GEOMETRIC AXIOMS FOR DIFFERENTIALLY CLOSED FIELDS WITH SEVERAL COMMUTING DERIVATIONS

Omar León Sánchez

University of Waterloo, ON, Canada March, 2011

Abstract. A geometric first-order axiomatization of differentially closed fields of characteristic zero with several commuting derivations, in the spirit of Pierce-Pillay [14], is formulated in terms of a relative notion of prolongation for Kolchin-closed sets.

Keywords: Differential Field, Prolongation, Model Theory

AMS 2010 Mathematics Subject Classification: 03C65, 12H05.

1. Introduction

An ordinary differential field is a field of characteristic zero equipped with a derivation, that is, an additive map δ : K → K such that δ(ab) = (δa)b + a(δb). A differentially closed field is a differential field (K, δ) such that every system of differential polynomial equations in several variables, with a solution in some differential extension, has a solution in K. An elegant first-order axiomatization of the class of ordinary differentially closed fields was given by Blum in [1]. In [14], Pierce and Pillay give a geometric axiomatization. Their axioms say that (K, δ) is differentially closed if and only if K is algebraically closed and whenever V and W are irreducible Zariski-closed sets with W contained in the prolongation of V and projecting dominantly onto V, then there is a K-point in W of the form (x̄, δx̄).

Similarly, a field K of characteristic zero equipped with m commuting derivations is differentially closed if every system of partial differential polynomial equations in several variables with a solution in some extension has a solution in K. A first-order axiomatization generalizing Blum’s was given by McGrail in [11]. Other work along these lines can be found in Tressl [16] and Yaffe [18].

If one tries to give a naive geometric axiomatization of differentially closed fields with several commuting derivations in terms of prolongations, the commutativity of the derivations might impose too many restrictions, so that a Pierce-Pillay type condition will not hold (see [13], Counterexample 6.2). Nonetheless, in [13], Pierce does manage to give an axiomatization (in arbitrary characteristic) that has a geometric flavor, though not exactly in the Pierce-Pillay sense. In this paper we take a different approach, establishing an axiomatization of differentially closed fields with (m + 1) commuting derivations which is geometric relative to the theory of differentially closed fields with m derivations. Our axioms are a precise generalization of the Pierce-Pillay axioms. Two complications arise in our setting that do not appear in the ordinary case: one has to do with extending commuting derivations and the other has to do with first-order axiomatizability. D! ifferential-algebraic results due to Kolchin are behind our solutions to both of these problems.

Suppose Δ = {δ_{1}, …, δ_{m}} are commuting derivations on a field K of characteristic zero and for each r = 0, …, m, let Δ_{r} = {δ_{1}, …, δ_{r}}. Also, suppose D : K → K is an additional derivation on K that commutes with Δ. If V is a Δ-closed set defined over the D-constants of K, then Kolchin constructs a Δ-tangent bundle of V which has x̄ → (x̄, Dx̄) as a section ([5], Chap. VIII, §2). In general, if V is not necessarily defined over the D-constants, then D gives a section of a certain torsor of the Δ-tangent bundle of V that we call the D/Δ-prolongation of V (cf. Definition 3.2). Our axioms will essentially say that (K, Δ ∪ {D}) is differentially closed if and only if K is algebraically closed and for each r = 0, …, m, whenever V and W are Δ_{r}-closed sets with W contained in the D/Δ_{r}-prolongation of V and projecting onto V, then there is a K-point in W of the fo! rm (x̄, Dx̄). “Essentially”, because in actual fact we also have to consider not just Δ and D but all their independent ℚ-linear combinations (cf. Theorem 4.3 below).

Pierce-Pillay type axiomatizations have been obtained in various other contexts: difference fields (Chatzidakis and Hrushovski [3]), difference-differential fields (Bustamante [2]), derivations of the Frobenius and commuting Hasse-Schmidt derivations in positive characteristic (Kowalski [6], [7]). However, the techniques used in these works do not seem to translate to our context.

The paper is organized as follows. In Section 2 we establish the differentialalgebraic facts that underpin our results. In Section 3 we introduce relative prolongations and prove a geometric characterization of differentially closed fields. Finally, in Section 4, we address the issue of first-order axiomatizability.

Acknowledgements: I would like to thank Rahim Moosa for all the useful discussions and support towards the completion of this article.

2. Extending Δ-derivations

In this paper the term ring is used for commutative ring with unity and the term field for field of characteristic zero.

Let us first recall some terminology from differential algebra. For details see [4]. Let R be a ring and S a ring extension. An additive map δ : R → S is called a derivation if it satifies the Leibniz rule; i.e., δ(ab) = (δa)b + a(δb). A ring R equipped with a set of derivations Δ = {δ_{1}, …, δ_{m}}, δ_{i} : R → R, such that the derivations commute with each other is called a Δ-ring. A Δ-ring which is also a field (of characteristic zero) is called a Δ-field.

