees with ψ on R and sends D^{n}X ↦ D^{n}s. That this is a map of D-rings is equivalent to the commutativity of the following diagram.

S ⟶ D_{e}(S)
ψ̃↑  ↑^{D_{e}(ψ̃)}
R⟨X⟩ ⟶ D_{e}(R⟨X⟩)

Since ψ : R → S is a map of D-rings, this is commutative when restricted to R. By construction, Dψ(D^{n}X) = D(D^{n}s) = D^{n+1}s = ψ(D^{n+1}X) = ψ(D(D^{n}X)). Thus, this diagram is commutative on all the generators of R⟨X⟩, and hence everywhere. ■

3. Notation and General Definitions

We refer to R⟨X⟩ as the ring of D-polynomials over R.

In general, if R is a D-ring and I ⊆ R is an ideal, then I is a D-ideal if D(I) ⊆ I. When I ⊆ R is a D-ideal, then the D_{e}-structure on R induces such a structure on R/I.

If R is a D-ring and Σ ⊆ R is a subset, then ⟨Σ⟩ is the D-ideal generated by Σ. Since the intersection of a set of D-ideals is a D-ideal this notion is well-defined. Concretely, ⟨Σ⟩ = ({D^{n}x}_{x∈Σ,n∈ω}).

Define the order and degree of a D-polynomial by: The zero D-polynomial has order and degree ∞. A nonzero constant polynomial is considered to have order −1 and degree −∞. Otherwise, ordP := min{n : P ∈ R[X, …, D^{n}X]}. If n is the order of P, then the degree is the degree of P as a polynomial in D^{n}X.

The D-polynomial P is simpler than the D-polynomial Q, written P ≪ Q, if in the lexico-graphic order, the order-degree of P is less than that of Q.

If P is a D-polynomial of the form P (X) = F (X, DX, …, D^{m}X), then define \frac{∂}{∂X_{i}} P to be the D-polynomial (\frac{∂}{∂X_{i}} F)(X, DX, …, D^{m}X). There are at least two natural ways to extend D to R⟨X⟩. In the first case, we simply treat the new variables as constants. That is, for P (X) = ∑ p_{α}X^{α_{0}}DX^{α_{1}} ⋯ D^{N}X^{α_{N}} we set P^{D}(X) := ∑ D(p_{α})X^{α_{1}} ⋯ D^{N}X^{α_{N}}. The other extension of D is the one coming from D(D^{i}X) = D^{i+1}X used to make R⟨X⟩ into the universal D-extension of R.

We may define a more refined degree: the total degree. T. degP := (deg_{X_{i}} P)_{i=0}^{∞} for nonzero P and T. deg0 := ∞. Notice that the image of T. deg on nonzero D-polynomials comprises the set ℕ^{(ω)} := {(n_{j})_{j=0}^{∞} : n_{j} ∈ ℕ, n_{j} = 0 for j ≫ 0}. Define an ordering on ℕ^{(ω)} by (n_{j})_{j=0}^{∞} < (m_{j})_{j=0}^{∞} iff there is some N such that n_{N} < m_{N} and n_{j} ≤ m_{j} for j > N. Observe that this ordering is a well-ordering of ℕ^{(ω)}. We define P ≺ Q if T. degP < T. degQ. The fact that this ordering on ℕ^{(ω)} is a well-ordering means that we can (and will) argue by induction with respect to ≺.

≺ has the properties

• \frac{∂}{∂X_{i}} P ⪯ P and

• If P ≺ Q and P̃ and Q̃ differ from P and Q respectively by linear changes of variables – that is, if P (X) = F ({D^{n}X}) and Q(X) = G({D^{n}X}) while P̃(X) = F ({a_{n}D^{n}X + b_{n}}) and Q̃(X) = G({c_{n}D^{n}X + d_{n}}) with the parameters a_{n}, b_{n}, c_{n}, d_{n} taken from R with a_{n}, c_{n} ≠ 0 – then P̃ ≺ Q̃.

If R is a D-ring then R^{D} := ker(D : R → R). Observe that R^{D} is a ring, D is R^{D}-linear, and that (R^{D})^{×} = R^{D} ∩ R^{×}.

If L/K is an extension of D-fields and a ∈ L, then K(⟨a⟩) denotes the D-subfield of L generated by K and a.

4. The Language

The models under consideration have three sorts (K, k, Γ).

• K is the valued field given with the signature of a D_{e}-ring: (+, ·, −, 0, 1, e, D).

• k is the residue field also given with the signature of a D_{e}-ring and possibly with some extra predicates needed to ensure quantifier elimination for k.

• Γ is the value group given with the signature of an ordered abelian group with divisibility predicates (+, −, ≤, 0, {n|·}_{n∈ℤ+}) and possibly with some extra predicates needed for quantifier elimination.

For convenience, an extra symbol ∞ is added to the language. For instance, one defines 0^{−1} = ∞ and (∀γ ∈ Γ)γ < ∞. The sorts are connected by the functions π : K → k ∪ ∞ (the residue map) and v : K → Γ ∪ ∞ (the valuation). Denote the first-order language described above by L.

If M is an L-structure and P is a predicate, then we denote the realization of P in M by P_{M}. If P is a particular sort, then S_{m,P} (A) denotes the space of m-types over A in the sort P. That is, each p(x_{1}, …, x_{m}) ∈ S_{m,P} (A) must contain the formula ⋀_{1≤i≤m} P (x_{i}). It will be proven that in the cases of P = Γ or k that A may be replaced by P_{A} when A is an L-substructure of a model of the theory described in Section 5.

5. Axioms

We restrict the models considered to those with a differential field of characteristic zero as residue field. The more general cases of positive residual characteristic or a difference field as residue field present technical problems. After the present paper was written the author found a way to treat some of these additional cases and will describe this argument in a forthcoming paper.

Let k be a differential field and G an ordered abelian group. We assume that k satisfies

1. char k = 0,

2. (k^{×})^{n} = k^{×} for each n ∈ ℤ_{+}, and

3. any non-zero linear differential operator L ∈ k[D] is surjective as a map L : k → k. We call a differential field satisfying this condition linearly differentially closed.

We also assume that enough predicates have been added to L on the sort k so that Th(k) admits elimination of quantifiers. Of course, one should take the language to be as simple as possible.

We also assume that the language for Γ is sufficiently rich so that Th(G) admits elimination of quantifiers. In many cases of interest (for example, Γ = ℤ) one may

achieve this with divisibility predicates defined by n|x ⟺ (∃y) \overbrace{y + ⋯ + y}^{n times} = x [13]. In general, more complicated predicates may be needed.

One can avoid cluttering L by taking k ⊧ DCF_{0} and G = ℚ.

The first axioms describe general valued (k, G)-D-fields.

Axiom 1. K and k are D-fields of characteristic zero and k ⊧ Th_{∀}(k).

Axiom 2. K is a valued field whose value group is a subgroup of Γ via the valuation v and whose residue field is a subfield of k via the residue map π and v(e) > 0.

Axiom 3. (∀x ∈ K) v(Dx) ≥ v(x) and π(Dx) = Dπ(x).

Axiom 4. Γ ⊧ Th_{∀}(G).

The next six axioms together with the first four describe (k, G)-D-henselian fields.

Axiom 5. (∀x ∈ K)[([∃y ∈ K] y^{n} = x) ⟺ n|v(x)].

Axiom 6. Γ = v((K^{D})^{×}).

Axiom 7. k = π(O_{K}).

Axiom 8 (D-Hensel’s Lemma). If P ∈ O_{K}⟨X⟩ is a D-polynomial, a ∈ O_{K}, and v(P (a)) > 0 = v(\frac{∂}{∂X_{i}} P (a)) for some i, then there is some b ∈ K with P (b) = 0 and v(a − b) ≥ v(P (a)).

If the hypotheses of the last axiom apply to P and a, then one says that DHL applies to P at a.

Axiom 9. Γ ≡ G.

Axiom 10. k ≡ k.

Remark 5.1. Axioms 1 and 2 imply that D is a derivation on k.

Remark 5.2. Axiom 8 may be strengthened to v(b − a) = v(P (a)).

We assumed that k is linearly differentially closed and is closed under roots in order to guarantee consistency of the theory of (k, G)-D-henselian fields.

Proposition 5.3. Axioms 1 - 8 together with G ≠ 0 imply that k is linearly differentially closed.

■ Let K be a model of the first eight axioms. Let L(X) = ∑_{i=1}^{n} a_{i}D^{i}X be a nonzero linear D-polynomial over k. Let y ∈ k be given. By Axiom 7 there are b_{i} ∈ O_{K} such that π(b_{i}) = a_{i} and z ∈ O_{K} such that π(z) = y. Since G ≠ 0, by Axiom 6 there is ϵ ∈ O_{K} with Dϵ = 0 and v(ϵ) > 0. Let P (X) = −ϵ · z + ∑_{i=1}^{n} b_{i}D^{i}X.

Evaluating at zero we see that v(P (0)) > 0 and for some i we have π(\frac{∂}{∂X_{i}} P (0)) = π(b_{i}) = a_{i} ≠ 0. So by DHL there is some x ∈ O_{K} such that P (x) = 0 and v(x) ≥ v(P (0)) = v(ϵ). Let x′ = \frac{x}{ϵ}. We have

0 = P (x)
= −ϵz + ∑_{i=1}^{n} b_{i}D^{i}x

= ϵ(−z + ∑_{i=1}^{n} b_{i}D^{i}x′)

Hence, z = ∑_{i=1}^{n} b_{i}D^{i}x′. Applying π, we find that y = L(π(x′)). ■

Proposition 5.4. Axioms 1 - 8 imply that (k^{×})^{n} = k^{×} for each positive integer n.

