FIELDS WITH SEVERAL COMMUTING DERIVATIONS

DAVID PIERCE

Abstract. For every natural number m, the existentially closed models of the theory of fields with m commuting derivations can be given a first-order geometric characterization in several ways. In particular, the theory of these differential fields has a model-companion. The axioms are that certain differential varieties determined by certain ordinary varieties are nonempty. There is no restriction on the characteristic of the underlying field.

How can we tell whether a given system of partial differential equations has a solution? An answer given in this paper is that, if we differentiate the equations enough times, and no contradiction arises, then it never will, and the system is soluble. Here, the meaning of ‘enough times’ can be expressed uniformly; this is one way of showing that the theory of fields with a given finite number of commuting derivations has a model-companion. In fact, this theorem is worked out here (as Corollary 4.6, of Theorem 4.5), not in terms of polynomials, but in terms of the varieties that they define, and the function-fields of these: in a word, the treatment is geometric.

The theory of fields with m commuting derivations will be called here m-DF; its model-companion, m-DCF. A specified characteristic can be indicated by a subscript. The model-companion of m-DF_{0} (in characteristic 0) has been axiomatized before, explicitly in terms of differential polynomials: see §3. The existence of a model-companion of m-DF (with no specified characteristic) appears to be a new result when m > 1 (despite a remark by Saharon Shelah [24, p. 315]: ‘I am quite sure that for characteristic p as well, [making m greater than 1] does not make any essential difference’).

The theory of model-companions and model-completions was worked out decades ago; perhaps for that very reason, it may be worthwhile to review the theory here, as I do in §1. In §2, I review the various known characterizations of existentially closed fields with single derivations. In fact, little of this work is of use in the passage to several derivations; but this near-irrelevance is itself interesting. In §3, I analyze the error of my earlier attempt, in [13], to axiomatize m-DCF_{0} in terms of differential forms. Something of value from this earlier work does remain: when we do have m-DCF_{0}, or more generally m-DCF, then we can obtain from it a modelcompanion of the theory of fields with m derivations whose linear span over the field is closed under the Lie bracket. In §4, I obtain m-DCF itself.

2000 Mathematics Subject Classification. 03C10 (03C60, 12H05).

Version of May 11, 2012.

1. Model-theoretic background

I try in this section to give original references, when I have been able to consult them. An exposition can also be found for example in Hodges [6] (particularly chapter 8).

Let Γ be a set of (first-order) sentences in some signature; the structures that have this signature and are models of Γ compose the class denoted by Mod(Γ). Every class K of structures of some signature has a theory, denoted by Th(K); this is the set of sentences in the signature that are true in each of the structures in K. Immediately K ⊆ Mod(Th(K)); in case of equality, K is called elementary. Similarly, Γ ⊆ Th(Mod(Γ)); in case of equality, that is, in case Γ is actually the theory of Mod(Γ), then Γ is called a theory, simply. So there is a Galois correspondence between elementary classes and theories.

Let 𝔐 be an arbitrary (first-order) structure; the theory of {𝔐} is denoted by Th(𝔐). The structure 𝔐 has the universe M. The structure denoted by 𝔐_{M} is the expansion of 𝔐 that has a name for every element of M. Then 𝔐 embeds in 𝔑 if and only if 𝔐_{M} embeds in an expansion of 𝔑. The class of structures in which 𝔐 embeds need not be elementary: for example, 𝔐 could be an uncountable model of a countable theory. However, the class of structures in which 𝔐_{M} embeds is elementary. The theory of the latter class is the diagram of 𝔐, or diag(𝔐): it is axiomatized by the quantifier-free sentences in Th(𝔐_{M}) [18, Thm 2.1.3, p. 24]. A model of Th(𝔐_{M}) itself is just a structure in which 𝔐_{M} embeds elementarily. Thus the class of such structures is elementary. The class of substructures of models of a theory T is elementary, and its theory is denoted by T_{∀}: this is axiomatized by the universal sentences of T [18, Thm 3.3.2, p. 71].

By a system over 𝔐, I mean a finite conjunction of atomic and negated atomic formulas in the signature of 𝔐_{M}; likewise, a system over a theory T is in the signature of T. A structure 𝔐 solves a system φ(x) if 𝔐 ⊧ ∃x φ(x). Note well here that x, in boldface, is a tuple of variables, perhaps (x^{0}, …, x^{n−1}). By an extension of a model of T, I mean another model of T of which the first is a substructure. Two systems over a model 𝔐 of T will be called equivalent if they are soluble in precisely the same extensions.

An existentially closed model of T is a model of T that solves every system over itself that is soluble in some extension. So a model 𝔐 of T is existentially closed if and only if T ∪ diag(𝔐) ⊢ Th(𝔐_{M})_{∀}, that is, every extension of 𝔐 is a substructure of an elementary extension ([5, §7] or [25, §2]).

A theory is model-complete if its every model is existentially closed. An equivalent formulation explains the name: T is model-complete if and only if T ∪ diag(𝔐) is complete whenever 𝔐 ⊧ T [19, Ch. 2].

Suppose every model of T has an existentially closed extension. Such is the case when T is inductive, that is, Mod(T) is closed under unions of chains [5, Thm 7.12]: equivalently, T = T_{∀∃} [9, 3]. Suppose further that we have a uniform first-order way to tell when systems over models of T are soluble in extensions: more precisely, suppose there is a function

φ(x, y) ⟼ φ̂(x, y),  (1)

where φ(x, y) ranges over the systems over T (with variables analyzed as shown), such that, for every model 𝔐 of T and every tuple a of parameters from M, the system φ(x, a) is soluble in some extension of 𝔐 just in case φ̂(x, a) is soluble in 𝔐. Then the existentially closed models of T compose an elementary class, whose theory T^{∗} is axiomatized by T together with the sentences

∀y (∃x φ̂(x, y) ⇒ ∃x φ(x, y)).  (2)

Immediately, T^{∗} is model-complete, so T^{∗} ∪ diag(𝔐) is complete when 𝔐 ⊧ T^{∗}. What is more, T^{∗} ∪ diag(𝔐) is complete whenever 𝔐 ⊧ T [18, Thm 5.5.1].

In general, T^{∗} is a model-completion of T if T^{∗}_{∀} ⊆ T ⊆ T^{∗} and T^{∗} ∪ diag(𝔐) is complete whenever 𝔐 ⊧ T. Model-completions are unique [17, (2.8)]. We have sketched the proof of part of the following; the rest is [17, (3.5)].

Lemma 1.1 (Robinson’s Criterion). Let T be inductive. Then T has a modelcompletion if and only if a function φ(x, y) ↦ φ̂(x, y) exists as in (1). In this case, the model-completion is axiomatized modulo T by the sentences in (2).

If T_{∀} = T^{∗}_{∀} and T^{∗} is model-complete, then T^{∗} is a model-companion of T ([1, §5]; cf. [5, §2]). Model-completions are model-companions, and model-companions are unique [1, Thm 5.3]. If T has a model-companion, then its models are just the existentially closed models of T [5, Prop. 7.10]. Conversely, if T is inductive, and the class of existentially closed models of T is elementary, then the theory of this class is the model-companion of T [5, Cor. 7.13].

2. Fields with one derivation

Let DF be the theory of fields with a derivation D, and let DPF be the theory of models of DF that, for each prime ℓ, satisfy also

∀x ∃y (ℓ = 0 & Dx = 0 ⇒ y^{ℓ} = x),

where the first occurrence of ℓ stands for 1+⋯+1 (with ℓ occurrences of 1); and y^{ℓ} stands for y ⋯ y (with ℓ occurrences of y). Models of DPF are called differentially perfect. A subscript on the name of one of these theories will indicate a required characteristic for the field. In particular, we have DPF_{0}, which is the same as DF_{0}.

Abraham Seidenberg [23] shows the existence of the function in Lemma 1.1 in case T is DPF_{p}, where p is prime or 0. Consequently:

Theorem 2.1 (Robinson). DF_{0} has a model-completion, called DCF_{0}.

Theorem 2.2 (Wood [30]). If p is prime, then DF_{p} has a model-companion, called DCF_{p}, which is the model-completion of DPF_{p}.