We fix for the rest of this section a Δ-ring R. Let Θ denote the free commutative monoid generated by Δ; that is,

Θ := {δ_{m}^{r_{m}} ⋯ δ_{1}^{r_{1}} : r_{m}, …, r_{1} ≥ 0}.

The elements of Θ are called the derivative operators of R. Let x̄ = (x_{1}, …, x_{n}) be a family of indeterminates, and define

θx̄ := {∂x_{j} : j = 1, …, n, ∂ ∈ Θ}.

The Δ-ring of Δ-polynomials over R in the differential indeterminates x̄ is R{x̄} := R[θx̄]; that is, the ring of polynomials in the algebraic indeterminates θx̄ with the canonical Δ-ring structure given by δ_{i}(δ_{m}^{r_{m}} ⋯ δ_{1}^{r_{1}}x_{j}) = δ_{m}^{r_{m}} ⋯ δ_{i}^{r_{i}+1} ⋯ δ_{1}^{r_{1}}x_{j}.

We fix an orderly ranking in θx̄ by:

δ_{m}^{r_{m}} ⋯ δ_{1}^{r_{1}}x_{i} ≤ δ_{m}^{r′_{m}} ⋯ δ_{1}^{r′_{1}}x_{j} ⟺ (∑ r_{l}, i, r_{m}, …, r_{1}) ≤ (∑ r′_{l}, j, r′_{m}, …, r′_{1}) in the lexicographical order. According to this ranking, we enumerate the algebraic indeterminates by θx̄ = (θ_{1}x̄, θ_{2}x̄, …). Therefore, if f ∈ R{x̄} there is a unique f̂ ∈ R[t_{1}, t_{2}, …] such that f(x̄) = f̂(θx̄).

We will be interested in adding an extra derivation on R. This amounts to the study of Δ-derivations (see Chapter 0 of [5]).

Definition 2.1. A Δ-derivation on R is a derivation D : R → S, where S is a Δ-ring extension of R, such that Dδ_{i} = δ_{i}D for i = 1, …, m.

If f ∈ R{x̄}, by f^{D} we mean the Δ-polynomial in S{x̄} obtained by applying D to the coefficients of f.

Remark 2.2.

(1) The map f ↦ f^{D} is itself a Δ-derivation from R{x̄} to S{x̄}. Indeed, it is clearly additive and to check the Leibniz rule and commutativity of D with Δ it suffices to show that (fg)^{D} = f^{D}g + fg^{D} and (δ_{i}f)^{D} = δ_{i}(f^{D}) for i = 1, …, m, where f(x̄) = f̂(θx̄) and g(x̄) = ĝ(θx̄) are such that f̂ and ĝ are monomials in R[t_{1}, t_{2}, …]. These computations are straightforward.

(2) The Δ-derivation D extends uniquely to a derivation

R{x̄} → S[Dθx̄] := S[Dθ_{k}x̄ : k = 1, …]

by D(θ_{k}x̄) = Dθ_{k}x̄, for k ≥ 1. Also, each δ_{i} extends uniquely to a derivation

S[Dθx̄] → S[δ_{i}Dθx̄] := S[δ_{i}Dθ_{k}x̄ : k = 1, …]

by δ_{i}(Dθ_{k}x̄) = δ_{i}Dθ_{k}x̄, for k ≥ 1.

In order to describe how D and compositions of D with elements of Δ act on R{x̄}, we introduce the following convenient terminology. Given f ∈ R{x̄}, the Jacobian of f is

df(x̄) := (\frac{∂f̂}{∂t_{i}}(θx̄))_{i∈ℕ}

viewed as an element of (R{x̄})^{ℕ}. Note that df is finitely supported, in the sense that all but finitely many coordinates are zero. A straightforward computation shows that for each δ_{i} ∈ Δ

δ_{i}f(x̄) = df(x̄) · δ_{i}θx̄ + f^{δ_{i}}(x̄),

where δ_{i}θx̄ = (δ_{i}θ_{1}x̄, δ_{i}θ_{2}x̄, …) and the dot product is well defined since df has finite support.

The Hessian of f is defined as

Hf(x̄) := (\frac{∂^{2}f̂}{∂t_{i}∂t_{j}} (θx̄))_{i,j∈ℕ}

viewed as an element of (R{x̄})^{ℕ×ℕ}. Again Hf is finitely supported.

Lemma 2.3. Let S be a Δ-ring extension of R and D : R → S a Δ-derivation. If f ∈ R{x̄} and δ ∈ Δ ∪ {D}, then

δf(x̄) = df(x̄) · δθx̄ + f^{δ}(x̄).