■ Let K be a model of the first eight axioms. Let x ∈ k^{×}. By Axiom 7 there exists y ∈ O_{K} such that π(y) = x. The valuation of y is zero so n|v(y) which implies by Axiom 5 that y has an n-th root z. Thus, π(z) is an n-th root of x. ■

6. Consistency and the Standard D-Henselian Fields

k and G continue to have the same meaning as in the previous section.

The generalized power series fields k((ϵ^{G})) provide canonical models for the theory of D-henselian fields. For the reader’s convenience, we recall the definition of these fields.

As a set, k((ϵ^{G})) = {f : G → k : supp(f) := {x ∈ G : f(x) ≠ 0} is well-ordered in the ordering induced by G}.

We think of an element f ∈ k((ϵ^{G})) as a formal power series.

f ↔ ∑_{γ∈G} f(γ)ϵ^{γ}
v(f) := min supp(f)
(f + h)(γ) := f(γ) + h(γ)
(fh)(γ) := ∑_{α+β=γ} f(α)h(β)

For the time being we denote the derivation on k by ∂.

If we wish to have e = 0, then define

(Df)(γ) = ∂(f(γ))

Otherwise, take for e any non-zero element of positive valuation and define an endomorphism σ from which we recover D by the formula Dx := \frac{σ(x)−x}{e}. On k define σ by

σ(x) := ∑_{n=0}^{∞} \frac{∂^{n}x}{n!}e^{n}

Extend to all of k((ϵ^{G})) by the rule

σ(f) = ∑ σ(f(γ))ϵ^{γ}

k((ϵ^{G})) is a maximally complete valued field [9, 12].

That this field is a D-henselian field is clear except perhaps for DHL. We prove DHL for K := k((ϵ^{G})) in a prima facie stronger form.

Proposition 6.1. If P ∈ O_{K}⟨X⟩ is a D-polynomial, a ∈ O_{K}, and v(P (a)) > 2v(\frac{∂}{∂X_{i}} P (a)) for some i, then there is b ∈ K such that P (b) = 0 and v(a − b) ≥ v(P (a)) − v(\frac{∂}{∂X_{i}} P (a)).

■ Let i be a non-negative integer such that v(\frac{∂}{∂X_{i}} P (a)) ≤ v(\frac{∂}{∂X_{j}} P (a)) for all j. Let γ_{0} := v(\frac{∂}{∂X_{i}} P (a)).

Inductively build an ordinal indexed Cauchy sequence of approximate solutions {x_{α}} from K. If at some point P (x_{α}) = 0, stop. At each point in the construction ensure that (∀β < α) v(P (x_{α})) > v(x_{α}−x_{β})+γ_{0} > 2v(\frac{∂}{∂X_{i}} P (a)) and that v(x_{α+1}− x_{α}) ≥ v(P (x_{α})) − γ_{0}. By starting with x_{0} = a, provided that one may construct the sequence so as to be cofinal in G, the result is proven.

For each j, choose c_{j} ∈ k such that v(c_{j}ϵ^{γ_{0}} − \frac{∂}{∂X_{j}} P (a)) > γ_{0}.

At successor stages, α+1, try to modify x_{α} slightly so as to increase the valuation of P. Let γ := v(P (x_{α}))−γ_{0} which by the inductive hypothesis in the case of α > 0 or by the hypothesis of the theorem in case α = 0 is greater than γ_{0}. Consider the expansion:

(1)  P (Xϵ^{γ} + x_{α}) = ∑_{m≥0} ∑_{|I|=m} ∂_{I}P (x_{α})ϵ^{mγ}X^{I}

Every coefficient on the right hand side has value ≥ γ + γ_{0}. For the constant term, this is because γ + γ_{0} = v(P (x_{α})) by definition. For the linear terms, one knows that each of v(\frac{∂}{∂X_{j}} P (x_{α})) is at least γ_{0}. For the higher order terms, note that mγ ≥ 2γ = 2v(P (x_{α})) − 2γ_{0} > v(P (x_{α})) + 2γ_{0} − 2γ_{0} = γ + γ_{0}. (The strict inequality follows from the fact that v(P (x_{α})) > 2γ_{0}.) Thus, we may divide the right hand side by ϵ^{γ+γ_{0}} and still have a D- polynomial with integral coefficients. In the residue field, the equation is

(2)  π(\frac{P (Xϵ^{γ} + x_{α})}{ϵ^{γ+γ_{0}}}) = π(\frac{P (x_{α})}{ϵ^{γ+γ_{0}}}) + ∑ c_{j}D^{j}X

As c_{i} ≠ 0, Equation 2 is a non-trivial inhomogeneous linear D-equation over k. As k is linearly differentially closed we may find x which is a solution to this equation and set x_{α+1} = xϵ^{γ} + x_{α}. The inductive hypothesis is still true at x_{α+1} as v(P (x_{α+1})) > v(P (x_{α})).

At limits, simply find any x_{λ} such that v(x_{λ} − x_{α}) < v(x_{λ} − x_{β}) for α < β < λ. Such exists by completeness of K. Let δ < α < λ and consider Equation 1. We compute by induction that v(P (x_{λ})) ≥ min{v(P (x_{α})), v(x_{λ} − x_{α}) + γ_{0}} > v(x_{α} − x_{δ}) + γ_{0} = v(x_{λ} − x_{δ}) + γ_{0} > 2γ_{0}.

Let b = lim x_{α}. ■

7. Quantifier Elimination

Theorem 7.1. With the restrictions imposed on k and G in Section 5, the theory of henselian (k, G)-D-fields eliminates quantifiers and is the model completion of the theory of valued (k, G)-D-fields. Its completions are determined by the isomorphism type of the valued D-field ℚ({D^{n}e}_{n∈ω}).

■ We prove this by a standard back-and-forth test.

Claim 7.2. Let M_{1} and M_{2} be two ℵ_{1}-saturated (k, G)-D-henselian fields. Let A_{1} ⊂ M_{1} and A_{2} ⊂ M_{2} be countable substructures. Let f : A_{1} → A_{2} be an isomorphism of L-structures. Let b ∈ M_{1}. Then f extends to a partial isomorphism from M_{1} to M_{2} having b in its domain.

See Theorem 8.4.1 of [5] for a proof that Claim 7.2 implies that the theory of (k, G)-D-henselian fields admits elimination of quantifiers.

We prove that the theory of D-henselian fields is the model completion of the theory of valued D-fields by showing that each of the constructions used in the proof of Claim 7.2 may actually be used to extend any valued D-field to a D-henselian field.

The description of the completions of the theory of (k, G)-D-henselian fields follows from quantifier elimination.

Our strategy for the proof of Claim 7.2 is to extend f a little bit at a time so that the type of b over dom(f) becomes transparent. We work mostly within the sort of the valued field so that by “x ∈ A_{1}” we mean that x is an element of the valued field. We treat the extension of f as an inductive process in which at each stage the current domain of f is equal to A_{1}.

Before extending f we fix a countable elementary submodel N_{1} ≺ M_{1} of M_{1} containing A_{1} and b. We actually extend f a little beyond N_{1}.

We start by enlarging A_{1} freely so that Γ_{N_{1}} is contained in the divisible hull of Γ_{A_{1}}. We then extend f so that one has n|γ ⟺ γ = nv(x) for some x ∈ A_{1} for each γ ∈ Γ_{A_{1}}. Once this is done we ensure that A_{1} has enough constants in the sense of Definition 7.3 below. At this point we extend A_{1} so that the residue map is surjective onto k_{N_{1}}. The model N_{1} will be an immediate extension of A_{1} once this step has been achieved. Finally, we extend f to a particular immediate extension of N_{1} by an inductive procedure. This step requires most of the work. ■

7.1. Extensions of k and of Γ.

Definition 7.3. The valued D-field (K, Γ) has enough constants if it satisfies Axiom 6.

Lemma 7.4. If K is a valued D-field with enough constants, then for any value γ ∈ Γ_{K} and any finite set of polynomials Q_{1}(X), …, Q_{n}(X) ∈ K[X] with Q_{i}(X) = ∑_{j=0}^{m} q_{i,j}X^{j} there is some ϵ ∈ K with v(ϵ) = γ, Dϵ = 0, and v(Q_{i}(ϵ)) = min{v(q_{i,j})+ jγ} for each Q_{i} from the above set.

■ As K has enough constants there is some η ∈ K with Dη = 0 and v(η) = γ. For each i, let δ_{i} ∈ K with v(δ_{i}) = min{v(q_{i,j}) + jγ}. If for each i it were the case that

(3)  π(\frac{Q_{i}(η)}{δ_{i}}) ≠ 0

then the desired result would be true with ϵ = η for Inequality 3 means simply that v(Q_{i}(η)) = v(δ_{i}) = min{v(q_{i,j})+jγ}. Alas, it may happen that with our first choice of η, some instance of Inequality 3 fails. We replace η with cη where v(c) = 0 and Dc = 0. We need only ensure π(c) is not a solution to any of ∑ π(\frac{q_{i,j}η^{j}}{δ_{i}})Y^{j} = 0. With finitely many exceptions, any choice from ℚ will work. ■

Corollary 7.5. Let K be an ℵ_{1}-saturated valued D-field with enough constants. Let L ⊂ K be a countable subfield of K. Let γ ∈ Γ_{K}. There is some ϵ ∈ K such that v(ϵ) = γ, Dϵ = 0 and for any polynomial Q(X) = ∑_{j=0}^{n} q_{j}X^{j} ∈ L[X] one has v(Q(ϵ)) = min{v(q_{j}) + jγ}.