The existence of a model-companion or model-completion of a theory does not necessarily tell us much about the existentially closed models of the theory. Since it involves all systems over a given theory, Robinson’s criterion yields the crudest possible axiomatization for a model-completion. In the case of DCF_{p} (again where p is prime or 0), there are two ways of refining the axiomatization—refining in the sense of finding seemingly weaker conditions on models of DF_{p} that are still sufficient for being existentially closed. It suffices to consider either systems in only one variable or systems involving only first derivatives. In the generalization to several derivations, the former refinement seems to be of little use; the latter refinement is of use indirectly, through its introduction of geometric ideas.

2.1. Single variables. Though the theory ACF of algebraically closed fields is the model-completion of the theory of fields, its axioms (modulo the latter theory) can involve only systems in one variable (indeed, single equations in one variable). A generalization of this observation is the following, which can be extracted from the proof of [21, Thm 17.2, pp. 89–91] (see also [2]):

Lemma 2.3 (Blum’s Criterion). Say T^{∗}_{∀} ⊆ T ⊆ T^{∗}.

(i) The theory T^{∗} is the model-completion of T if and only if the commutative

diagram

\xymatrix{𝔐 \\ 𝔄 \ar[u] \ar[r] & 𝔅 \ar@{.>}[ul]}

of structures and embeddings can be completed as indicated when 𝔄 and 𝔅 are models of T and 𝔐 is a |B|^{+}-saturated model of T^{∗}.

(ii) If T = T_{∀}, it is enough to assume that 𝔅 is generated over 𝔄 by a single element.

This allows a refinement of Lemma 1.1 in a special case:

Lemma 2.4. Suppose T = T_{∀}. Then Lemma 1.1 still holds when φ(x, y) is replaced with φ(x, y) (where x is a single variable).

From Lemma 2.3, Lenore Blum obtains Theorem 2.5 below in characteristic 0, in which case the first two numbered conditions amount to K ⊧ ACF ([21, pp. 298 ff.] or [2]). If p > 0, then DPF_{p} is not universal, so part (ii) of Blum’s criterion does not apply; Carol Wood instead uses a primitive-element theorem of Seidenberg [22] to obtain new axioms for DCF_{p} [31]. These can be combined with Blum’s axioms for DCF_{0} to yield the following. (Here SCF is the theory of separably closed fields.)

Theorem 2.5 (Blum, Wood). A model (K, D) of DF is existentially closed if and only if

(i) K ⊧ SCF;

(ii) (K, D) ⊧ DPF;

(iii) (K, D) ⊧ ∃x (f(x, Dx, …, D^{n+1}x) = 0 & g(x, Dx, …, D^{n}x) ≠ 0) whenever f and g are ordinary polynomials over K in tuples (x^{0}, …, x^{n+1}) and (x^{0}, …, x^{n}) of variables respectively such that g ≠ 0 and ∂f/∂x^{n+1} ≠ 0.

Hence DF has a model-companion, called DCF.

There is a similar characterization of the existentially closed ordered differential fields [26].

2.2. First derivatives. Alternative simplified axioms for DCF are parallel to those found for the model-companion ACFA of the theory of fields with an automorphism [10, 4]. Suppose (K, D) ⊧ DPF and K ⊧ SCF. Given a system over (K, D), we can rewrite it so that D is applied only to variables or derivatives of variables; then we can replace each derivative with a new variable, obtaining a system

⋀_{f} f = 0 & g ≠ 0 & Dx = y,  (3)

where f, g ∈ K[x, y] and Dx = y stands for ⋀_{i} Dx^{i} = y^{i}. We can also incorporate this condition into the rest of the system, writing ⋀_{f} f(x, Dx) = 0 & g(x, Dx) ≠ 0.

Suppose (3) has the solution (a, b). Then K(a, b)/K is separable [14, Lem. 1.5, p. 1328]. Let V and W be the varieties over K with generic points a and (a, b) respectively, let T_{D}(V) be the twisted tangent bundle of V, and let U be the open subset of W defined by the inequation g ≠ 0. In characteristic 0, the model (K, D) of DF is existentially closed if and only if, in every such geometric situation, U contains a K-rational point (c, Dc); this yields the so-called geometric axioms for DCF_{0} found with Anand Pillay [15]. In positive characteristic, it is still true that, if (a, b) is a generic point of V, then D extends to K(a) so that Da = b. However, an additional condition is needed to ensure that D extends to all of K(a, b); it is enough to require that the projection of T_{D}(W) onto T_{D}(V) contain a generic point of W; this yields Piotr Kowalski’s geometric axioms for DCF_{p} [8].

By the trick of replacing g ≠ 0 with z · g = 1, we may assume that there is no inequation in (3). In an alternative geometric approach to DCF, we can then consider (3) as a special case of

⋀_{f} f = 0 & ⋀_{i<k} Dx^{i} = g^{i},  (4)

where f ∈ K[x^{0}, …, x^{n−1}] and g^{i} ∈ K(x^{0}, …, x^{n−1}). Suppose this has solution a, that is, (a^{0}, …, a^{n−1}), which is a generic point of a variety V. Then we know that D extends so as to map K(a) into some field. We know too that this extension of D maps the subfield K(a^{0}, …, a^{k−1}) of K(a) into K(a) itself; indeed, this extension is given by the equations Da^{i} = g^{i}(a). Now we can extend D further to all of K(a) so that this becomes a differential field [14, Lem. 1.2 (3), p. 1326]. Going back and picking a new generating tuple for K(a) as needed, we may assume that (a^{0}, …, a^{k−1}) is a separating transcendence-basis of K(a)/K. Then we have a dominant, separable rational map x ↦ (x^{0}, …, x^{k−1}) or φ from V onto 𝔸^{k}, and another rational map x ↦ (g^{0}(x), …, g^{k−1}(x)) or ψ from V to 𝔸^{k}. So (K, D) is existentially closed if and only if V always has a K-rational point P such that D(φ(P)) = ψ(P) [14, Thm 1.6, p. 1328].

3. Fields with several derivations

Let m-DF be the theory of fields with m commuting derivations. Tracey McGrail [12] axiomatizes the model-completion, m-DCF_{0}, of m-DF_{0}. Alternative axiomatizations arise as special cases in work of Yoav Yaffe [32] and Marcus Tressl [28]. There is a common theme: A differential ideal has a generating set of a special form; in the terminology of Joseph Ritt [16, §I.5, p. 5] (when m = 1) and Ellis Kolchin [7, §I.10, pp. 81 ff.], this is a characteristic set. There is a first-order way to tell, uniformly in the parameters, whether a given set of differential polynomials is a characteristic set of some differential ideal, and then to tell, if it is a characteristic set, whether it has a root. In short, the function φ ↦ φ̂ in Robinson’s criterion (Lemma 1.1) is defined for sufficiently many systems φ. (Applying Blum’s criterion, McGrail and Yaffe consider only systems in one variable, so they must include inequations in these systems; Tressl uses only equations, in arbitrarily many variables.)

I do not give the definition of a characteristic set, as not all ingredients of the definition are needed for the arguments presented in §4. However, some of the ingredients are needed; these are in 4.1.

3.1. Spaces of derivations. In [13] I attempted to apply the geometric approach described in 2.2 to m-DF_{0}. I worked more generally with DF_{0}^{m}, where DF^{m} is the theory of structures (K, D_{0}, …, D_{m−1}) such that (K, D_{i}) ⊧ DF for each i, and each bracket [D_{j}, D_{k}] is a K-linear combination of the D_{i}. (This is roughly what Yaffe did too.) In [14, §2] I made some minor corrections and otherwise adapted the argument to arbitrary characteristic. Nonetheless, in May, 2006, Ehud Hrushovski showed me a counterexample to [13, Thm A, p. 926], a theorem that was an introductory formulation of [13, Thm 5.7, p. 942]. Then I found an error at the end of the proof of the latter theorem. That theorem is simply wrong; the present paper does not so much correct the theorem as replace it.

The developments leading up to the wrong theorem are still of some use. The general situation is as follows. Let (K, D_{0}, …, D_{m−1}) ⊧ DF^{m}, and let E be the K-linear span of the D_{i}. Then E is a Lie-ring, as well as a vector-space over K. As a vector-space, E has a dual, E^{∗}; and there is a derivation d from K into E^{∗} given by D(d x) = Dx. Then E^{∗} has a basis (d t^{i}: i < ℓ) for some t^{i} in K and some ℓ no greater than m [13, Lem. 4.4, p. 932], and this basis is dual to a basis (∂_{i}: i < ℓ) of E, where [∂_{i}, ∂_{j}] = 0 in each case, and d can be given by

d x = ∑_{i<ℓ} d t^{i} · ∂_{i}x  (5)

[13, Lem. 4.7, p. 934]. We can use these ideas to prove the following.