Also, if δ, ζ ∈ Δ ∪ {D} with δ ≠ ζ, then

δζf(x̄) = df(x̄) · δζθx̄ + δθx̄ · Hf(x̄) · (ζθx̄)^{t} + f^{δζ}(x̄)
+ df^{δ}(x̄) · ζθx̄ + df^{ζ}(x̄) · δθx̄.

For any choice of δ and ζ the elements δf and δζf are well defined by part (2) of Remark 2.2. Note that the dot product df(x̄) · δθx̄ and the matrix product δθx̄ · Hf(x̄) · (ζθx̄)^{t} are well defined because df and Hf have finite support.

Proof. For the first equation, by additivity, it suffices to prove it for monomials f(x̄) = c ∏_{i}(θ_{i}x̄)^{α_{i}} where c ∈ R and α_{i} = 0 except finitely many times:

δf(x̄) = ∑_{k} (c ∏_{i≠k}(θ_{i}x̄)^{α_{i}} α_{k}(θ_{k}x̄)^{α_{k}−1}δθ_{k}x̄) + δ(c) ∏_{i}(θ_{i}x̄)^{α_{i}}
= df(x̄) · δθx̄ + f^{δ}(x̄).

For the second equation,

δζf(x̄) = δ (df(x̄) · ζθx̄ + f^{ζ}(x̄))
= δ (∑_{k} \frac{∂f̂}{∂t_{k}}(θx̄)ζθ_{k}x̄ + f^{ζ}(x̄))

= ∑_{k} \frac{∂f̂}{∂t_{k}}(θx̄)δζθ_{k}x̄ + ∑_{k} δ (\frac{∂f̂}{∂t_{k}}(θx̄)) ζθ_{k}x̄ + δ (f^{ζ}(x̄))

= df(x̄) · δζθx̄ + ∑_{k} (d (\frac{∂f̂}{∂t_{k}}(θx̄)) (x̄) · δθx̄ + (\frac{∂f̂}{∂t_{k}}(θx̄))^{δ} (x̄)) ζθ_{k}x̄

+ df^{ζ}(x̄) · δθx̄ + f^{δζ}(x̄)

= df(x̄) · δζθx̄ + ∑_{k} ∑_{l} \frac{∂^{2}f̂}{∂t_{l}∂t_{k}} (θx̄) δθ_{l}x̄ ζθ_{k}x̄

+ ∑_{k} (\frac{∂f̂}{∂t_{k}}(θx̄))^{δ} (x̄) ζθ_{k}x̄ + df^{ζ}(x̄) · δθx̄ + f^{δζ}(x̄)

= df(x̄) · δζθx̄ + δθx̄ · Hf(x̄) · (ζθx̄)^{t} + f^{δζ}(x̄)

+ ∑_{k} (\frac{∂f̂^{δ}}{∂t_{k}} ) (θx̄)ζθ_{k}x̄ + df^{ζ}(x̄) · δθx̄

= df(x̄) · δζθx̄ + δθx̄ · Hf(x̄) · (ζθx̄)^{t} + f^{δζ}(x̄)

+ df^{δ}(x̄) · ζθx̄ + df^{ζ}(x̄) · δθx̄.

Where the sixth equality uses (\frac{∂f̂}{∂t_{k}} (θx̄))^{δ} = \frac{∂f̂^{δ}}{∂t_{k}} (θx̄). This follows by additivity and the fact that

[\frac{∂(c t_{1}^{n_{1}} ⋯ t_{l}^{n_{l}})}{∂t_{k}}(θx̄)]^{δ} = (n_{k}c (θ_{1}x̄)^{n_{1}} ⋯ (θ_{k}x̄)^{n_{k}−1} ⋯ (θ_{l}x̄)^{n_{l}})^{δ}
= n_{k}δ(c) (θ_{1}x̄)^{n_{1}} ⋯ (θ_{k}x̄)^{n_{k}−1} ⋯ (θ_{l}x̄)^{n_{l}}
= \frac{∂(δ(c)t_{1}^{n_{1}} ⋯ t_{l}^{n_{l}})}{∂t_{k}}(θx̄).

□

Corollary 2.4. Let S be a Δ-ring extension of R and D : R → S a Δ-derivation. Suppose ā is a tuple in S and D′ : R{ā} → S is a derivation extending D such that D′θā = θD′ā. Then D′ commutes with Δ on R{ā}.

Proof. Let f ∈ R{x̄} we must show that for any δ_{i} ∈ Δ, δ_{i}D′f(ā) = D′δ_{i}f(ā). By the second equality of Lemma 2.3, we have that

δ_{i}D′f(ā) − D′δ_{i}f(ā) = df(ā) · (δ_{i}D′θā − D′δ_{i}θā).