Remark 7.6. In the case that k already admits elimination of quantifiers in the natural language of differential rings, Corollary 7.5 may be used to give a quick proof that the isomorphism may be extended so that A_{i} has enough constants. Unfortunately, in general the adjunction of constants may enlarge the residue field so that there may be some ambiguity as to the extension unless the types of the new elements of the residue field are controlled.

Lemma 7.7. If K is a valued field which is also a D_{e}-ring in which v(e) > 0 and a, b ∈ K^{×} satisfy v(Da) ≥ v(a) and v(Db) ≥ v(b), then v(D(\frac{a}{b})) ≥ v(\frac{a}{b}).

■

v(D(\frac{a}{b})) = v(\frac{D(a)b − D(b)a}{bσ(b)})
≥ min{v(Da) − v(σ(b)), v(a) + v(Db) − (v(b) + v(σ(b)))}
= min{v(Da) − v(b), v(a) + v(Db) − 2v(b)}
≥ min{v(a) − v(b), v(a) + v(b) − 2v(b)}
= v(a) − v(b)
= v(\frac{a}{b})
■

Lemma 7.8. Let K be a valued D-field. Let p ∈ S_{1,Γ}(Γ_{K}) be a 1-type in the value group sort extending {nx ≠ γ : n ∈ ℤ_{+}, γ ∈ v(K^{×})}. There is a unique (up to L_{K}- isomorphism) structure of a valued D-field on K(x), the field of rational functions over K in the indeterminate x, such that v(x) ⊧ p, Dx = 0, and such that for any polynomial P (x) = ∑_{i=0}^{n} p_{i}x^{i} ∈ K[x] one has v(∑ p_{i}x^{i}) = min{v(p_{i}) + iv(x)}.

■ The hypotheses completely describe the D_{e}-structure and the valuation structure. Since there is no extension of the residue field, we need not consider the extra structure on it. Let us check now that this prescription actually gives a valued D-field. We need to check that v(Dy) ≥ v(y) for y ∈ K(x). By Lemma 7.7, it suffices to consider y = P (x) ∈ K[x]. Write P (x) = ∑ p_{i}x^{i}. Then

DP (x) = ∑ D(p_{i}x^{i})
= ∑ D(p_{i})x^{i} + σ(p_{i})D(x^{i})
= ∑ D(p_{i})x^{i}

Since K is a valued D-field, v(D(p_{i})) ≥ v(p_{i}). Therefore,

v(DP (x)) = min{v(D(p_{i})) + iv(x)}
≥ min{v(p_{i}) + iv(x)}
= v(P (x))
■

Lemma 7.9. If K is a valued D-field and L/K is an unramified valued field extension of K given with an extension of the D_{e}-structure with D(O_{L}) ⊆ O_{L}, then L is also a valued D-field.

■ Let x ∈ L^{×}. Let y ∈ K such that v(x) = v(y). Let α = \frac{y}{x}. The hypothesis that D(O_{L}) ⊆ O_{L} means that v(Dα) ≥ v(α) = 0. Since y ∈ K, v(Dy) ≥ v(y). As x = \frac{y}{α}, Lemma 7.7 shows v(Dx) ≥ v(x). ■

Remark 7.10. If in Lemma 7.9 one drops the requirement that the extension is unramified, then the result is not true. For an example take K = ℚ with the trivial valuation and derivation. Let L = ℚ((x)) with the order of vanishing at the origin valuation and the derivation ∂ = \frac{d}{dx}. \frac{d}{dx}(ℚ[[x]]) ⊆ ℚ[[x]], but ord_{x}(\frac{d}{dx}x) = ord_{x}(1) = 0 < 1 = ord_{x}(x).

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

■ Let φ : O_{K} → D_{e}(O_{K}) be the map x ↦ (x, Dx) expressing the D-structure on O_{K}. Let φ̃ : O_{K} → D_{e}(R) be the composition of φ with the inclusion D_{e}(O_{K}) ↪ D_{e}(R). By the universal property of the henselization, there is a unique map of local rings ψ : R → D_{e}(R) compatible with the inclusion O_{K} ↪ R and the map φ̃. Define D : R → R as the function for which ψ(x) = (x, Dx). Then R is a Dring with respect to this function. By Proposition 2.8, L has a unique D-structure extending that on R. By Lemma 7.9, L is a valued D-field with respect to this structure. ■

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

• if x ⊧ p, then π(P) is of minimal total degree among nonzero Q(X) ∈ π(O_{K})⟨X⟩ with Q(x) = 0 and

• T. degP = T. degπ(P), there is a unique (up to L_{K}-isomorphism) D-field L = K(⟨a⟩) such that P (a) = 0 and π(a) ⊧ p.

Remark 7.13. Lemma 7.12 applies equally well in the case that P = 0. Recall that in this case, T. degP = ∞.

■ We begin by proving uniqueness and then prove existence. We analyze L = K(⟨a⟩) as a direct limit of valued field extensions K_{n} := K(a, Da, …, D^{n}a). Let m := ordP and b := π(a).

• (n ≤ m) Every element of K_{m} is of the form \frac{Q_{1}(a)}{Q_{2}(a)} with Q_{i} ≪ P so in order to pin down the valuation on K_{m} it suffices to compute v(Q(a)) for Q(X) ∈ K⟨X⟩ with Q ≪ P. Let α ∈ K such that αQ ∈ O_{K}⟨X⟩ and π(αQ) ≠ 0. Since π(αQ)(b) ≠ 0, we must have v(Q(a)) = −v(α).

• (n > m) If e = 0, then L = K_{m} so there is nothing more to do.

We assume now that e ≠ 0. Let P (X) = F (X, …, D^{m}X). Let G(X) := F (a, …, D^{m−1}a, X). In the case e ≠ 0, σy and Dy are inter-definable, so it suffices to consider the extension

K(a, Da, …, D^{n−1}a, σ^{n−m}D^{m}a)/K(a, Da, …, D^{n−1}a)

σ^{n−m}D^{m}a satisfies σ^{n−m}G. So the minimal polynomial Q_{n} of σ^{n−m}D^{m}a over K(a, …, D^{n−1}a) divides σ^{n−m}G. Since σ reduces to the identity automorphism, π(σ^{n−m}D^{m}a) = D^{m}b which is a simple root of π(G) = π(σ^{n−m}G) (as \frac{∂}{∂X_{m}} P ≪ P). Thus, π(G′)(D^{m}b) = π(\frac{∂}{∂X_{m}} (P))(b) ≠ 0. Thus, b is a simple root of π(Q_{n}) so the extension is completely determined as an extension of valued fields.

We check now that the process used above to analyze such extensions may be used to produce them. Let b ⊧ p be a realization of p.

Let K′ be the field of fractions of K[X, DX, …, D^{m}X]/(F (X, …, D^{m}X)). Let a denote the image of X in K′. K′ is given a valuation structure by setting v(Q(a)) := max{−v(α) : αQ ∈ O_{K}⟨X⟩} for Q(X) ∈ K⟨X⟩ with Q ≪ P. In the case that e = 0, K′ is already a differential field. In the case that e ≠ 0, let Q_{1} be the unique (up to multiplication by a unit) factor of σ(G) (over O_{K′}[Y]) for which π(Q_{1})(D^{m}b) = 0 and π(Q_{1}) ≠ 0. This was proven to exist in the course of the uniqueness proof. Since D^{m}b is a simple root of π(Q_{1})(Y) = 0, the equation Q_{1}(Y) = 0 has a unique solution y in R a henselization of O_{K′} such that π(y) = D^{m}b. Define σ on K as usual by σ(x) := eDx + x and extend to σ : K′ → K′(y) via σ(D^{i}a) := eD^{i+1}a + D^{i}a for i < m and σ(D^{m}a) := y. We formally define Dz := \frac{σ(z)−z}{e} as a function K′ → K′(y). At this point, we could continue to incrementally define D as the analysis in the uniqueness part of the proof might suggest. This works and the reader is invited to finish the argument this way. We take a different tack. First we check that the required inequalities continue to hold at least for z ∈ K′.

Claim 7.14. v(D^{m+1}a) ≥ 0 ✠ Since σ reduces to the identity on O_{K}[a, …, D^{m−1}a]/(e), D^{m}a is a root to σ(G) modulo e. Thus, v(y − D^{m}a) ≥ v(e). Since D^{m+1}a = \frac{y−D^{m}a}{e}, the result is now clear. ✠

The next claim is valid in general. That is, there is no restriction on e.

Claim 7.15. If Q(X) ≪ P (X), then v(Q(a)) ≤ v(DQ(a)).

✠ By Lemma 7.7 we may assume that Q(X) ∈ O_{K}⟨X⟩ and π(Q) ≠ 0. This implies v(Q(a)) = 0. Write Q(X) = ∑ q_{i}(D^{m}X)^{i} where q_{i} ∈ O_{K}[a, …, D^{m−1}a]. Then DQ(X) = ∑ D(q_{i})(D^{m}X)^{i} + σ(q_{i})D(D^{m}X)^{i} ∈ O_{K}⟨X⟩. For j ≤ m, it is clear that v(D^{j}a) ≥ 0. Claim 7.14 shows that v(D^{m+1}a) ≥ 0 in the case that e ≠ 0. In the case that e = 0, we observe that D^{m+1}a = \frac{−P^{D}}{\frac{∂}{∂Xm} P} and \frac{∂}{∂X_{m}} P ≪ P so that its

valuation is zero.