Theorem 3.1. If m-DF has a model-companion, then so does DF^{m}.

Proof. In characteristic 0, the result is implicit in [13, Thm 5.3 and proof], explicit in [27, §3]; but the proof works generally. The main point is to find, for any model (K, D_{0}, …, D_{m−1}) of DF^{m}, an extension in which the named derivations are linearly independent over the larger field. As above, the space spanned over K by the D_{i} has a basis (∂_{i}: i < ℓ) of commuting derivations of K. If ℓ < m, then let L = K(α^{ℓ}, …, α^{m−1}), where (α^{ℓ}, …, α^{m−1}) is algebraically independent over K. Extend the ∂_{i} to L so that they are 0 at the α^{j}; then, if ℓ ⩽ k < m, define ∂_{k} to be 0 on K and to be δ_{k}^{j} at α^{j}. Then (∂_{i}: i < m) is a linearly independent m-tuple of commuting derivations on L; from this, we obtain linearly independent extensions D̃_{i} of the D_{i} to L such that the brackets [D̃_{j}, D̃_{k}] are the same linear combinations of the D̃_{i} that the [D_{j}, D_{k}] are of the D_{i} (by [13, Lem. 5.2, p. 937]— or [27, Lem. 2.1, p. 1930], by a different method—in characteristic 0; generally, [14, Lem. 2.4, p. 1334]). Then (K, D_{i}, …, D_{m−1}) ⊆ (L, D̃_{0}, …, D̃_{m−1}), and the latter is a model of DF^{m}. Moreover, (L, D̃_{0}, …, D̃_{m−1}) is an existentially closed model if and only if (L, ∂_{0}, …, ∂_{m−1}) is an existentially closed model of DF^{m}; so a model-companion of DF^{m} can be derived from a model-companion of m-DF. □

That much stands, and differential forms are convenient for establishing it. The theorem, combined with the results of §4, will yield a model-companion, DCF^{m}, of DF^{m}.

3.2. A new approach. In [13] I tried also to obtain DCF_{0}^{m} independently as follows. Suppose now we have a separably closed field K, along with a Lie-ring and finite-dimensional space E of derivations of K; as a space, E has (for some m) a basis (∂_{i}: i < m), whose dual is (d t^{i}: i < m), so that the ∂_{i} commute. We may assume that (K, ∂_{0}, …, ∂_{m−1}) is differentially perfect [14, Lem. 2.4]. Every system over (K, ∂_{0}, …, ∂_{m−1}) is equivalent to a system of the form of (4), generalized to

⋀_{f} f = 0 & ⋀_{j<k} ⋀_{i<m} ∂_{i}x^{j} = g_{i}^{j}.  (6)

Here again f ∈ K[x^{0}, …, x^{n−1}], and g_{i}^{j} ∈ K(x^{0}, …, x^{n−1}). By means of (5), we can also write the system as

⋀_{f} f = 0 & ⋀_{j<k} d x^{j} = ∑_{i<m} d t^{i} · g_{i}^{j}.  (7)

If (a^{0}, …, a^{n−1}) or a is a solution (from some extension), we may assume that (a^{0}, …, a^{k−1}) is a separating transcendence-basis of K(a^{0}, …, a^{k−1})/K. However, we can no longer assume that (a^{0}, …, a^{k−1}) is a separating transcendence-basis of K(a)/K itself. In characteristic 0, this is shown by the example referred to in [13, Exple 1.2, p. 927]; we can adapt the example to positive characteristic p by letting K = 𝔽_{p}(b^{σ}: σ ∈ ω^{2}) and defining ∂_{0}b^{(i,j)} = b^{(i+1,j)} and ∂_{1}b^{(i,j)} = b^{(i,j+1)}. Let (a^{1}, a^{2}) be algebraically independent over K, but a^{0} = (a^{2})^{p}, and define ∂_{0}a^{0} = 0 = ∂_{1}a^{0} and ∂_{0}a^{1} = b^{(0,0)} and ∂_{1}a^{1} = a^{2}. Then the ∂_{i} are commuting derivations mapping K(a^{0}, a^{1}) into K(a^{0}, a^{1}, a^{2}), and they extend as commuting derivations to the latter field, but not so as to map this field into itself. This is an important difference from the case of one derivation; it is what causes the difficulties in the case of several derivations.

In our example, K(a^{0}, a^{1}, a^{2})/K(a^{0}, a^{1}) is not separable. However, if we let a^{3} be a new transcendental and define ∂_{0}a^{2} = b^{(0,1)} and ∂_{1}a^{2} = a^{3}, then the ∂_{i} are still commuting derivations, now mapping K(a^{0}, a^{1}, a^{2}) into K(a^{0}, a^{1}, a^{2}, a^{3}), and the larger field is indeed separable over the subfield. This will turn out to be possible in general. That is, in (7), we shall be able to assume that (a^{0}, …, a^{k−1}) is a separating transcendence-basis of K(a^{0}, …, a^{k−1})/K and it extends to a separating transcendence-basis (a^{0}, …, a^{ℓ−1}) of K(a)/K. This is obvious in characteristic 0.

The solution a to (7) then can be understood as follows. First we have the field K(a), and then (7) can be be written as

⋀_{j<k} d a^{j} = ∑_{i<m} d t^{i} · g_{i}^{j}(a).  (8)

A solution of this can be understood as a model (L, ˜∂_{0}, …, ˜∂_{m−1}) of m-DF extending (K, ∂_{0}, …, ∂_{m−1}) such that K(a) ⊆ L and also (8) holds when d a^{j} = ∑_{i<m} d t^{i} · ˜∂_{i}a^{j}. This last condition is

⋀_{j<k} ⋀_{i<m} ˜∂_{i}a^{j} = g_{i}^{j}(a).  (9)

Since the ˜∂_{i} commute, it is necessary that

⋀_{j<k} ⋀_{i<m} ⋀_{h<i} ˜∂_{h}(g_{i}^{j}(a)) = ˜∂_{i}(g_{h}^{j}(a))  (10)

[13, §1, p. 926]. Any derivative with respect to ˜∂_{i} of an element of K(a) is a constant plus a linear combination of the derivatives ˜∂_{i}a^{j}, where j < ℓ (by [14, Fact 1.1 (0, 2)], for example); we know what these derivatives ˜∂_{i}a^{j} are when j < k, by (9); so (10) becomes a linear system over K(a) in the unknowns ˜∂_{i}a^{j} where k ⩽ j < ℓ.

If k = ℓ, then this linear system has no variables, so it is true or false; its truth is a sufficient condition for (8) to have a solution. If k < ℓ, then the linear system is soluble or not. If it is soluble, then it is possible to extend the ∂_{i} to derivations ˜∂_{i} as required by (9) that commute on K(a^{0}, …, a^{k−1}); but these derivations need not commute on all of K(a). In [13] I claimed that they could commute, and that the solubility of (10) was sufficient for solubility of (8) in the sense above. I was wrong.

A generic solution to the linear system (10) generates an extension of K(a); then we have to check extensibility of the commuting derivations to this. That is, we are back in the same kind of situation we started with. However, it turns out that there is a bound on the number of times that we need to repeat this process in order to ensure solubility of the original differential system. This is what is shown in §4 below; differential forms are apparently not useful for this after all.

3.3. A counterexample. Over a model of DF^{2}, let (a, b, c) be an algebraically independent triple. The counterexample supplied by Hrushovski is the system

d a = d t^{0} · c^{2} + d t^{1} · c,  d b = d t^{0} · 2a + d t^{1} · c  (11)

(where c^{2} is the square of c; the constants (k, ℓ) of §3.1 are now (2, 3)). Equivalently, by (5), the system comprises the equations

∂_{0}a = c^{2},  ∂_{1}a = c,  ∂_{0}b = 2a,  ∂_{1}b = c.

From these, we compute

∂_{1}∂_{0}a = 2c · ∂_{1}c, ∂_{0}∂_{1}a = ∂_{0}c, ∂_{1}∂_{0}b = 2 · ∂_{1}a = 2c, ∂_{0}∂_{1}b = ∂_{0}c.