But by assumption δ_{i}D′θā = δ_{i}θD′ā = D′δ_{i}θā, as desired. □

Definition 2.5. Let f ∈ R{x̄}. We define the Δ-polynomial τ_{D/Δ}f ∈ S{x̄, ȳ} by

τ_{D/Δ}f(x̄, ȳ) := df(x̄) · θȳ + f^{D}(x̄).

When Δ and D are understood we simply write τf. If ā ∈ S, we write τ(f)ā(ȳ) for τf(ā, ȳ) ∈ S{ȳ}. Note that τθx̄ = θȳ and if c ∈ R then τc = Dc.

Lemma 2.6. Suppose S is a Δ-ring extension of R and D : R → S is a Δderivation. Then τ : R{x̄} → S{x̄, ȳ} is a Δ-derivation extending D.

Proof. The map f ↦ τf is additive and, by part (1) of Remark 2.2 and the fact that d(fg)(x̄) · θȳ = (df(x̄) · θȳ) g(x̄) + f(x̄) (dg(x̄) · θȳ), we have that τ(fg) = (τf)g + f(τg). Hence, since τc = Dc for all c ∈ R, τ is a derivation extending D. For commutativity, note that τθx̄ = θȳ = θτx̄. Now just apply Corollary 2.4 with τ, x̄ and S{x̄, ȳ} in place of D′, ā and S. □

We can now give the desired criterion for when a Δ-derivation can be extended to a finitely generated Δ-ring extension. The analogue of the following proposition when Δ = ∅ (i.e. m = 0) can be found in ([8], Chap. VII, §5), and it is the main point in the Pierce-Pillay geometric axiomatization of ordinary differentially closed fields.

Proposition 2.7. Suppose S is a Δ-ring extension of R and D : R → S is a Δderivation. Let ā be a tuple of S and A ⊆ R{x̄} such that [A] = I(ā/R). Here [A] denotes the Δ-ideal generated by A and I(ā/R) = {f ∈ R{x̄} : f(ā) = 0}. Suppose there is a tuple b̄ of S such that

(2.1)  τ(g)ā(b̄) = 0, for all g ∈ A.

Then there is a unique Δ-derivation D′ : R{ā} → S extending D such that D′ā = b̄.

Proof. First we show that (2.1) holds for all elements in I(ā/R). For each ∂ ∈ Θ, g ∈ A and h ∈ R{x̄}, by Lemma 2.6 we have

(2.2)  τ(h∂g)(x̄, ȳ) = τh(x̄, ȳ)∂g(x̄) + h(x̄)∂(τg(x̄, ȳ)).

By assumption τg(ā, b̄) = τ(g)ā(b̄) = 0 and since g ∈ I(ā/R) we get ∂g(ā) = 0, so evaluating (2.2) at (ā, b̄) yields τ(h∂g)ā(b̄) = 0. It follows that for each f ∈ [A] = I(ā/R), τ(f)ā(b̄) = 0.

Now let α ∈ R{ā}, then α = f(ā) for some f ∈ R{x̄}. Define

D′α = τ(f)ā(b̄).

This does not depend on the choice of f, since if α = f(ā) = f′(ā) then f − f′ ∈ I(ā/R) and hence by assumption τ(f)ā(b̄) − τ(f′)ā(b̄) = τ(f − f′)ā(b̄) = 0. It is clear that D′ā = τ(x̄)ā(b̄) = b̄ and, for all c ∈ R, D′c = τ(c)ā(b̄) = Dc. By Lemma 2.6 this is a Δ-derivation extending D to R{ā} → S. To show uniqueness suppose D′′ : R{ā} → S is another Δ-derivation extending D such that D′′ā = b̄. Then, for any f ∈ R{x̄}, by the first equality of Lemma 2.3 we get

D′′f(ā) = df(ā) · D′′θā + f^{D′′}(ā) = df(ā) · θD′′ā + f^{D}(ā)
= df(ā) · θb̄ + f^{D}(ā) = τ(f)ā(b̄) = D′f(ā)

so D′ = D′′. □

Thus if we want to extend a Δ-derivation we need to find solutions to certain Δ-equations. The following result of Kolchin’s can then be used to see that there is always such a solution in some Δ-field extension.

Fact 2.8 ([5], Chap. 0, §4). Suppose R is a Δ-subring of a Δ-field K and D : R → K a Δ-derivation. Let ā be a tuple of K. Then

[τ(f)ā(ȳ) : f ∈ I(ā/R)]

is a proper Δ-ideal of K{ȳ}.