Since each of the D^{j}a have non-negative valuation, v(DQ(a)) ≥ 0 = v(Q(a)). ✠

Let R be the henselization of O_{K′} and let L be the field of fractions of R. Let φ : K′ → D_{e}(L) be the map x ↦ (x, Dx). By Claim 7.15, φ(O_{K′}) ⊆ D_{e}(R). By Lemma 2.10, D_{e}(R) is a local ring with maximal ideal π_{0}^{−1}𝔪_{R}. Since π_{0} ∘ φ = id_{R}, φ is a local homomorphism. By Lemma 2.11, D_{e}(R) is henselian. Thus, there is a unique extension of φ to a local homomorphism φ̃ : R → D_{e}(R). By Lemma 2.7, φ̃ extends uniquely to a ring homomorphism ψ : L → D_{e}(L). Let D denote the function D : L → L for which ψ(x) = (x, Dx). Since D(R) ⊆ R, by Lemma 7.9, L is a valued D-field. ■

Proposition 7.16. Let K be a valued D-field. Let a ∈ K^{×}. There is an unramified extension L/K of valued D-fields of the form L = K(x) for which v(x) = v(a) and Dx = 0. Moreover, the L_{K}-isomorphism type of L is determined by tp(π(\frac{a}{x})/k_{K}).

■ We wish to find x so that Dx = 0 and v(x) = v(a). This is equivalent to finding y = “\frac{a}{x}”. Such a y would have to satisfy Dy = \frac{Da}{x} = \frac{Da}{a} \frac{a}{x} = \frac{Da}{a}y. Conversely, if y satisfies Dy = \frac{Da}{a}y and we define x by x := \frac{a}{y}, then Dy = \frac{Da}{x} + σ(a)D(\frac{1}{x}) as well so that 0 = D(\frac{1}{x}) which implies that Dx = 0.

One would also need v(y) = 0 in order for v(\frac{a}{y}) = v(a). That is, we need y to be a solution to DY = \frac{Da}{a}Y with v(y) = 0. By Lemma 7.12, such extensions exist and they are determined by tp(y/k_{K}). ■

Proposition 7.17. Let K be a valued D-field. Assume that v(K^{×}) = v((K^{D})^{×}). Let η ∈ K^{D} such that n|v(η). Assume that v(η) ∉ m · v(K^{×}) for each positive integer m dividing n. There exists a unique (up to L_{K}-isomorphism) extension of valued D-fields of the form K( \sqrt[n]{η}).

■ Let ϵ = \sqrt[n]{η}. Since the extension is totally ramified, the valuation structure is completely determined. I claim that the D_{e}-structure is determined by Dϵ = 0. This fact would certainly fully specify the D_{e}-structure, the content of my claim is that one must have Dϵ = 0. When e = 0, this follows from the fact that 0 = D(η) = D(ϵ^{n}) = nϵ^{n−1}Dϵ. When e ≠ 0, the assertion Dϵ ≠ 0 is equivalent to σ(ϵ) = ωϵ for some nontrivial n-th root of unity ω. But then Dϵ = \frac{σ(ϵ)−ϵ}{e} = ϵ\frac{ω−1}{e}. Since ω ≠ 1, v(\frac{ω−1}{e}) = −v(e) < 0. So that v(Dϵ) < v(ϵ) which violates Axiom 3.

We check that this prescription correctly defines a valued D-field. Let x = ∑_{i=0}^{n−1} x_{i}ϵ^{i} ∈ L. Then v(x) = min\{v(x_{i}) + \frac{i}{n}v(η)\} and Dx = ∑_{i=0}^{n−1} Dx_{i}ϵ^{i} so that visibly, the inequality v(Dx) ≥ v(x) holds. ■

Proposition 7.18. The isomorphism may be extended so that v(A_{1}^{×}) = v((A_{1}^{D})^{×}).

■ Let a ∈ A_{1} such that there is no constant in A_{1} having the same value as that of a. Let p ∈ S_{1,k}(k_{A_{1}}) be some type containing the formula Dx = π(\frac{Da}{a})x as well as the formulas x ≠ b for each b ∈ k_{A_{1}}. By the saturation hypotheses, p is realized in k_{M_{1}} by some b_{1} and f(p) is realized in k_{M_{2}} by some b_{2}. By the surjectivity of the residue map and DHL there is some c_{1} ∈ M_{1} and some c_{2} ∈ M_{2} such that π(c_{i}) = b_{i}, Dc_{1} = \frac{Da}{a}c_{1} and Dc_{2} = f(\frac{Da}{a})c_{2}. By Proposition 7.16, the extension of f given by c_{1} ↦ c_{2} (and therefore \frac{a}{c_{1}} ↦ \frac{f(a)}{c_{2}}) is an isomorphism of L-structures and the element \frac{a}{c_{1}} is a constant with value equal to that of a. ■

Proposition 7.19. The isomorphism extends so that A_{1} has enough constants.

■ By Proposition 7.18 we may now assume that v(A_{1}^{×}) = v((A_{1}^{D})^{×}). Let γ ∈ Γ_{A_{1}} \ v(A_{1}^{×}). In the case that tp(γ/v(A_{1}^{×})) ⊢ {nγ ≠ v(a) : n ∈ ℤ_{+}, a ∈ A_{1}^{×}}, we find ϵ_{1} ∈ N_{1} with v(ϵ_{1}) = γ and Dϵ_{1} = 0 and ϵ_{2} ∈ M_{2} with v(ϵ_{2}) ⊧ f(tp(γ/v(A_{1}^{×})) and Dϵ_{1} = Dϵ_{2} = 0 by Axiom 6. Lemma 7.8 shows that the extension given by ϵ_{1} ↦ ϵ_{2} is an isomorphism of L-structures.

In the case that nγ ∈ v(A_{1}^{×}) for some n ∈ ℤ_{+}, take n minimal with this property and find some η ∈ A_{1}^{D} with v(η) = nγ. By Axiom 5, we may find ϵ_{i} ∈ M_{i} such that ϵ_{1}^{n} = η and ϵ_{2}^{n} = f(η). By Proposition 7.17, this gives an isomorphism of L-structures. ■

Proposition 7.20. f extends so that Γ_{N_{1}} ⊆ v((A_{1}^{D})^{×}).

■ As we have assumed quantifier elimination in the value group sort, we may fix some elementary embedding \overline{f} : Γ_{N_{1}} → Γ_{M_{2}} extending f. By Proposition 7.19, we may extend f over \overline{f}. ■

Proposition 7.21. The map extends so that k_{N_{1}} ⊆ π(O_{A_{1}}).

■ As the theory of the residue field of M_{1} has quantifier elimination by assumption we may fix some isomorphism \overline{f} between k_{N_{1}} and some countable elementary substructure of k_{M_{2}} extending f restricted to k_{A_{1}}.

Take a ∈ O_{N_{1}} so that π(a) is a new element of k_{N_{1}} \ π(O_{A_{1}}). If π(a) is differentially transcendental over π(O_{A_{1}}), then let b_{1} = a. Let b_{2} ∈ O_{M_{2}} such that π(b_{2}) = \overline{f}(a). Necessarily, b_{1} is D-transcendental over A_{1} while b_{2} is Dtranscendental over A_{2}. Then Lemma 7.12 shows that we may extend the isomorphism by setting f(b_{1}) = b_{2}.

Otherwise, Let P ∈ O_{A_{1}}⟨X⟩ be such that π(P)(a) = 0, T. degP = T. degπ(P), π(P) ≠ 0 and T. degP is minimal with these properties. The minimality condition on P implies that π(P) is a minimal D-polynomial for π(a) over the residue field of A_{1} so that for some i one has v(\frac{∂}{∂X_{i}} P (a)) = 0. By DHL in both N_{1} and M_{2} there is some b_{1} ∈ N_{1} and b_{2} ∈ M_{2} such that P (b_{1}) = 0, f(P)(b_{2}) = 0, π(b_{1}) = π(a) and π(b_{2}) = \overline{f}(π(a)).

By Lemma 7.12, A_{1}(⟨b_{1}⟩) ≅ A_{2}(⟨b_{2}⟩) via an isomorphism extending f. ■

7.2. Immediate Extensions. The rest of this section is devoted to proving that the isomorphism may be extended to immediate extensions.

Definition 7.22.

1. A pseudo-convergent sequence is a limit ordinal indexed sequence {x_{α}}_{α<κ} of elements of K such that (∀α < β < γ < κ) v(x_{α} − x_{β}) < v(x_{β} − x_{γ}).

2. If L/K is an extension of valued D-fields and {x_{α}} is a pseudo-convergent sequence from K, then the set of pseudo-limits of {x_{α}} in L is the set of c ∈ L such that (∀α < β < κ) v(x_{α} −c) < v(x_{β} −c). In this case, one writes x_{α} ⇒ c and says that {x_{α}} pseudo-converges to c.

3. The pseudo-convergent sequence {x_{α}} pseudo-solves the D-polynomial P if either P (x_{α}) ⇒ 0 or P (x_{α}) = 0 for α ≫ 0.

4. A pseudo-convergent sequence from K is strict if it has no pseudo-limits in K.

Remark 7.23. If {x_{α}}_{α<κ} is a pseudo-convergent sequence and one restricts to {x_{α}}_{α∈I} where I is cofinal in κ, then the new sequence is also pseudo-convergent with the same pseudo-limits and D-polynomials which it pseudo-solves as the original sequence. As the value groups considered in this proof are countable, one could always assume that κ = ω by making such a restriction.

We will repeatedly use this process of restricting to cofinal subsequences. The next lemma shows that we may pass from one strict pseudo-convergent sequence to a subsequence of another sequence known to well-approximate the first without changing the set of pseudo-limits.

Lemma 7.24. Let {x_{α}}_{α<κ} be a strict pseudo-convergent sequence from K. Suppose that {y_{α}}_{α<κ} is another sequence from K and that v(y_{α} − x_{α}) ≥ v(x_{α+1} − x_{α}) for all α < κ. Then there is a cofinal S ⊆ κ such that {y_{α}}_{α∈S} is a strict pseudoconvergent sequence with the same pseudo-limits as those of {x_{α}}_{α<κ}.