Equating ∂_{0}∂_{1} and ∂_{1}∂_{0} yields the linear system

∂_{0}c − 2c · ∂_{1}c = 0,  ∂_{0}c = 2c,  (12)

which has the solution (∂_{0}c, ∂_{1}c) = (2c, 1). But then we must have ∂_{1}∂_{0}c = 2·∂_{1}c = 2, while ∂_{0}∂_{1}c = ∂_{0}1 = 0, which means (11) has no solution, contrary to my claim in [13].

For the record, the mistake is at the end of the proof of [13, Thm 5.7, p. 942] and can be seen as follows. Write the system (11) as d a = α, d b = β; then

d α = d c ∧ (d t^{0} · 2c + d t^{1}),

d β = d a ∧ d t^{0} · 2 + d c ∧ d t^{1}

= (d c − d t^{0} · 2c) ∧ d t^{1}. (13)

Since also d β = d^{2} b = 0, we now have a condition on d c ∧ d t^{1}, hence on ∂_{0}c; in particular, ∂_{0}c = 2c, which is what we found above. But there is no apparent condition on ∂_{1}c, so I try introducing a new transcendental, d, for this derivative. By (12) then,

d c = d t^{0} · 2c + d t^{1} · d,

which by (13) yields d α = d t^{0} ∧ d t^{1} · 2c(1 − d). But we must have d α = 0, so d = 1, contrary to assumption. In short, the next to last sentence of the proof of [13, Thm 5.7] (beginning ‘This ideal is linearly disjoint from’) is simply wrong. (I had not attempted to argue that it was correct.)

4. Resolution

For a correct understanding of the existentially closed differential fields, it is better not to introduce differential forms from the beginning, but to allow equations to involve any number of applications of the derivations. In contrast to 2.1 above, there does not seem to be an advantage now in restricting attention to equations in one variable.

4.1. Terminology. I shall now avoid working with differential polynomials as such, but shall work instead with the algebraic dependencies that they determine.

Let (K, ∂_{0}, …, ∂_{m−1}) ⊧ m-DF. Higher-order derivatives with respect to the ∂_{i} can be indexed by elements of ω^{m}: so, for ∂_{0}^{σ(0)} ⋯ ∂_{m−1}^{σ(m−1)}x, we may write ∂^{σ}x. Let ⩽ be the product ordering of ω^{m}. Then the derivative ∂^{σ}x is below ∂^{τ} x (and the latter is above the former) if σ ⩽ τ. (In particular, a derivative is both below and above itself.) If n ∈ ω, then two elements of ω^{n} × n will be related by ⩽ only if they agree in the last coordinate, so that

(σ, k) ⩽ (τ, ℓ) ⟺ σ ⩽ τ & k = ℓ;

we may use the corresponding terminology of ‘above’ and ‘below’, so that ∂^{σ}x_{k} is below ∂^{τ} x_{ℓ} if (and only if) (σ, k) ⩽ (τ, ℓ).

If σ ∈ ω^{m}, let the sum ∑_{i<m} σ(i) be denoted by |σ|: this is the height of σ or of ∂^{σ}x. (Kolchin [7, §I.1, p. 59] uses the word order.) If n is a positive integer, let ω^{m} × n be (totally) ordered by ⊴, which is taken from the left lexicographic ordering of ω^{m+1} by means of the embedding

(ξ, k) ⟼ (|ξ|, k, ξ(0), …, ξ(m − 2))

of ω^{m} × n in ω^{m+1}. Then (ω^{m} × n, ⊴) is isomorphic to (ω, ⩽). We may write (σ, k) ◁ ∞ for all (σ, k) in ω^{m}×n. Suppose (x_{h}: h < n) is a tuple of indeterminates. By ordering the formal derivatives ∂^{σ}x_{k} in terms of (σ, k) and ⊴, we have Kolchin’s example of an orderly ranking of derivatives [7, §I.8, p. 75]. If (σ, k) ◁ (τ, ℓ), I shall say that the derivative ∂^{σ}x_{k} is less than ∂^{τ} x_{ℓ} or is a predecessor of ∂^{τ} x_{ℓ}, and ∂^{τ} x_{ℓ} is greater than ∂^{σ}x_{k}; likewise for the expressions a_{k}^{σ} and a_{ℓ}^{τ}, introduced in (15) below. (So, the terms just defined refer to the strict total ordering ◁, while ‘below’ and ‘above’ refer to the partial ordering ⩽.)

Addition and subtraction on ω induce corresponding operations on ω^{m}. Then

τ ⩽ σ + τ,
∂^{σ}∂^{τ} x_{k} = ∂^{σ+τ} x_{k},
(σ, k) ⊴ (σ + τ, k),  (14)
(σ, k) ⊴ (τ, ℓ) ⟺ (σ + ρ, k) ⊴ (τ + ρ, ℓ).

If i < m, let i denote the characteristic function of {i} in ω^{m}, so that ∂^{i} = ∂_{i}, and more generally ∂_{i}∂^{σ} = ∂^{σ+i}, and also ∂_{i}∂^{σ−i} = ∂^{σ} if σ(i) > 0.

Let L be an extension of K with generators that are indexed by an initial segment of (ω^{m} × n, ⊴); that is,

L = K(a_{h}^{ξ}: (ξ, h) ◁ (τ, ℓ)),  (15)

where (τ, ℓ) ∈ ω^{m} × n, or possibly (τ, ℓ) = ∞, in which case L = K(a_{h}^{ξ}: (ξ, h) ∈ ω^{m} × n). It could happen that, in the generating tuple (a_{h}^{ξ}: (ξ, h) ◁ (τ, ℓ)) of L/K, the same element of L may appear twice, with different indices. In this case, when writing a_{h}^{ξ}, we may mean not just a particular element of L, but that element together with the pair (ξ, h) of indices. For example, by (14), if (σ + i, k) ◁ (τ, ℓ), then (σ, k) ◁ (τ, ℓ); hence we may say that, if a_{k}^{σ+i} is one of the generators of L/K, then so is a_{k}^{σ}. Let us say that L, with the tuple of generators given in (15), meets the differential condition if there is no obstacle to extending each derivation ∂_{i} to a derivation D_{i} on K(a_{h}^{ξ}: (ξ + i, h) ◁ (τ, ℓ)) such that

D_{i}a_{k}^{σ} = a_{k}^{σ+i}  (16)

whenever (σ + i, k) ◁ (τ, ℓ). (If the right-hand member of (16) is not defined, then the left need not be defined.) To be precise, if f is a rational function over K in variables (x_{h}^{ξ}: (ξ, h) ⊴ (σ, k)) for some (σ, k) in ω^{m} ×n, and D is a derivation of K, then f has a derivative Df, which is the linear function over K(x_{h}^{ξ}: (ξ, h) ⊴ (σ, k)) given by

Df = ∑_{(ξ,h)⊴(σ,k)} \frac{∂f}{∂x_{h}^{ξ}} · y_{h}^{ξ} + f^{D}.

Then the differential condition is that for all such f, if (σ + i, k) ◁ (τ, ℓ) for some i in m, and if

f(a_{h}^{ξ}: (ξ, h) ⊴ (σ, k)) = 0,  (17)

then ∂_{i}f(a_{h}^{ξ}, a_{h}^{ξ+i}: (ξ, h) ⊴ (σ, k)) = 0, that is,

∑_{(η,g)⊴(σ,k)} \frac{∂f}{∂x_{g}^{η}} (a_{h}^{ξ}: (ξ, h) ⊴ (σ, k)) · a_{g}^{η+i} + f^{∂_{i}}(a_{h}^{ξ}: (ξ, h) ⊴ (σ, k)) = 0. (18)

(Note well the assumption that (σ + i, k) ◁ (τ, ℓ). In (18), each of the a_{g}^{η+i} must exist, even though the coefficient (∂f/∂x_{g}^{η})(a_{h}^{ξ}: (ξ, h) ⊴ (σ, k)) might be 0.) So the differential condition is necessary for the extensibility of the ∂_{i} as desired (see for example [14, Fact 1.1 (0)]); sufficiency is part of Lemma 4.1 below.

An extension (M, D_{0}, …, D_{m−1}) of (K, ∂_{0}, …, ∂_{m−1}) is compatible with the extension L of K given in (15) if L ⊆ M, and (16) holds whenever (σ+i, k) ◁ (τ, ℓ).