Putting Proposition 2.7 and Fact 2.8 together we get the following result on extending Δ-derivations to differentially closed fields, which is essentially Corollary 1 of §0.4 of Kolchin [5].

Corollary 2.9. Suppose R is a Δ-subring of a differentially closed field (K, Δ), and D : R → K a Δ-derivation. Then there is a Δ-derivation D′ : K → K extending D.

Proof. Consider the set

S = {(S, D′) : R ⊆ S ⊆ K is a Δ-subring of K and

D′ : S → K is a Δ-derivation extending D}.

partially ordered by containment. The union of any chain of S gives an upper bound, and so, by Zorn’s lemma, there is a maximal element, call it (S, D′). Towards a contradiction, suppose S ≠ K. Then there is a ∈ K\S, and, by Fact 2.8, [τ(f)_{a}(y) : f ∈ I(a/S)] is a proper Δ-ideal of K{x̄}. By the Nullstellensatz for differentially closed fields (see for example theorem 3.1.10 of [11]), there is b ∈ K such that τ(f)_{a}(b) = 0 for all f ∈ I(a/S). By Proposition 2.7, there is a Δ-derivation D′′ : S{a} → K extending D′. This contradicts the maximality of (S, D′). Hence S = K and D′ : K → K is a Δ-derivation extending D. □

We conclude this section with an improvement on Proposition 2.7. We would like to only have to check condition (2.1) for a set of Δ-polynomials A ⊆ R{x̄} such that {A} = I(ā/R), where {A} denotes the radical Δ-ideal generated by A. As the reader may expect this will be useful when dealing with issues of first-order axiomatizability (see Proposition 3.3 below).

First we need a lemma. For each i = 1, 2, …, let x̄_{i} be an n-tuple of differential indeterminates. Suppose D : R → R is a Δ-derivation. Then τ : R{x̄_{1}} → R{x̄_{1}, x̄_{2}}. Thus we can compose τ with itself, for each k ≥ 1 and f ∈ R{x̄_{1}}, τ^{k}f = τ ⋯ τf ∈ R{x̄_{1}, x̄_{2}, …, x̄_{2^{k}}}. Define ∇x̄ := (x̄, Dx̄) and note that, for each k ≥ 1, the composition ∇^{k}x̄ = ∇ ⋯ ∇x̄ is a tuple of length n2^{k}.

Lemma 2.10. Suppose D : R → R is a Δ-derivation and f ∈ R{x̄}.

(1) If ā is a tuple of R, then for each k ≥ 1,

τ^{k}f(∇^{k}ā) = D^{k}f(ā)

In particular, if f(ā) = 0 then τ^{k}f(∇^{k}ā) = 0.

(2) For each k ≥ 1, we have

τ^{k}f^{k} = k!(τf)^{k} + f p

for some p ∈ R{x̄_{1}, x̄_{2}, …, x̄_{2^{k}}}.

Proof. (1) By induction on k. The first equality of Lemma 2.3 gives us

τf(∇ā) = df(ā) · θDā + f^{D}(ā) = df(ā) · Dθā + f^{D}(ā) = Df(ā).

The induction step follows easily:

τ^{k+1}f(∇^{k+1}ā) = τ(τ^{k}f)(∇(∇^{k}ā)) = Dτ^{k}f(∇^{k}ā) = DD^{k}f(ā) = D^{k+1}f(ā).

(2) We prove that for each l = 1, …, k we have

(2.3)  τ^{l}(f^{k}) = \frac{k!}{(k − l)!}f^{k−l}(τf)^{l} + f^{k−l+1} p_{l}

for some p_{l} ∈ K{x̄_{1}, x̄_{2}, …, x̄_{2^{l}}}. From which the results follows when l = k. By Lemma 2.6, we have τf^{k} = kf^{k−1}τf, so (2.3) holds for l = 1 with p_{1} = 0. Assume it holds for 1 ≤ l < k, then

τ^{l+1}f^{k} = ττ^{l}f^{k} = τ (\frac{k!}{(k − l)!}f^{k−l}(τf)^{l} + f^{k−l+1}p_{l})
= \frac{k!}{(k − l)!} ((k − l)f^{k−l−1}(τf)^{l+1} + lf^{k−l}(τf)^{l−1}τ^{2}f)
+ (k − l + 1)f^{k−l}(τf) p_{l} + f^{k−l+1}τp_{l}
= \frac{k!}{(k − l − 1)!}f^{k−l−1}(τf)^{l+1} + f^{k−l} p_{l+1}

where

p_{l+1} = \frac{k! l}{(k − l)!}(τf)^{l−1}τ^{2}f + (k − l + 1)(τf) p_{l} + fτp_{l}.