■ We define S by transfinite recursion. We start with 0 ∈ S. At a stage α where we have decided that α ∈ S, look for the least β > α such that v(x_{β+1} − x_{β}) > v(y_{α} − x_{β}) and let β be the next element of S.

Such a β must exist for if it did not, then y_{α} would be a pseudo-limit of {x_{γ}}_{γ<κ} which is impossible as we assumed this sequence to be strict. In detail, if such a β failed to exist, then for each γ > β > α, we would have v(y_{α} −x_{γ}) ≥ v(x_{γ+1} −x_{γ}) > v(x_{β+1} − x_{β}) = v(x_{γ} − x_{β}) = min{v(y_{α} − x_{γ}), v(x_{γ} − x_{β})} = v(y_{α} − x_{β}) so that y_{α} would be a pseudolimit of this sequence.

At limit stages, we simply take the least λ < κ not yet in S.

Let us check that this choice of S works. First we show that {y_{α}}_{α∈S} is a pseudoconvergent sequence. Let α < β < γ be in S. Then v(y_{γ} − y_{β}) ≥ v(y_{γ} − x_{γ} + x_{γ} − x_{β} + x_{β} − y_{β}) ≥ min{v(x_{γ+1} − x_{γ}), v(x_{β+1} − x_{β})} = v(x_{β+1} − x_{β}) > v(y_{α} − x_{β}) = min{v(x_{β} − y_{α}), v(y_{β} − x_{β})} = v(y_{β} − y_{α}).

Let now a be a pseudo-limit of {x_{α}}_{α<κ} in some extension of K. Let α < β ∈ S. Let α′ be the successor of α in S and β′ the successor of β in S. Then v(a − y_{β}) = v(a − x_{β′} + x_{β′} − y_{β}) = v(x_{β′+1} − x_{β′}) > v(x_{α′+1} − x_{α′}) = v(a − y_{α}). So every pseudo-limit of {x_{α}}_{α<κ} is a pseudo-limit of {y_{α}}_{α∈S}.

Let now a be a pseudo-limit of {y_{α}}_{α∈S}. Let α < κ be given. Let β ∈ S be minimal with β > α. Then v(a−x_{α}) = v(a−y_{β} +y_{β} −x_{β} +x_{β} −x_{α}) = v(x_{β} −x_{α}) = v(x_{α+1} − x_{α}) as v(a − y_{β}) = v(y_{β′} − y_{β}) ≥ v(x_{β} − y_{β}) ≥ v(x_{β+1} − x_{β}) > v(x_{β} − x_{α}). So, a is also a pseudo-limit of {x_{α}}_{α<κ}.

Since {x_{α}}_{α<κ} and {y_{α}}_{α∈S} have the same pseudo-limits and {x_{α}}_{α<κ} has no pseudo-limits in K, {y_{α}}_{α∈S} is also strict. ■

Lemma 7.25. Assume that the residue field of K is linearly differentially closed and that K has enough constants. If DHL applies to P ∈ O_{K}⟨X⟩ at a ∈ O_{K}, then either there is some b ∈ O_{K} with P (b) = 0 and v(a−b) = v(P (a)) or there is a strict pseudo-convergent sequence {x_{α}}_{α<λ} from K pseudo-solving P with v(x_{α} − a) = v(P (a)).

■ This is proven exactly as in the proof of DHL for complete fields so we give only a sketch here referring the reader to the proof of Proposition 6.1 for the detailed computations.

We produce either a solution to P (b) = 0 and v(a − b) = v(P (a)) or a sequence with the properties:

• x_{0} = a,

• v(x_{α+1} − x_{α}) = v(P (x_{α})), and

• v(P (x_{α})) is strictly increasing. Start with x_{0} = a.

At a limit stage, look for x_{λ} such that x_{α} ⇒ x_{λ}. If no x_{λ} exists, then stop – the sequence {x_{α}}_{α<λ} is a strict pseudo-solution of P.

At a successor stage, since K has enough constants, there is some ϵ ∈ K^{D} with v(ϵ) = v(P (x_{α})). We look to solve

P (x_{α} + ϵY) = 0

As in the proof of Proposition 6.1, we let y ∈ O_{K} be a lifting of a solution to

0 = π(\frac{P (x_{α})}{ϵ}) + ∑_{i=0}^{ordP} π(\frac{∂}{∂X_{i}} P (a))D^{i}Y

Set x_{α+1} = x_{α} + ϵy. ■

Definition 7.26. Let N ∈ ℕ^{(ω)}∪{∞}. The valued D-field K is N- full if whenever {x_{α}} is a pseudo-convergent sequence from K and there is an immediate extension K(⟨a⟩) with

• x_{α} ⇒ a,

• Q(a) = 0 for some Q ∈ K⟨X⟩ with T. degQ < N, and

• {x_{α}} pseudo-satisfies Q, then {x_{α}} has a pseudo-limit in K.

Remark 7.27. Another way of stating the definition of N-fullness is to say that no strict pseudo-convergent sequences satisfying the three listed hypotheses exist.

Definition 7.28. Let A ∈ K⟨X⟩ be a non-zero D-polynomial. A refinement of A at a is a D-polynomial G(Y) = \frac{A(ϵY +a)−A(a)}{c} where c, ϵ ∈ (K^{D})^{×} and G ∈ O_{K}⟨X⟩ but π(G) ≠ 0. ϵ is called the internal scale and c is the external scale. The Dpolynomial H(Y) = \frac{1}{c}A(a + ϵY) = G(Y) + \frac{A(a)}{c} is called the rescaling of A at a with scales ϵ and c. Note that H need not have integral co-efficients.

Remark 7.29. Notice that G may be expressed as

G(Y) = ∑_{|I|>0} \frac{ϵ^{|I|}}{c} ∂_{I}A(a)Y^{I}

Observe that the valuation of the external scale is entirely determined by the conditions G(Y) ∈ O_{K}⟨Y ⟩ and π(G) ≠ 0.

Definition 7.30. Let {x_{α}} be a pseudo-convergent sequence. Let A be a Dpolynomial. A refinement of A along {x_{α}} is a sequence of refinements of A at x_{α} having internal scale ϵ_{α} where v(ϵ_{α}) = v(x_{α+1} − x_{α}). Likewise, a rescaling of A along {x_{α}} is a sequence of rescalings of A at x_{α} having internal scale ϵ_{α} as above.

Definition 7.31. The non-zero D-polynomial A(X) ∈ O_{K}⟨X⟩ is residually linear if π(A) ∈ k⟨X⟩ is a non-constant linear D-polynomial. A is potentially residually linear if some refinement of A is residually linear. We will stipulate that the zero polynomial is residually linear.

Proposition 7.32. Let K be a valued D-field. Let P (X) ∈ K⟨X⟩ be an irreducible D-polynomial. Assume that K has enough constants, has a linearly differentially closed residue field, and is T. degP -full. Let {x_{α}}_{α<κ} be a strict pseudo-convergent sequence from O_{K}.

1. If {x_{α}} is a pseudo-solution of P, then there is an immediate extension of valued D-fields of the form K(⟨a⟩) in which P (a) = 0 and x_{α} ⇒ a.

2. If K(⟨a⟩) is an extension in which x_{α} ⇒ a and P (a) = 0, then K(⟨a⟩) is unique up to L_{K}-isomorphism.

3. P is potentially residually linear. In fact, for α ≫ 0 any refinement of P along {x_{α}} is residually linear.

Remark 7.33. Our proof of Proposition 7.32 is more complicated than one might expect it needs to be. The idea behind the proof is fairly simple, but technical problems arose for us. For the existence proof, one would like to take some sort of limit. Of course, {x_{α}} may be merely pseudo-convergent rather than convergent, so that there will not be a good notion of a completion. One might try to find the limit by working in some saturated extension and then specializing so as to eliminate excess infinitesimals. Instead, we employ an algebraic construction. For the uniqueness proof, one might like to argue that for Q ≺ P the sequence v(Q(x_{α})) is non-decreasing so that either {x_{α}} pseudo-solves Q and hence pseudo-converges to some a ∈ K by the inductive hypothesis and fullness (contradicting the strictness of the sequence) or the value settles down. Again some technical problems arise, notably with controlling the qualitative behavior of {v(Q(x_{α}))}, so that our actual proof is a bit more involved.

Remark 7.34. Proposition 7.32 allows us to finish extending f. We will arrange that the hypotheses are true of A_{1} by an inductive argument. We use potential residual linearity to see that after a linear change of variables, any solution to a Dpolynomial may be analyzed as an instance of DHL so that we can find the relevant solutions on both sides.

Definition 7.35. If {γ_{α}}_{α<κ} is a sequence of elements of Γ, then we say that the limit of the sequence exists iff the sequence is eventually constant. In that case, we write lim γ_{α} = γ where γ_{α} = γ for α ≫ 0.

Most of the lemmata proved in what follows will be employed to prove Proposition 7.32 and they depend inductively on Proposition 7.32. We indicate this by the condition

†: The hypotheses of Proposition 7.32 are assumed to hold and we assume inductively on T. degP that Proposition 7.32 is true.

Lemma 7.36 (†). Let Q ∈ O_{K}⟨X⟩. Suppose that Q ≺ P and DHL applies to Q at a ∈ O_{K}. There is some b ∈ K such that Q(b) = 0 and v(b − a) = v(Q(a)).