Borrowing some terminology used for differential polynomials [16, §IX.1, p. 163], let us say that a generator a_{k}^{σ} of L/K is a leader if it is algebraically dependent over K on its predecessors, that is,

a_{k}^{σ} ∈ K(a_{h}^{ξ}: (ξ, h) ◁ (σ, k))^{alg}.

Then a_{k}^{σ} is a separable leader if it is separably algebraic over K(a_{h}^{ξ}: (ξ, h) ◁ (σ, k)); otherwise, it is an inseparable leader. A separable leader a_{k}^{σ} is minimal if there is no separable leader strictly below it—no separable leader a_{k}^{ρ} such that ρ < σ.

Lemma 4.1. Suppose (K, ∂_{0}, …, ∂_{m−1}) ⊧ m-DF and L meets the differential condition, where L is an extension K(a_{h}^{ξ}: (ξ, h) ◁ (τ, ℓ)) of K. Then the derivations ∂_{i} extend to derivations D_{i} from K(a_{h}^{ξ}: (ξ + i, h) ◁ (τ, ℓ)) into L such that (16) holds when (σ + i, k) ◁ (τ, ℓ). If a_{k}^{σ} is a separable leader, and (σ + i, k) ◁ (τ, ℓ), then

a_{k}^{σ+i} ∈ K(a_{h}^{ξ}: (ξ, h) ◁ (σ + i, k))  (19)

(that is, a_{k}^{σ+i} is a rational function over K of its predecessors); in particular, a_{k}^{σ+i} is a separable leader. Therefore generators of L/K that are above separable leaders are themselves separable leaders.

Proof. The claim follows from the basic properties of derivations, such as are gathered in [14, §1]. We extend the derivations to each generator in turn, according to the ordering ◁. Suppose D_{i} has been defined as desired on K(a_{h}^{ξ}: (ξ, h) ◁ (k, σ)) (so that (16) holds whenever it applies). If a_{k}^{σ} is not a leader, then we are free to define the derivatives D_{i}a_{k}^{σ} as we like. Now suppose a_{k}^{σ} is a separable leader, and (σ +i, k) ◁ (τ, ℓ). Then D_{i}a_{k}^{σ} is obtained by differentiating the minimal polynomial of a_{k}^{σ} over K(B). That is, D_{i}a_{k}^{σ} is obtained by differentiating an equation like (17); by the differential condition, D_{i}a_{k}^{σ} must be a_{k}^{σ+i} as given by (18); this shows that a_{k}^{σ+i} is a rational function over K of its predecessors.

Finally, in a positive characteristic p, a_{k}^{σ} may be an inseparable leader. Then (a_{k}^{σ})^{p^{r}} ∈ K(a_{h}^{ξ}: (ξ, h) ◁ (σ, k))^{sep} for some positive r. If (σ + i, k) ◁ (τ, ℓ), then we are free to define D_{i}a_{k}^{σ} as a_{k}^{σ+i}, provided D_{i}((a_{k}^{σ})^{p^{r}}) = 0. But again this condition is ensured by the differential condition. Indeed, we may suppose (17) shows the separable dependence of (a_{k}^{σ})^{p^{r}} over the predecessors of a_{k}^{σ}. That is, we can understand f(a_{h}^{ξ}: (ξ, h) ⊴ (σ, k)) as g((a_{k}^{σ})^{p^{r}}) for some separable polynomial g over K(a_{h}^{ξ}: (ξ, h) ◁ (σ, k)). Then D_{i}((a_{k}^{σ})^{p^{r}}) is obtained from (18), provided we replace the term (∂f/∂x_{k}^{σ})(a_{h}^{ξ}: (ξ, h) ⊴ (σ, k)) · a_{k}^{σ+i} with g′((a_{k}^{σ})^{p^{r}}) · D_{i}((a_{k}^{σ})^{p^{r}}). But in the present case, the former term is 0. Since, after the replacement, the resulting equation still holds, we must have D_{i}((a_{k}^{σ})^{p^{r}}) = 0. □

4.2. A solubility condition. If (K, ∂_{0}, …, ∂_{m−1}) ⊧ m-DF, then this model has an extension whose underlying field is the separable closure of K (as by [13, Lem. 3.4, p. 930] and [14, Lem. 2.4, p. 1334]). We shall need this in a more general form:

Lemma 4.2. Suppose a field M has two subfields L_{0} and L_{1}, which in turn have a common subfield K. For each i in 2, suppose there is a derivation D_{i} mapping K into L_{i} and L_{1−i} into M. Then the bracket [D_{0}, D_{1}] is a well-defined derivation on K. Suppose it is the 0-derivation. Suppose also that a is an element of M that is separably algebraic over K. Then each D_{i} extends uniquely to K(a), and D_{i}a ∈ L_{1−i}(a), so D_{1−i}D_{i}a is also well-defined. Moreover, [D_{0}, D_{1}]a = 0.

Proof. Obvious from [14, Fact 1.1 (2)]. □

In a positive characteristic, the possibility of inseparably algebraic extensions presents a challenge, which however is handled by the following.

Theorem 4.3. Suppose (K, ∂_{0}, …, ∂_{m−1}) ⊧ m-DF, and K has an extension K(a_{h}^{ξ}: |ξ| ⩽ 2r & h < n) meeting the differential condition for some positive integers r and n. Suppose further that, whenever a_{k}^{σ} is a minimal separable leader, then |σ| ⩽ r. Then (K, ∂_{0}, …, ∂_{m−1}) has an extension (M, D_{0}, …, D_{m−1}) compatible with K(a_{h}^{ξ}: |ξ| < 2r & h < n).

Proof. The claim can be compared to and perhaps derived from a differential-alge-braic lemma of Rosenfeld [20, §I.2], at least in characteristic 0. Here I give an independent argument, for arbitrary characteristic. We shall obtain M recursively as K(a_{h}^{ξ}: (ξ, h) ∈ ω^{m} × n), at the same time proving inductively that the ∂_{i} can be extended to D_{i} so that (16) holds in all cases.

Let L = K(a_{h}^{ξ}: |ξ| < 2r & h < n); this is K(a_{h}^{ξ}: (ξ, h) ⊴ ((2r−1, 0, …, 0), n−1)). Then by (18), the differential condition requires of the tuple (a_{h}^{ξ}: |ξ| = 2r & h < n) only that it solve some linear equations over L. The hypothesis of our claim is that there is a solution, namely (a_{h}^{ξ}: |ξ| = 2r & h < n). We may therefore assume that this tuple is a generic solution of these equations. In particular, no entry of this tuple is an inseparable leader. (If, instead of being chosen generically, the entries of (a_{h}^{ξ}: |ξ| = 2r & h < n) were chosen from the field L, then this field would be closed under the desired extensions D_{i} of ∂_{i}, and the derivations D_{i} would commute on the subfield K(a_{h}^{ξ}: |ξ| + 1 < 2r & h < n); but they might not commute on all of L.)

Now, as an inductive hypothesis, suppose we have the extension K(a_{h}^{ξ}: (ξ, h) ◁ (τ, ℓ)) of K meeting the differential condition, so that there are derivations D_{i} as given by Lemma 4.1; suppose also that

(i) if a_{k}^{σ} is a minimal separable leader, then |σ| ⩽ r; (ii) if a_{k}^{σ} is an inseparable leader, then |σ| < 2r.

We need to choose a_{ℓ}^{τ} in such a way that these conditions still hold for K(a_{h}^{ξ}: (ξ, h) ⊴ (τ, ℓ)). The inductive hypothesis is correct when |τ| ⩽ 2r, and then the desired conclusion follows; so we may assume |τ| > 2r. Hence, if τ(i) > 0, so that τ − i is defined, then |τ − i| ⩾ 2r, so a_{ℓ}^{τ−i} is not an inseparable leader.

If a_{ℓ}^{τ−i} is not a leader at all, for any i in m, then we may let a_{ℓ}^{τ} be a new transcendental, and we may define each derivative D_{i}a_{ℓ}^{τ−i} as this [14, Fact 1.1 (1)].

In the other case, a_{ℓ}^{τ−i} is a separable leader for some i. Then D_{i}a_{ℓ}^{τ−i} is determined (Lemma 4.1). We want to let a_{ℓ}^{τ} be this derivative. However, possibly also a_{ℓ}^{τ−j} is a separable leader, where i ≠ j. In this case, we must check that

D_{j}a_{ℓ}^{τ−j} = D_{i}a_{ℓ}^{τ−i},  (20)

that is, [D_{i}, D_{j}]a_{ℓ}^{τ−i−j} = 0.