□

Proposition 2.11. Suppose R is a reduced ℚ-algebra and D : R → R is a Δderivation. Let ā a tuple of R and A ⊆ I(ā/R). Suppose there is a tuple b̄ of R such that

(2.4)  τ(g)ā(b̄) = 0, for all g ∈ A.

Then τ(f)ā(b̄) = 0 for all f ∈ {A}.

Proof. By the first argument in the proof of Proposition 2.7, equation (2.4) holds for all elements in [A]. Let f ∈ \{A\}, since R\{x̄\} is also a ℚ-algebra \{A\} = \sqrt{[A]}, and so there is k ≥ 1 such that f^{k} ∈ [A] and hence τf^{k}(ā, b̄) = 0. By part (1) of Lemma 2.10, τ^{k−1}(τf^{k})(∇^{k−1}(ā, b̄)) = 0. Thus, by part (2) of Lemma 2.10, we have

k!(τf)^{k}(ā, b̄) + f(ā)p(∇^{k−1}(ā, b̄)) = 0,

for some p ∈ R{x̄_{1}, x̄_{2}, …, x̄_{2^{k}}}. Since f(ā) = 0, we get k!(τf)^{k}(ā, b̄) = 0. Thus, since R is a reduced ℚ-algebra, τ(f)ā(b̄) = 0. □

Corollary 2.12. If S is a Δ-field, then Proposition 2.7 holds even if we replace the assumption that [A] = I(ā/R) by {A} = I(ā/R).

Proof. Suppose {A} = I(ā/R) and τ_{D/Δ}(g)ā(b̄) = 0 for all g ∈ A. Let (K, Δ) be a differentially closed field extending S. By Corollary 2.9, we can extend D to a derivation D′ : K → K. Now, by Proposition 2.11, τ_{D′/Δ}(g)ā(b̄) = 0 holds for all g ∈ {A}_{K}, where {A}_{K} denotes the radical Δ-ideal in K{x̄} generated by A. But {A} ⊆ {A}_{K}, so that τ_{D/Δ}(g)ā = 0 for all g ∈ I(ā/R). Now apply Proposition 2.7. □

3. Relative prolongations and a characterization of DCF_{0,m+1}

From now on we use freely the basic notions and terminology of model theory; we suggest [9] as a general reference. We work in the language of differential rings L_{m} = {0, 1, +, −, ×, δ_{1}, …, δ_{m}}, and we let L_{0} be the language of rings. We denote by DF_{0,m} the theory of differential fields of characteristic zero with m commuting derivations, and by DCF_{0,m} its model-completion, the theory of differentially closed fields. In the ordinary case, when m = 1, we write DF_{0} and DCF_{0} in place of DF_{0,1} and DCF_{0,1}. The theory of algebraically closed fields of characteristic zero is denoted by ACF_{0}.

Let us recall the geometric axioms of DCF_{0} given by Pierce and Pillay in [14]. Given a δ-field K and V a Zariski-closed set of K^{n}, the prolongation of V, τV, is the Zariski-closed subset of K^{2n} defined by the equations f = 0 and ∑_{i=1}^{n} \frac{∂f}{∂x_{i}} (x̄)y_{i} + f^{δ}(x̄) = 0 for each polynomial f ∈ K[x̄] vanishing on V. Note that, in terms of our notation from Definition 2.5, the last equation is just τ_{δ/∅}f(x̄, ȳ) = 0.

Fact 3.1 (Pierce-Pillay Axioms). Let (K, δ) ⊧ DF_{0}. Then (K, δ) ⊧ DCF_{0} if and only if K ⊧ ACF_{0} and for each pair of irreducible Zariski-closed sets V ⊆ K^{n} and W ⊆ τV such that W projects dominantly onto V, there is ā ∈ V such that (ā, δā) ∈ W.

The Pierce-Pillay characterization of DCF_{0} is indeed first-order. Expressing irreducibility of a Zariski-closed set as a definable condition on the parameters uses the existence of bounds to check primality of ideals in polynomial rings in finitely many variables [17]. Also, if the field is algebraically closed, one can find a first-order formula, in the language of rings, describing for which parameters a Zariski-closed set projects dominantly onto some fixed irreducible Zariski-closed set. This uses the fact that in a model of ACF_{0} the Morley rank of a definable set agrees with the dimension of its Zariski-closure.

The goal of this section is to extend the Pierce-Pillay axioms, in an appropriate sense, to the context of several commuting derivations. Our approach is to accomplish this by characterizing DCF_{0,m+1} in terms of the geometry of DCF_{0,m}. The Pierce-Pillay axioms are then the m = 0 case (under the convention DCF_{0,0} = ACF_{0}).

For the rest of this section we fix a differential field (K, Δ ∪ {D}) with Δ = {δ_{1}, …, δ_{m}}, and V ⊆ K^{n} a Δ-closed set.