■ By Lemma 7.25 either the lemma is true or there is a strict pseudo-convergent sequence {y_{β}}_{β<κ} from K pseudo-solving Q with y_{0} = a and v(y_{β} − a) = v(Q(a)). By the inductive hypothesis (via †), there is an immediate extension K(⟨c⟩) in which Q(c) = 0 and y_{β} ⇒ c. By T. degP -fullness of K, {y_{β}} is not strict. This is a contradiction. ■

Lemma 7.37 (†). Let Q ∈ O_{K}⟨X⟩. If Q ≺ P, then lim v(Q(x_{α})) exists.

■ We prove this lemma by ≺-induction on Q. When Q is a constant D-polynomial, the result is obvious. We may now assume that Q is not constant. Thus, there is some i for which \frac{∂}{∂X_{i}} Q is not the zero D-polynomial. By the inductive hypothesis, lim v(\frac{∂}{∂X_{i}} Q(x_{α})) exists. We finish this proof by a series of lemmata. In each of these lemmata, we assume inductively † as well as

‡: Lemma 7.37 is true for Q̃ ≺ Q. ■

Lemma 7.38 (†, ‡). Let Q ≺ P. There is a pseudo-convergent sequence {y_{α}} having the same pseudo-limits as {x_{α}} such that if H_{α} is a refinement of Q at y_{α} with internal scale η_{α} with v(η_{α}) = v(y_{α+1} − y_{α}), then v(H_{α}(\frac{y_{α+1}−y_{α}}{η_{α}})) = 0.

Remark 7.39. While the lemma and certainly the proof are stated in terms of the refinements of Q, the point of this lemma is to understand Q itself. In terms of Q, we find a pseudo-convergent sequence {y_{α}} having the same pseudo-limits as {x_{α}} such that for any α one has v(Q(y_{α+1})−Q(y_{α})) = min\{v(\frac{∂}{∂X_{i}} Q(y_{α}))+v(y_{α+1}−y_{α}) : 0 ≤ i ≤ ord(Q)}.

■ We may assume that no cofinal sequence in {x_{α}} already works. For each α let G_{α} be a refinement of Q at x_{α} with internal scale ϵ_{α} having v(ϵ_{α}) = v(x_{α+1} − x_{α}). So for α ≫ 0 we have that v(G_{α}(\frac{x_{α+1}−x_{α}}{ϵ_{α}})) > 0.

We construct the sequence {y_{α}} allowing repetition and later thin using Lemma 7.24 to get an actual pseudo-convergent sequence.

Using the inductive hypotheses we may assume that the valuations of the partials of Q have stabilized (via ‡) and any refinement of Q along x_{α} is residually linear (via †).

Claim 7.40. For each α, there is some β > α minimal with the properties that v(G_{α}(\frac{x_{β}−x_{α}}{ϵ_{α}})) < v(ϵ_{β}) − v(ϵ_{α}) and v(G_{α}(\frac{x_{γ}−x_{α}}{ϵ_{α}})) ≤ v(G_{α}(\frac{x_{β}−x_{α}}{ϵ_{α}})) for γ ≥ β.

✠ There cannot be a cofinal sequence of γ’s on which v(G_{α}(\frac{x_{γ}−x_{α}}{ϵ_{α}})) is increasing for if this were to occur, by the inductive hypothesis for existence and fullness the sequence {x_{α}} would not be strict.

If the first condition were to fail, then since G_{α} is residually linear, DHL would apply at \frac{x_{β}−x_{α}}{ϵ_{α}} so that Lemma 7.36 would produce w_{β} with G_{α}(w_{β}) = 0 and v(w_{β} − \frac{x_{β}−x_{α}}{ϵ_{α}}) ≥ v(ϵ_{β}) − v(ϵ_{α}). Set z_{β} := x_{α} + ϵ_{α}w_{β}. We may restrict to a subsequence of {z_{β}} which is pseudo-convergent and has the same pseudo-limits as {x_{β}} by Lemma 7.24.

For each β, one has Q̃(z_{β}) := Q(z_{β}) − Q(x_{α}) = 0 so that by the inductive hypothesis, there is some extension K(ζ) of K in which z_{β} ⇒ ζ and Q̃(ζ) = 0. By fullness, {z_{β}} is not strict. Consequently {x_{α}} is not strict. This is a contradiction. ✠

We construct the sequence {y_{α}} now.

By Claim 7.40, there is a cofinal S ⊆ κ such that for any α ∈ S if α′ denotes the successor of α in S, then v(G_{α}(\frac{x_{α′} −x_{α}}{ϵ_{α}})) < v(ϵ_{α′}) − v(ϵ_{α}) and for any β ≥ α′ one has v(G_{α}(\frac{x_{β}−x_{α′}}{ϵ_{α}})) ≤ v(G_{α}(\frac{x_{α′} −x_{α}}{ϵ_{α}}). We restrict to S and omit it from the notation.

As in the proof of Claim 7.40, find z_{α} such that G_{α}(z_{α}) = 0 and v(z_{α} − \frac{x_{α+1}−x_{α}}{ϵ_{α}}) = v(G_{α}(\frac{x_{α+1}−x_{α}}{ϵ_{α}}). Define y_{α} := x_{α} +z_{α}ϵ_{α}. Since v(y_{α} −x_{α}) = v(z_{α}ϵ_{α}) ≥ v(ϵ_{α}) = v(x_{α+1} − x_{α}), Lemma 7.24 applies and we can find a cofinal pseudoconvergent subsequence of {y_{α}}_{α∈S} having the same pseudo-limits as those {x_{α}}.

Let us compute the valuation of y_{β} − y_{α} for β > α both from S.

v(y_{β} − y_{α}) = v(x_{β} + ϵ_{β}z_{β} − x_{α} − ϵ_{α}z_{α})
= v(ϵ_{β}z_{β} + ϵ_{α}(\frac{x_{β} − x_{α}}{ϵ_{α}} − z_{α}))
= v(ϵ_{α}) + v(G_{α}(\frac{x_{α+1} − x_{α}}{ϵ_{α}}))

The last equality uses the definition of z_{α} and the fact that v(ϵ_{β}) − v(ϵ_{α}) > v(G_{α}(\frac{x_{α+1}−x_{α}}{ϵ_{α}})) = v(G_{α}(\frac{x_{β}−x_{α}}{ϵ_{α}})).

To finish the proof, let θ ∈ Γ be min_{i}\{v(\frac{∂}{∂X_{i}} Q(x_{α}))\}. (Recall that these values do not depend on α.) In what follows, H_{β} denotes a refinement of Q at y_{β} with internal scale η_{β} having v(η_{β}) = v(y_{β+1} − y_{β}). Since H_{β} is residually linear, the minimal valuation of a coefficient of Q(y_{α} + η_{α}Y) − Q(y_{α}) is θ + v(η_{α}).

Let now α < β ∈ S

v(H_{α}(\frac{y_{β} − y_{α}}{η_{α}})) = v(Q(y_{β}) − Q(y_{α})) − θ − v(η_{α})
= v(Q(x_{β})) − Q(x_{α})) − θ − v(ϵ_{α}) − v(G_{α}(\frac{x_{α+1} − x_{α}}{ϵ_{α}}))
= v(G_{α}(\frac{x_{β} − x_{α}}{ϵ_{α}})) − v(G_{α}(\frac{x_{α+1} − x_{α}}{ϵ_{α}}))
= 0

The last equality is justified by the observations that H_{α} and its arguments are integral so that the final expression can be no less than zero, but it can be no more then zero because of the second condition of Claim 7.40. ■

Lemma 7.41 (†, ‡). If Q ≺ P, then there is some β < κ such that if L/K is any extension of valued D-fields and c ∈ L with v(x_{β+1} − c) > v(x_{β+1} − x_{β}), then v(Q(c)) = v(Q(x_{β})). Moreover, Q(x_{α}) ⇒ Q(c).

■ Let L be an extension of K (as a valued D-field) and in L take c with x_{α} ⇒ c. Let \{y_{α}\}, H_{α} and θ be as in Lemma 7.38. Since v(H_{β}(\frac{y_{β+1}−y_{β}}{η_{β}})) = 0 and y_{β} ⇒ c

(so that π(\frac{c−y_{β}}{η_{β}}) = π(\frac{y_{β+1}−y_{β}}{η_{β}})), we have

0 = v((H_{β}(\frac{c − y_{β}}{η_{β}}))
= v(Q(c) − Q(y_{β})) − θ − v(η_{β})

So v(Q(c) − Q(y_{β})) = θ + v(η_{β}). The right-hand side is growing with β so Q(x_{β}) ⇒ Q(c). This implies that either v(Q(c)) = v(Q(y_{β})) eventually (and we’re done) or v(Q(y_{β})) is increasing cofinally. In this second case, by † and fullness of K, {y_{β}} is not strict. This contradicts the strictness of {x_{α}}.

By the compactness theorem, there is some β < κ for which we have v(c−x_{β+1}) > v(x_{β} − x_{β+1}) ⇒ v(Q(c)) = v(Q(x_{β})). ■

Lemma 7.41 finishes the proof of Lemma 7.37.

Lemma 7.42 (†). For α ≫ 0, any refinement of P along x_{α} is residually linear.

■ By Lemma 7.37, lim v(∂_{I}P (x_{α})) exists for any non-zero multi-index I. From now on, work only with α large enough so that this common value has been attained.

For each α, let G_{α} be a refinement of P along x_{α} with scales ϵ_{α} and c_{α}. Write G_{α}(Y) = ∑_{I} g_{I,α}Y^{I}.

Claim 7.43. The set of multi-indices {I : v(g_{I,α}) = 0} does not depend on α for α sufficiently large. In fact, all such I have the same length.