There are minimal separable leaders a_{ℓ}^{π} and a_{ℓ}^{ρ} below a_{ℓ}^{τ−i} and a_{ℓ}^{τ−j} respectively. Let ν be π ∨ ρ, the least upper bound of {π, ρ} with respect to ⩽. Then ν ⩽ τ. But |ν| ⩽ |π| + |ρ| ⩽ 2r < |τ|; so ν < τ. Hence ν ⩽ τ − k for some k in m, which means a_{ℓ}^{ν} is below a_{ℓ}^{τ−k}. Consequently,

(i) a_{ℓ}^{π} is below both a_{ℓ}^{τ−i} and a_{ℓ}^{τ−k};
(ii) a_{ℓ}^{ρ} is below both a_{ℓ}^{τ−j} and a_{ℓ}^{τ−k}.

If k = j, then a_{ℓ}^{π} is below a_{ℓ}^{τ−i−j}, so this is a separable leader. As D_{i} and D_{j} commute on K(a_{h}^{ξ}: (ξ, h) ◁ (τ − i − j, ℓ)) by the differential condition, they must commute also at a_{ℓ}^{τ−i−j} (Lemma 4.2), so (20) is established. The argument is the same if k = i. If k is different from i and j, then again the same argument yields D_{j}a_{ℓ}^{τ−j} = D_{k}a_{ℓ}^{τ−k} and D_{k}a_{ℓ}^{τ−k} = D_{i}a_{ℓ}^{τ−i}, so (20) holds.

In no case did we introduce a new minimal separable leader or an inseparable leader. This completes the induction and the proof. □

In terms of differential polynomials and ideals, the theorem can be understood as follows. Given the hypothesis of the theorem, let S be the set of differential polynomials f(∂^{ξ}x_{h}: |ξ| < 2r & h < n), where f ranges over the ordinary polynomials over K such that f(a_{h}^{ξ}: |ξ| < 2r & h < n) = 0. Then S includes a characteristic set for the differential ideal that it generates.

We can now characterize the existentially closed models of m-DF by means of the following lemma. The lemma follows from unproved statements in [7, §0.17, p. 49]; let’s just prove it here.

Lemma 4.4. For every m in ω and positive integer n, every antichain of (ω^{m} × n, ⩽) is finite.

Proof. The general case follows from the case when n = 1, since if S is an antichain of (ω^{m} × n, ⩽), then

S = ⋃_{j<n}{(ξ, h) ∈ S: h = j},

and each component of the union is in bijection with an antichain of (ω^{m}, ⩽). As an inductive hypothesis, suppose every antichain of (ω^{ℓ}, ⩽) is finite; but suppose also, if possible, that there is an infinite antichain S of (ω^{ℓ+1}, ⩽). Then S contains some σ. By inductive hypothesis, the subset

⋃_{j⩽ℓ} ⋃_{i⩽σ(j)}{ξ ∈ S: ξ(j) = i}

of S is a finite union of finite sets, so its complement in S has infinitely many elements τ; but then σ < τ, so S was not an antichain. □

Theorem 4.5. Suppose (K, ∂_{0}, …, ∂_{m−1}) ⊧ m-DF. Then the following are equivalent:

(i) The model (K, ∂_{0}, …, ∂_{m−1}) of m-DF is existentially closed.

(ii) For all positive integers r and n, if K has an extension K(a_{h}^{ξ}: |ξ| ⩽ 2r & h < n) meeting the differential condition such that |σ| ⩽ r whenever a_{k}^{σ} is a minimal separable leader, then the tuple (a_{h}^{ξ}: |ξ| < 2r & h < n) has a specialization (∂^{ξ}b_{h}: |ξ| < 2r & h < n) for some tuple (b_{h}: h < n) of elements of K.

Proof. Assume (i) and the hypothesis of (ii). Let S be a (finite) generating set of the ideal of (a_{h}^{ξ}: |ξ| < 2r & h < n) over K. By Theorem 4.3, the system

⋀_{f∈S} f(∂^{ξ}x_{h}: |ξ| < 2r & h < n) = 0

has a solution in some extension, hence it has a solution in K itself, which means the conclusion of (ii) holds. So (ii) is necessary for (i).

Every system over (K, ∂_{0}, …, ∂_{m−1}) is equivalent to a system of equations. Suppose such a system has a solution (a_{h}: h < n) in some extension (L, D_{0}, …, D_{m−1}). Then the extension K(∂^{ξ}a_{h}: (ξ, h) ∈ (ω^{m} × n)) has a finite set of minimal separable leaders, by Lemma 4.4, since this set is indexed by an antichain of (ω^{m} × n, ⩽). Hence there is r large enough that all of these minimal separable leaders are also generators of K(D^{ξ}a_{h}: |ξ| ⩽ r & h < n). We may assume also that r is large enough that |σ| ⩽ r for every derivative ∂^{σ}x_{k} that appears in the original system. The hypothesis of (ii) is now satisfied when each a_{k}^{σ} is taken as D^{σ}a_{k}. If the conclusion of (ii) follows, then (b_{h}: h < n) is a solution of the original system. Thus, (ii) is sufficient for (i). □

Corollary 4.6. The theory m-DF has a model-companion, m-DCF.

Proof. Let (K, ∂_{0}, …, ∂_{m−1}) be a model of m-DF, let L be an extension K(a_{h}^{ξ}: |ξ| ⩽ 2r & h < n) of K meeting the differential condition, and suppose |σ| ⩽ r whenever a_{k}^{σ} is a minimal separable leader. That is, assume the hypothesis of Condition (ii) of the theorem. Write a for (a_{h}^{ξ}: |ξ| < 2r & h < n) and b for (a_{h}^{ξ}: |ξ| = 2r & h < n), so that L = K(a, b). The ideal of K[x, y] comprising the polynomials that are 0 at (a, b) is generated by a set {f(p, x, y): f ∈ T }, where T is a finite subset of ℤ[z, x, y], and p is a (finite) list of parameters from K. We may assume that T has a subset S, where S ⊆ ℤ[z, x] and {g(p, x): g ∈ S} generates the ideal of a. We need to ensure that there is a formula φ(z) such that

(i) φ(p) holds in (K, ∂_{0}, …, ∂_{m−1}); (ii) for every model (K, ∂_{0}, …, ∂_{m−1}) of m-DF and every tuple q from K, if φ(q) holds in the model, then the polynomials f(q, x, y) (where f ∈ T) generate over K a prime ideal, a generic zero of which generates an extension of K as in the hypothesis of Condition (ii) of the theorem.

In this case, by the theorem, m-DCF will have, as axioms, the axioms of m-DF, along with one sentence of the form

φ(z) ⇒ ∃x ( ⋀_{g∈S} g(z, x) = 0 & ⋀_{i<m} ⋀_{h<n} ⋀_{|ξ+i|<2r} x_{h}^{ξ+i} = ∂_{i}x_{h}^{ξ}).

for each model (K, ∂_{0}, …, ∂_{m−1}) of m-DF and each extension K(a_{h}^{ξ}: |ξ| ⩽ 2r & h < n) as above.

So now we must show that the formula φ exists as desired; that is, we must show that there are first-order conditions on the parameters p as required. This we can do as follows.

That the ideal I generated by some finite set of polynomials is a prime ideal— this condition is a first-order condition on the parameters of the polynomials, by van den Dries and Schmidt [29]. (As they point out, the results of theirs that we shall use are not original with them; the proofs are original.) In particular, there is some N depending only on the degrees of the generating polynomials and their numbers of variables such that if fg ∈ I ⇒ f ∈ I ∨ g ∈ I for all polynomials f and g of degree less than N, then the implication holds for all f and g.

Here I may be the ideal of (a, b) or of a, defined in terms of T or S as above. The extension K(a, b) meets the differential condition, because the derivatives of each g(p, x) (where g ∈ S) are certain combinations (which can be made explicit) of the f(p, x, y) (where f ∈ T). Thus there is a sufficient first-order condition on the parameters p for the meeting of the differential condition; and p does meet this condition.