Definition 3.2. We define the D/Δ-prolongation of V, τ_{D/Δ}V ⊆ K^{2n}, to be the Δ-closed set defined by

f = 0 and τ_{D/Δ}f = 0, for all f ∈ I(V/K).

Here I(V/K) = {f ∈ K{x̄} : f vanishes on V }. When Δ and D are understood, we just write τf and τV. For ā ∈ V, τ(V)ā denotes the fibre of τV at ā. Note that when m = 0 this is just the usual prolongation.

By the first equality of Lemma 2.3, if ā is in V then (ā, Dā) ∈ τV. In particular the projection π : τV → V given by π(ā, b̄) = ā is surjective.

Suppose A ⊆ K{x̄} is such that [A] = I(V/K). Then τV is defined by f = 0 and τf = 0 for all f ∈ A. Indeed, this follows from the first argument in the proof of Proposition 2.7. Moreover, the following consequence of Proposition 2.11 implies that the D/Δ-prolongation varies uniformly with V.

Proposition 3.3. Suppose (K, Δ) ⊧ DCF_{0,m}. If V = V(f_{1}, …, f_{s}) := {ā ∈ K^{n} : f_{i}(ā) = 0, i = 1, …, s}, then τV = V(f_{i}, τf_{i} : i = 1, …, s).

Proof. Clearly τV ⊆ V(f_{i}, τf_{i} : i = 1, …, s). Let (ā, b̄) ∈ V(f_{i}, τf_{i} : i = 1, …, s). By Proposition 2.11, τf(ā, b̄) = 0 for all f ∈ {f_{1}, …, f_{s}}. Since (K, Δ) ⊧ DCF_{0,m}, we have {f_{1}, …, f_{s}} = I(V(f_{1}, …, f_{s})) = I(V). Hence, (ā, b̄) ∈ τV. □

Remark 3.4.

(1) Suppose (K, Δ) ⊧ DCF_{0,m}. If V is defined over the D-constants, that is, V = V(f_{1}, …, f_{s}) where f_{i} ∈ C_{D}{x̄}, then τV is just Kolchin’s Δ-tangent bundle of V. Indeed, by Proposition 3.3, the equations defining τV become f_{i}(x̄) = 0 and τf_{i}(x̄, ȳ) = df_{i}(x̄) · θȳ = 0, i = 1, …, s. These are exactly the equations for Kolchin’s Δ-tangent bundle T_{Δ}V ([5], Chap.VIII, §2).

(2) In general, τV is a torsor under T_{Δ}V. Indeed, from the equations one sees that τ(V)ā is a translate of T_{Δ}(V)ā, and so the map T_{Δ}V ×_{V} τV → τV given by ((ā, b̄), (ā, c̄)) ↦ (ā, b̄+c̄) is a regular action of T_{Δ}V on τV over V.

Note that in case Δ = ∅, part (2) of Remark 3.4 reduces to the fact that the prolongation of a Zariski-closed set is a torsor under its tangent bundle.

Here is our extension of the Pierce-Pillay characterization to several commuting derivations.

Theorem 3.5. Suppose (K, Δ∪{D}) ⊧ DF_{0,m+1}. Then (K, Δ∪{D}) ⊧ DCF_{0,m+1} if and only if

(1) (K, Δ) ⊧ DCF_{0,m}

(2) For each pair of irreducible Δ-closed sets V ⊆ K^{n}, W ⊆ τV such that W projects Δ-dominantly onto V. If O_{V} and O_{W} are nonempty Δ-open subsets of V and W respectively, then there exists ā ∈ O_{V} such that (ā, Dā) ∈ O_{W}.

As we will see in the proof, it would have been equivalent in condition (2) to take O_{V} = V and O_{W} = W. Also note that when m = 0, the theorem is exactly Fact 3.1.

Proof. Suppose (K, Δ ∪ {D}) ⊧ DCF_{0,m+1}, and V, W, O_{V} and O_{W} are as in condition (2). Let (𝕌, Δ) be a |K|^{+}-saturated elementary extension of (K, Δ). If X is an (L_{m}-)definable subset of K^{n}, by X(𝕌) we mean the interpretation of X in 𝕌^{n}. Let (ā, b̄) ∈ 𝕌^{2n} be a Δ-generic point of W over K; that is, I(ā, b̄/K) = I(W (𝕌)/K). Then (ā, b̄) ∈ O_{W} (𝕌). Since (ā, b̄) ∈ τV (𝕌) we have that τ(f)ā(b̄) = 0 for all f ∈ I(V (𝕌)/K). The fact that W projects Δ-dominantly onto V implies that ā is a Δ-generic point of V over K, so ā ∈ O_{V} (𝕌) and I(ā/K) = I(V (𝕌)/K). Hence, τ(f)ā(b̄) = 0 for all f ∈ I(ā/K). By Proposition 2.7, there is a unique Δ-derivation D′ : K{ā} → 𝕌 extending D such that D′ā = b̄. By Corollary 2.9, we can extend D′ to all of 𝕌, call it D′′. Hence, 𝕌 becomes a Δ ∪ {D′′}-field extending the Δ ∪D-closed field K. Since ā ∈ O_{V} (𝕌), (ā, b̄) ∈ O_{W} (𝕌) and D′′ā = b̄, we get a point (ā′, b̄′) in K such that ā′ ∈ O_{V}, (ā′, b̄′) ∈ O_{W} and Dā′ = b̄′.