✠ If it ever happens that for some I and J with |I| < |J| that v(g_{α,I}) ≤ v(g_{α,J}), then v(g_{β,I}) < v(g_{β,J}) for β > α. To see this, observe that the hypothesis is that

v(\frac{ϵ_{α}^{|I|}}{c_{α}}) + v(∂_{I}P (x_{α})) = v(g_{I,α})
≤ v(g_{J,α})
= v(\frac{ϵ_{α}^{|J|}}{c_{α}}) + v(∂_{J}P (x_{α}))

That is,

v(∂_{I}P (x_{α})) ≤ (|J| − |I|)v(ϵ_{α}) + v(∂_{J}P (x_{α}))

Since the valuations of the partials are the same whether evaluated at x_{α} or x_{β}, |J| − |I| > 0, and v(ϵ_{β}) > v(ϵ_{α}), we conclude

v(∂_{I}P (x_{β})) < (|J| − |I|)v(ϵ_{β}) + v(∂_{J}P (x_{β}))

Reversing the above manipulations, the claim follows. ✠

If the lemma were not true, then for some non-zero multi-indices I and J and cofinal sequence of α’s we would have

• J = (i_{0}, …, i_{j−1}, i_{j} + 1, i_{j+1}, …, i_{n}) where I = (i_{0}, …, i_{n}) and

• v(g_{I,α}) > 0 = v(g_{J,α}) Using the chain rule, we calculate

(4)  ∂_{I}G_{α}(Y) = \frac{ϵ_{α}^{|I|}}{c_{α}} ∂_{I}P (ϵ_{α}Y + x_{α})

By the expansion for G_{α} and the fact that v(∂_{I}P (x_{β})) is stable, we have

(5)  v(∂_{I}G_{α}(\frac{x_{β} − x_{α}}{ϵ_{α}})) = v(g_{I,α})

for β ≥ α. So by Equation 4 with J playing the rôle of I and Equation 5, we have

v(\frac{∂}{∂X_{j}} ∂_{I}G(\frac{x_{β} − x_{α}}{ϵ_{α}})) = v(g_{J,α})
= 0
< v(g_{I,α})
= v(∂_{I}G(\frac{x_{β} − x_{α}}{ϵ_{α}}))

for β > α.

Claim 7.44. There is another pseudo-convergent sequence {y_{δ}} having the same pseudo-limits as {x_{α}}_{α<κ} but pseudo-satisfying ∂_{I}P.

✠ We build {y_{δ}} while following along the sequence {x_{α}}.

Start with y_{0} := x_{0}. At stage α, if y_{β} = y is constant for α > β ≫ 0 (N.B.: If α is a successor, this condition will always be true.) and v(x_{α+1} − y) ≥ v(x_{α+1} − x_{α}), then set y_{α} := y.

(N.B.: Since {x_{α}} is strict, cofinally we will not be in this case.)

Otherwise, since DHL applies to ∂_{I}G_{α} at \frac{x_{α+1}−x_{α}}{ϵ_{ϵ}}, by Lemma 7.36 there is some w_{α} ∈ K such that ∂_{I}G_{α}(w_{α}) = 0 and v(w_{α} − \frac{x_{α+1}−x_{α}}{ϵ_{α}}) ≥ v(∂_{I}G_{α}(\frac{x_{α+1}−x_{α}}{ϵ_{α}})) > 0. Define y_{α} := ϵ_{α}w_{α} + x_{α}.

Let S := {α < κ : y_{β} is not constant for 0 ≪ β < α}. S is cofinal in κ, so {x_{α}}_{α∈S} is pseudo-convergent with the same pseudo limits as those of {x_{α}}_{α<κ}.

Equation 4 shows that if α ∈ S, then ∂_{I}P (y_{α}) = 0. By the construction, v(y_{α} − x_{α}) ≥ v(ϵ_{α}) so that by Lemma 7.24 there is a cofinal J ⊆ S such that {y_{α}}_{α∈J} is pseudo-convergent with the same pseudo-limits as those of {x_{α}}_{α∈S} and hence of {x_{α}}_{α<κ}. ✠

By the inductive hypothesis since ∂_{I}P ≺ P and {y_{α}} is a pseudo-solution to ∂_{I}P (X) = 0, there is some b in some immediate valued D-field extension of K such that y_{α} ⇒ b and ∂_{I}P (b) = 0. By T. degP -fullness of K, {y_{α}} is not strict. This is a contradiction. ■

Lemma 7.45 (†). Suppose that K(⟨a⟩) is a valued D-field extending K in which P (a) = 0 and x_{α} ⇒ a. Then every rescaling of P along x_{α} has integral co-efficients for α ≫ 0.

■ By Lemma 7.41 for any non-zero multi-index I we have v(∂_{I}P (a)) = v(∂_{I}P (x_{α})) for α ≫ 0. By Lemma 7.42 any refinement of P along x_{α} is residually linear. Work now only with α large enough so that the stable value of the partials has been achieved and so that the refinements are residually linear. A rescaling of P at x_{α}, with internal scale ϵ_{α} having v(ϵ_{α}) = v(x_{α+1} − x_{α}) must have an external scale c with valuation min\{v(\frac{∂}{∂X_{i}} P (x_{α})) + v(ϵ_{α})\} = min\{v(\frac{∂}{∂X_{i}} P (a)) + v(ϵ_{α})\}. Thus, the D-polynomial H(Y) := \frac{1}{c}[P (a + ϵ_{α}Y) − P (a)] = \frac{1}{c}P (a + ϵ_{α}Y) is a refinement of P at a. In particular, H(Y) ∈ O_{L}⟨Y ⟩. Since v(a − x_{α}) = v(x_{α+1} − x_{α}) = v(ϵ_{α}), the element \frac{x_{α}−a}{ϵ_{α}} is integral. Thus, 0 ≤ v(H(\frac{x_{α}−a}{ϵ_{α}})) = v(\frac{1}{c}P (x_{α})). That is, the rescaling of P at x_{α} with scales c and ϵ_{α} is integral. ■

Lemma 7.46 (†). The structure of a valued D-field on K(⟨a⟩) is determined by P (a) = 0 and x_{α} ⇒ a.

■ When e = 0, this is an immediate consequence of Lemma 7.41 since every element of K(⟨a⟩) is of the form \frac{Q(a)}{R(a)} with Q, R ≪ P.

Take now e ≠ 0. Lemma 7.41 shows that K(a, …, D^{m}a) is an immediate extension of K. So the extension K(a, …, D^{m}a)/K(a, …, D^{m−1}a) is generated by some single element b satisfying F (b) = 0, π(b) = 0 and v(F′(b)) = 0 for some irreducible F ∈ O_{K(a,... ,D^{m−1}a)}[X] by Proposition 1.1. Write σ(F) = ∏_{i=1}^{ℓ} G_{i} as a factorization into irreducibles of σ(F) in O_{K(a,... ,D^{m}a)}[X]. As zero is a simple root of the reduction of F, it is also a simple root of σ(F). Hence, π(G_{i}(0)) = 0 for exactly one choice of i, say i = 1, and zero is a simple root of π(G_{1}). The extension K(a, …, D^{m+1}a)/K(a, …, D^{m}a, b) is then generated by σ(b) which is the unique root of G_{1} which reduces to zero. As in the proof of Lemma 7.12, the extension of D to the henselization of O_{K(a,... ,D^{m}a)} is now determined. ■

At this point we would like to show that this analysis may be used to produce a valued D-field extending K determined by the data x_{α} ⇒ a and P (a) = 0.

Lemma 7.47 (†). Assume that {x_{α}} pseudo-satisfies P. There is an immediate extension of K of the form K(⟨a⟩) in which x_{α} ⇒ a and P (a) = 0.

■

We break the argument into cases depending on whether or not m := ordP is zero.

Consider first the case that m = 0. Let R be the henselization of O_{K}. Let L be the field of fractions of R. By Proposition 7.32 and Lemma 7.45 all rescalings of P along {x_{α}} are residually linear for α ≫ 0. If Q(Y) is one such rescaling at x_{α} with internal scale ϵ_{α} (having v(ϵ_{α}) = v(x_{α+1} − x_{α})), we set y_{β} := \frac{x_{β}−x_{α}}{ϵ_{α}}, and we find a solution b ∈ R to Q(b) = 0 and y_{β} ⇒ b, then a := x_{α} + bϵ_{α} is a solution to P and x_{β} ⇒ a. So, it suffices to find such a b.

Since Q is residually linear, there is a unique solution, call it b, to Q(X) = 0 in R with π(y_{β}) = π(b). Taylor expand Q about b to see that y_{β} ⇒ b. By Lemma 7.11, the D-structure on K extends to a D-structure on L with respect to which L is a valued D-field.

Work now in the case that m > 0. Write P (X) = F (X, DX, …, D^{m}X) for some F (X_{0}, …, X_{m}) ∈ O_{K}[X_{0}, …, X_{m}].

We let K′ be the field of fractions of K[X_{0}, …, X_{m}]/(F). Denote by D^{i}a the image of X_{i} in K′. We define a valuation on K′ by v(G(a, Da, …, D^{m}a)) := lim v(G(x_{α}, Dx_{α}, …, D^{m}x_{α})) for G ∈ K[X_{0}, …, X_{m}] whose degree in X_{m} is less than that of F. By Lemma 7.41 these limits exist and one checks easily that this defines a valuation extending v on K having the same value group. In particular, if R denotes the henselization of O_{K(a,Da,... ,D^{m−1}a)} and L is its field of fractions, then K′ embeds into L as a valued field.