Alternatively, if a polynomial F and finitely many additional polynomials G with parameters p are given, the condition that F be a member of the ideal I generated by the G is a first-order condition on p. This follows from the existence of a uniform bound on the degrees of the polynomial coefficients needed to obtain F from the polynomials G if indeed F ∈ I. This bound is uniform in the sense that it depends only on the degrees of F and the G and the number of their variables. The existence of this bound is again shown by van den Dries and Schmidt [29].

Moreover, for a list w of variables that appear in the polynomials G, the condition that no non-zero polynomial in w alone belongs to I is also a first-order condition on p. Indeed, since this condition is invariant under replacement of the underlying field by its algebraic closure, we may appeal to the general result that Morley rank is definable in algebraically closed fields and more generally in strongly minimal sets [11, §6.2]. For some formula ψ, the condition holds if and only if ψ(p) is true in the algebraic closure of the underlying field; but by quantifier-elimination in algebraically closed fields, we may assume also ψ is quantifier-free, so ψ(p) is true in the algebraic closure of a field if and only if it is true in the field itself.

For each leader in (a, b), we may assume that some irreducible polynomial in T shows that it is a leader. The irreducibility of this polynomial is a first-order condition on p (since in general primeness of ideals is first-order). The leader is separable if and only if the formal derivative of its irreducible polynomial is not zero, that is, not all of its coefficients belong to the ideal generated by {f(p, x, y) : f ∈ T }; as noted above, this is a first-order condition. The condition that an entry in (a, b) is not a leader at all is also a first-order condition on the parameters, since this condition is just that no non-zero polynomial in certain variables belongs to the ideal generated by {f(p, x, y): f ∈ T }.

Now we can arrange that φ(z) establishes all of the conditions discussed; and this is enough. □

By Theorem 3.1, DF^{m} now also has a model-companion.

4.3. Differential forms again. The condition in Theorem 4.3 can be adjusted to yield the following:

Theorem 4.7. Suppose (K, ∂_{0}, …, ∂_{m−1}) ⊧ m-DF, and K has an extension K(a_{h}^{ξ}: |ξ| ⩽ |μ| & h < n) meeting the differential condition for some μ in ω^{m} and some positive integer n. Suppose further that, if a_{k}^{σ} is a minimal separable leader, then σ ⩽ μ. Then (K, ∂_{0}, …, ∂_{m−1}) has an extension compatible with K(a_{h}^{ξ}: |ξ| < |μ| & h < n).

Proof. The proof is as for Theorem 4.3, mutatis mutandis. What needs adjusting is the choosing of a_{ℓ}^{τ} in case both a_{ℓ}^{τ−i} and a_{ℓ}^{τ−j} are separable leaders. Again we have minimal separable leaders a_{ℓ}^{π} and a_{ℓ}^{ρ} below a_{ℓ}^{τ−i} and a_{ℓ}^{τ−j} respectively. Since we may assume |μ| < |τ|, there is some k in m such that μ(k) < τ(k). If k = j, then π(j) ⩽ μ(j) < τ(j), so π(j) ⩽ (τ − j)(j) = (τ − i − j)(j). Then π ⩽ τ − i − j, so a_{ℓ}^{π} is below a_{ℓ}^{τ−i−j}. Now we can proceed as before. □

As Theorem 4.3 yields Theorem 4.5, so Theorem 4.7 yields a characterization of the existentially closed models of m-DF. Moreover, Theorems 4.3 and 4.7 can be combined in the following way:

Theorem 4.8. Suppose (K, ∂_{0}, …, ∂_{m−1}) ⊧ m-DF, and K has an extension K(a_{h}^{ξ}: |ξ| ⩽ 2r & h < n) meeting the differential condition for some positive integers n and r. Suppose further that, for each k in m, either |σ| ⩽ r whenever a_{k}^{σ} is a minimal separable leader, or else there is some τ in ω^{m} such that |τ| = 2r, and |σ| ⩽ |τ| whenever a_{k}^{σ} is a minimal separable leader. Then (K, ∂_{0}, …, ∂_{m−1}) has an extension compatible with K(a_{h}^{ξ}: |ξ| < 2r & h < n).

Proof. Combine the proofs of Theorems 4.3 and 4.7. □

There is a corresponding first-order characterization of the models of m-DCF, parallel to Theorem 4.5 and Corollary 4.6.

4.4. Another sufficient condition. If (K, ∂_{0}, …, ∂_{m−1}) ⊧ m-DF, and K(a_{h}^{ξ}: |ξ| ⩽ |π| & h < n) is an extension L of K meeting the differential condition, this by itself is not enough to ensure that (K, ∂_{0}, …, ∂_{m−1}) has an extension compatible with L. However, if such an extension does exist, then its existence can be shown by means of Theorem 4.3, provided |π| can be made large enough: this is Theorem 4.10 below, which relies on the existence of bounds as in the following.

Lemma 4.9. For all positive integers m and n, for all sequences (a_{i}: i ∈ ω) of positive integers, there is a bound on the length of strictly increasing chains

S_{0} ⊂ S_{1} ⊂ S_{2} ⊂ ⋯  (21)

of antichains S_{k} of (ω^{m} × n, ⩽), where also S_{k} ⊆ {(ξ, h): |ξ| ⩽ a_{k}}.

Proof. Divide and conquer. First reduce to the case when n = 1. Indeed, suppose the claim does hold in this case. Suppose also, as an inductive hypothesis, that the claim holds when n = ℓ. Now fix m and the sequence (a_{i}: i ∈ ω) or rather (a(i): i ∈ ω), and consider arbitrary chains as in (21), where n = ℓ + 1. Analyze each S_{k} as S′_{k} ∪ S′′_{k}, where

S′_{k} = {(ξ, h) ∈ S_{k}: h < ℓ},  S′′_{k} = {(ξ, h) ∈ S_{k}: h = ℓ}.

For each k such that S_{k+1} exists, at least one of the inclusions S′_{k} ⊆ S′_{k+1} and S′′_{k} ⊆ S′′_{k+1} is strict; also, by our assumption, there is an upper bound f(k) on those r such that

S′′_{k} ⊂ S′′_{k+1} ⊂ ⋯ ⊂ S′′_{r−1}.  (22)

The function f depends only on m and (a_{i}: i ∈ ω)), not on the choice of chain in (21).

Let k(0) = 0, and if k(i) has been chosen, let k(i + 1) be the least r, if it exists, such that S′_{k(i)} ⊂ S′_{r}. Here k(i) does depend on the chain. But if r is maximal in (22), and S′_{r} exists, then S′_{k} ⊂ S′_{r}. Hence k(i + 1) ⩽ f(k(i)). Since the function f is not necessarily increasing, we derive from it the increasing function g, where g(k) = max_{i⩽k} f(i). Then x ⩽ y ⟹ g(x) ⩽ g(y), so

k(r) ⩽ f(k(r−1)) ⩽ g(k(r−1)) ⩽ g∘g(k(r−2)) ⩽ ⋯ ⩽ \overbrace{g ∘ ⋯ ∘ g}^{r}(0) = g^{r}(0). (23)

In particular, S_{k(r)} ⊆ {(ξ, h): |ξ| ⩽ a(g^{r}(0))}. The sequence (a(g^{i}(0)): i ∈ ω) does not depend on the original chain. Hence the inductive hypothesis applies to the chain

S′_{k(0)} ⊂ S′_{k(1)} ⊂ ⋯,  (24)

showing that there is s (independent of the original chain) such that k(s) is defined, and r ⩽ s for all entries S′_{k(r)} in (24). Hence also, by (23), if S′_{r} is an entry in (24), then r ⩽ k(s) ⩽ g^{s}(0).

Now suppose S′_{r} is the final entry in (24). Then S′′_{r} ⊂ S′′_{r+1} ⊂ ⋯; but if S′′_{t} is an entry of this chain, then t < f(r) ⩽ g(r) ⩽ g(g^{s}(0)) = g^{s+1}(0).

Therefore the original chain in (21) has a final entry S_{t}, where t < g^{s+1}(0). Thus the claim holds when n = ℓ + 1. By induction, the claim holds for all positive n, provided it holds when n = 1.