We define φ : O_{K(a,Da,... ,D^{m−1}a)} → D_{e}(R) by φ(x) := (x, Dx) for x ∈ K and φ(D^{i}a) := (D^{i}a, D^{i+1}a) for 0 ≤ i < m. By the universal property of the henselization, this extends uniquely to ψ : R → D_{e}(R). Let D : R → R be the function defined by ψ(x) = (x, Dx). By Lemma 7.9, L is a valued D-field with respect to this valuation and D-structure. Moreover, P (a) = 0 in L and x_{α} ⇒ a. ■

We note now that the †-lemmata give a full proof of Proposition 7.32

■ Via † we proceed by induction on T. degP. Lemma 7.46 proves uniqueness. Lemma 7.47 proves existence. Lemma 7.42 proves potential residual linearity. ■

Proposition 7.32 is the last step needed to prove that any (k, G)-D-field may be embedded into a (k, G)-D-henselian field. We give the details of this in the next subsection.

To finish proving Theorem 7.1 it remains to show that for any particular b ∈ M_{1} we can enlarge A_{1} so that A_{1}(⟨b⟩) is an immediate extension of A_{1} and if P ∈ A_{1}⟨X⟩ is of minimal T. deg such that P (b) = 0, then A_{1} is T. degP -full. We show this using the following lemmata.

Lemma 7.48. If K ⊆ M_{1} is a sub-valued D-field whose value group is countable, then there is an unramified ∞-full extension L of K in M_{1}. If K has a linearly differentially closed residue field, one may take L to be an immediate extension.

■ Extend K to some K′ first so that the residue field of K′ is linearly differentially closed using Lemma 7.21. Let L be a maximal immediate extension of K′ inside M_{1}. We show now that this L works. Take a minimal counter-example to ∞- fullness of L. That is, find some strict pseudo-convergent sequence {x_{α}}_{α<κ} from L pseudo-solving some D-polynomial P (X) ∈ K⟨X⟩ and some a in an extension valued D-field for which P (a) = 0, x_{α} ⇒ a and T. degP is minimal.

Since v(K^{×}) is countable, we may assume that κ = ω. Replacing a with \frac{a}{ϵ}, x_{α} with \frac{x_{α}}{ϵ}, and P with P (ϵX) for some ϵ ∈ K^{D} with v(ϵ) = v(a), we may assume that v(a) = 0 = v(x_{α}). By Proposition 7.32 and Lemma 7.45, by replacing P with a rescaling we may assume that P has integral coefficents and is residually linear. We note that DHL applies to P at each x_{α} (α > 0): {x_{α}} pseudo-solves P, so v(P (x_{α})) is increasing. As everything in sight is integral, v(P (x_{α})) ≥ 0. Hence, for α > 0 we have v(P (x_{α})) > v(P (x_{0})) ≥ 0. As P is residually linear, there is some i such that v(\frac{∂}{∂X_{i}} P (x_{α})) = 0 or any α.

As M_{1} is ℵ_{1}-saturated and satisfies DHL, the partial type {P (y) = 0} ∪ {v(y − x_{n+1}) > v(y − x_{n})}_{n∈ω} is realized by some element b ∈ M_{1}. By Proposition 7.32 again the extension L(⟨b⟩) is immediate. As L is a maximal immediate extension of K′ inside M_{1}, b ∈ L. This contradicts the strictness of {x_{α}}. ■

Lemma 7.49. If L/K is an immediate extension of valued D-fields, L is ∞-full and has a linearly differentially closed residue field, and K has enough constants, then L may be realized as a direct limit L = ∪_{i<|L\K|}K_{i} where K_{0} = K, K_{λ} := ∪_{i<λ}K_{i} for λ a limit, and K_{i+1} = K_{i}(⟨a_{i}⟩) where K_{i} is T. degP -full for P (X) ∈ K_{i}⟨X⟩ a minimal D-polynomial for a_{i} over K_{i}.

■ Well-order L \ K in order type |L \ K|. Build K_{i} inductively. Set K_{0} := K. For i = λ, a limit ordinal, let K_{λ} := ∪_{i<λ}K_{i}. At stage i + 1, let N be min{T. deg(P) : P (X) ∈ K_{i}⟨X⟩, P (a) = 0 for some a ∈ L \ K_{i}}. Let a_{i} be minimal in the wellordering of L \ K such that the T. degP = N for P a minimal D-polynomial for a_{i} over K_{i}.

Claim 7.50. K_{i} is N-full.

✠ Consider a minimal counter-example to N-fullness:

• a strict pseudo-convergent sequence {x_{α}}_{α<κ} from K_{i},

• a in some immediate extension of K_{i} and

• Q(X) ∈ O_{K_{i}}⟨X⟩ satisfying

• T. degQ < N,

• Q(a) = 0,

• {x_{α}} pseudo-solves Q,

• x_{α} ⇒ a, and

• T. degQ is minimal subject to these conditions.

By Proposition 7.32 and ∞-fullness of L, {x_{α}} cannot continue to be strict in L. Let b ∈ L such that x_{α} ⇒ b. Since Q was chosen to have minimal T. deg among those D-polynomials witnessing the failure of N-fullness, K_{i} is T. degQ-full. Thus, Proposition 7.32 shows that Q is potentially residually linear along {x_{α}}. As {x_{α}} pseudo-solves Q, v(Q(x_{α})) is eventually increasing. Fix α large enough so that the valuations of the partials have stabilized, the refinements are residually linear, and for γ > β > α we have v(Q(x_{γ})) > v(Q(x_{β})) or Q(x_{β}) = 0. Let A := min\{v(\frac{∂}{∂X_{i}} Q(a))\}.

Let β > α. Taylor expand Q around x_{β} to conclude that v(Q(b)) ≥ min{v(Q(x_{β}), A+ v(ϵ_{β})} where as usual ϵ_{β} is a D-constant with v(ϵ_{β}) = v(x_{β+1} − x_{β}). If v(Q(b)) < A + v(ϵ_{β}), then we have v(Q(b)) = v(Q(x_{β})). This can happen at most once since either v(Q(x_{γ})) > v(Q(x_{β})) or Q(x_{γ}) = 0 for γ > β and our calculation used only the hypothesis that β > α. Thus, v(Q(b)) > A + v(ϵ_{β}) for any β > α. It follows that if H(Y) = \frac{1}{c}Q(x_{α} + ϵ_{α}Y) is a rescaling of Q at x_{α}, then v(H(\frac{b−x_{α}}{ϵ_{α}})) ≥ v(ϵ_{β}) − v(ϵ_{α}) > 0 for every β > α. By Lemma 7.36 there is some b′ ∈ L with H(b′) = 0 and v(b′ − \frac{b−x_{α}}{ϵ_{α}}) = v(H(\frac{b−x_{α}}{ϵ_{α}})). Let d := x_{α} + ϵ_{α}b′. Then Q(d) = 0 and x_{β} ⇒ d. The T. degQ-fullness of K_{i} contradicts the strictness of {x_{α}}. ✠ Set K_{i+1} := K_{i}(⟨a_{i}⟩). ■

We use these lemmata to finish the extension of f to b.

Proposition 7.51. If A_{1} has a countable value group, has a linearly differentially closed residue field and has enough constants and A_{1}(⟨b⟩) is an immediate extension of A_{1}, then f extends so that b is in the domain of f.

■ Let L be an ∞-full immediate extension of A_{1}(⟨b⟩) produced by Lemma 7.48. We will extend f to all of L. Express L = ⋃_{i<|L\A_{1}|} K_{i} as in Lemma 7.49. In extending f, the only issue comes at successor stages. At stage i + 1, we need to define f(a_{i}). Since K_{i} is T. degP -full where P is a minimal D-polynomial for a_{i} over K_{i} and K_{i} has enough constants and linearly differentially closed residue field, Proposition 7.32 implies that the extension K_{i}(⟨a_{i}⟩) is entirely determined by a strict pseudo-convergent approximation {x_{α}} to a_{i} (which can be taken to be countable) and the equation P (a_{i}) = 0. By Proposition 7.32 and Lemma 7.45, any rescaling of P along x_{α} for α ≫ 0 is integral and residually linear, so that DHL applies to the rescaling. Thus, by DHL in M_{2} and ℵ_{1}-saturation, there is some b_{i} ∈ M_{2} satisfying f(P)(b_{i}) = 0 and f(x_{α}) ⇒ b_{i}. Extend f by defining f(a_{i}) = b_{i}. ■

This completes the proof of completeness and quantifier elimination, though we recap the argument in the next subsection.

7.3. Recap. With the necessary lemmata now proven, let us go through the proof of Theorem 7.1 again.

Let us first prove quantifier elimination for the theory of (k, G)-D-henselian fields.

We fix a countable elementary substructure N_{1} ≺ M_{1} of M_{1} containing A_{1} and b. Using Proposition 7.20 we extend f so that its domain has enough constants and has value group equal to the value group of N_{1}. We then use Proposition 7.21 to extend f so that its domain has residue field containing k_{N_{1}}. Finally, we use Proposition 7.51 to extend f to an ∞-full immediate extension of N_{1}. In particular, the extension will be defined at b.

As to proving that the theory of (k, G)-D-henselian fields is the model completion of the theory of (k, G)-D-fields we use the existence parts of the lemmata cited above to show that any (k, G)-D-field may be embedded into a (k, G)-D-henselian field.

Let K be a (k, G)-D-field. We use the existence part of Proposition 7.19 to prove that K may be enlarged so as to have enough constants. We use Lemma 7.17 to enlarge K to satisfy Axiom 5 and so that its value group is a model of Th(G). We use the existence part of Lemma 7.12 to enlarge K so that its residue field is a model of Th(k). Using Lemma 7.25 we produce the necessary pseudo-convergent sequence in K and then use Proposition 7.32 to actually find a solution in an immeditate valued D-field extension.

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University of California, Berkeley, Department of Mathematics, Evans Hall #3840, Berkeley, California 94720-3840

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