It remains to show that, for all positive m, for all sequences (a_{i}: i ∈ ω), there is a bound on the length of chains

S_{0} ⊂ S_{1} ⊂ S_{2} ⊂ ⋯  (25)

of antichains S_{k} of (ω^{m}, ⩽), where S_{k} ⊆ {ξ: |ξ| ⩽ a_{k}}. The claim is trivially true when m = 1. Suppose it is true when m = ℓ. Now let m = ℓ + 1, and suppose we have a chain as in (25). We may assume that S_{0} contains some σ. If i < m and j ∈ ω, let

S_{k}^{i, j} = {ξ ∈ S_{k}: ξ(i) = j}.

Then the inductive hypothesis applies to chains of the form

S_{k(0)}^{i, j} ⊂ S_{k(1)}^{i, j} ⊂ S_{k(2)}^{i, j} ⊂ · · · .

Moreover, if τ ∈ S_{k}, then τ(i) ⩽ σ(i) for some i in m (since σ is also in S_{k}, and this is an antichain). Hence

S_{k} = ⋃_{i<m} ⋃_{j⩽σ(i)} S_{k}^{i, j},

a union of no more than |σ| + m-many sets, hence no more than a_{0} + m-many sets. So the proof can proceed as in the reduction to n = 1: for each k such that S_{k+1} exists, one of the inclusions S_{k}^{i, j} ⊆ S_{k+1}^{i, j} is strict, and so forth. □

Theorem 4.10. Suppose m, r, and n are positive integers. Then there is a positive integer s, where r ⩽ s, such that, if (K, ∂_{0}, …, ∂_{m−1}) ⊧ m-DF, and K(a_{h}^{ξ}: |ξ| ⩽ s & h < n) meets the differential condition, then (K, ∂_{0}, …, ∂_{m−1}) has an extension that is compatible with K(a_{h}^{ξ}: |ξ| ⩽ r & h < n).

Proof. Suppose K(a_{h}^{ξ}: |ξ| ⩽ 2^{t}r & h < n) meets the differential condition for some t. When u ⩽ t, let K_{u} = K(a_{h}^{ξ}: |ξ| ⩽ 2^{u}r & h < n), and let S_{u} be the set of minimal separable leaders of K_{u}. Then we have an increasing chain S_{0} ⊆ S_{1} ⊆ … ⊆ S_{t}. By the preceding lemma, there is a value of t, depending only on m, r, and n, large enough that this chain cannot be strictly increasing. Then S_{u} = S_{u+1} for some u less than this t. Then K_{u+1} satisfies the hypothesis of Theorem 4.3. So (K, ∂_{0}, …, ∂_{m−1}) has an extension compatible with K(a_{h}^{ξ}: |ξ| < 2^{u+1}r & h < n), and a fortiori with K(a_{h}^{ξ}: |ξ| ⩽ r & h < n). In short, the desired s is 2^{t}r. □

This theorem yields yet another first-order characterization of the models of m-DCF.

References

[1] Jon Barwise and Abraham Robinson, Completing theories by forcing, Ann. Math. Logic 2 (1970), no. 2, 119–142. MR MR0272613 (42 #7494)

[2] Lenore Blum, Differentially closed fields: a model-theoretic tour, Contributions to algebra (collection of papers dedicated to Ellis Kolchin), Academic Press, New York, 1977, pp. 37–61. MR MR0491149 (58 #10415)

[3] Chen Chung Chang, On unions of chains of models, Proc. Amer. Math. Soc. 10 (1959), 120–127. MR MR0103812 (21 #2576)

[4] Zoé Chatzidakis and Ehud Hrushovski, Model theory of difference fields, Trans. Amer. Math. Soc. 351 (1999), no. 8, 2997–3071. MR 2000f:03109

[5] Paul Eklof and Gabriel Sabbagh, Model-completions and modules, Ann. Math. Logic 2 (1970/1971), no. 3, 251–295. MR MR0277372 (43 #3105)

[6] Wilfrid Hodges, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993. MR 94e:03002

[7] E. R. Kolchin, Differential algebra and algebraic groups, Academic Press, New York, 1973, Pure and Applied Mathematics, Vol. 54. MR 58 #27929

[8] Piotr Kowalski, Derivations of the Frobenius map, J. Symbolic Logic 70 (2005), no. 1, 99–110. MR MR2119125 (2005m:03069)

[9] Jerzy Łoś and Roman Suszko, On the extending of models (IV): Infinite sums of models, Fund. Math. 44 (1957), 52–60. MR MR0089813 (19,724c)

[10] Angus Macintyre, Generic automorphisms of fields, Ann. Pure Appl. Logic 88 (1997), no. 2-3, 165–180, Joint AILA-KGS Model Theory Meeting (Florence, 1995). MR 99c:03046

[11] David Marker, Model theory: an introduction, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002. MR 1 924 282

[12] Tracey McGrail, The model theory of differential fields with finitely many commuting derivations, J. Symbolic Logic 65 (2000), no. 2, 885–913. MR 2001h:03066

[13] David Pierce, Differential forms in the model theory of differential fields, J. Symbolic Logic 68 (2003), no. 3, 923–945. MR 2 000 487

[14] , Geometric characterizations of existentially closed fields with operators, Illinois J. Math. 48 (2004), no. 4, 1321–1343. MR MR2114160

[15] David Pierce and Anand Pillay, A note on the axioms for differentially closed fields of characteristic zero, J. Algebra 204 (1998), no. 1, 108–115. MR 99g:12006

[16] Joseph Fels Ritt, Differential algebra, Dover Publications Inc., New York, 1966, originally published in 1950. MR MR0201431 (34 #1315)

[17] A. Robinson, Some problems of definability in the lower predicate calculus, Fund. Math. 44 (1957), 309–329. MR MR0091922 (19,1032f)

[18] Abraham Robinson, Introduction to model theory and to the metamathematics of algebra, North-Holland Publishing Co., Amsterdam, 1963. MR MR0153570 (27 #3533)

[19] , Complete theories, second ed., North-Holland Publishing Co., Amsterdam, 1977, With a preface by H. J. Keisler, Studies in Logic and the Foundations of Mathematics, first published 1956. MR MR0472504 (57 #12202)

[20] Azriel Rosenfeld, Specializations in differential algebra, Trans. Amer. Math. Soc. 90 (1959), 394–407. MR MR0107642 (21 #6367)

[21] Gerald E. Sacks, Saturated model theory, W. A. Benjamin, Inc., Reading, Mass., 1972, Mathematics Lecture Note Series. MR MR0398817 (53 #2668)

[22] A. Seidenberg, Some basic theorems in differential algebra (characteristic p, arbitrary), Trans. Amer. Math. Soc. 73 (1952), 174–190. MR MR0049174 (14,130c)

[23] , An elimination theory for differential algebra, Univ. California Publ. Math. (N.S.) 3 (1956), 31–65. MR MR0082487 (18,558a)

[24] Saharon Shelah, Differentially closed fields, Israel J. Math. 16 (1973), 314–328. MR MR0344116 (49 #8856)

[25] H. Simmons, Existentially closed structures, J. Symbolic Logic 37 (1972), 293–310. MR 51 #12518

[26] Michael F. Singer, The model theory of ordered differential fields, J. Symbolic Logic 43 (1978), no. 1, 82–91. MR MR495120 (80a:03044)

[27] , Model theory of partial differential fields: from commuting to noncommuting derivations, Proc. Amer. Math. Soc. 135 (2007), no. 6, 1929–1934 (electronic). MR MR2286106 (2007m:03077)

[28] Marcus Tressl, The uniform companion for large differential fields of characteristic 0, Trans. Amer. Math. Soc. 357 (2005), no. 10, 3933–3951 (electronic). MR MR2159694

[29] L. van den Dries and K. Schmidt, Bounds in the theory of polynomial rings over fields. A nonstandard approach, Invent. Math. 76 (1984), no. 1, 77–91. MR MR739626 (85i:12016)

[30] Carol Wood, The model theory of differential fields of characteristic p ≠ 0, Proc. Amer. Math. Soc. 40 (1973), 577–584. MR 48 #8227

[31] , Prime model extensions for differential fields of characteristic p ≠ 0, J. Symbolic Logic 39 (1974), 469–477. MR 50 #9577

[32] Yoav Yaffe, Model completion of Lie differential fields, Ann. Pure Appl. Logic 107 (2001), no. 1-3, 49–86. MR 1 807 840

Mathematics Department, Mimar Sinan Fine Arts University, Istanbul, Turkey

E-mail address: dpierce@msgsu.edu.tr
URL: http://mat.msgsu.edu.tr/~dpierce