The converse is essentially as in [14]. Let ϕ(x̄) be a conjunction of atomic L_{m+1}- formulas over K. Suppose ϕ has a realisation ā in some (F, Δ ∪ {D}) ⊧ DF_{0,m+1} extending of (K, Δ ∪ {D}). Let

ϕ(x̄) = ψ(x̄, δ_{m+1}x̄, …, δ_{m+1}^{r}x̄)

where ψ is a conjunction of atomic L_{m}-formulas over K and r > 0. Let c̄ = (ā, Dā, …, D^{r−1}ā) and X ⊆ F^{nr} be the Δ-locus of c̄ over K. Let Y ⊆ F^{2nr} be the Δ-locus of (c̄, Dc̄) over K. Let

χ(x̄_{0}, …, x̄_{r−1}, ȳ_{0}, …, ȳ_{r−1}) := ψ(x̄_{0}, …, x̄_{r−1}, ȳ_{r−1}) ∧ (∧_{i=1}^{r−1}x̄_{i} = ȳ_{i−1})

then χ is realised by (c̄, Dc̄). Since (c̄, Dc̄) is a Δ-generic point of Y over K and its projection c̄ is a Δ-generic point of X over K, we have that Y projects Δdominantly onto X over K. Thus, since (K, Δ) ⊧ DCF_{0,m}, Y (K) projects Δdominantly onto X(K). Also, since (c̄, Dc̄) ∈ τX, we have Y (K) ⊆ τ(X(K)). Applying (2) with V = O_{V} = X(K) and W = O_{W} = Y (K), there is d̄ in V such that (d̄, Dd̄) ∈ W. Let d̄= (d̄_{0}, …, d̄_{r−1}) then (d̄_{0}, …, d̄_{r−1}, Dd̄_{0}, …, Dd̄_{r−1}) realises χ. Thus, (d̄_{0}, Dd̄_{0}, …, D^{r}d̄_{0}) realises ψ. Hence, d̄_{0} is a tuple of K realising ϕ. This proves that (K, Δ ∪ {D}) ⊧ DCF_{0,m+1}. □

4. Making the geometric characterization first-order

It is not known to the author if condition (2) of Theorem 3.5 can be expressed in a first-order way. One issue is to express irreducibility of Δ-closed sets as a definable condition. This seems to be an open problem related to the generalized Ritt problem [12]. The other issue is how to express when a Δ-closed set projects Δ-dominantly onto another Δ-closed set as a definable condition. Unlike the algebraic case, in differentially closed fields, Morley rank does not concide with the Krull-Noetherian geometric dimension (see the example given in §2.5 of [10]).

We resolve the problem in this section by modifying the characterization of Theorem 3.5 so that it no longer mentions irreducibility or dominance. The first of these can be handled rather easily by the following lemma.

Lemma 4.1. Let K be a Δ ∪ {D}-field. Let V ⊆ K^{n} be a Δ-closed set with K-irreducible components {V_{1}, …, V_{s}}. If ā ∈ V_{i}\ ⋃_{j≠i} V_{j}, then τ(V)ā = τ(V_{i})ā.

Proof. Clearly τ(V_{i})ā ⊆ τ(V)ā. Let b̄ ∈ τ(V)ā and f ∈ I(V_{i}/K). Since ā is not in V_{j}, for j ≠ i, we can pick a g_{j} ∈ I(V_{j}/K) such that g_{j}(ā) ≠ 0. Then, if g = ∏_{j} g_{j}, we get fg ∈ I(V/K) and so

0 = τ(fg)ā(b̄) = τ(f)ā(b̄)g(ā) + f(ā)τ(g)ā(b̄) = τ(f)ā(b̄)g(ā)

where the third equality holds because ā ∈ V_{i}. Since g(ā) ≠ 0, we have τ(f)ā(b̄) = 0, and so b̄ ∈ τ(V_{i})ā. □

It follows that if W ⊆ τV projects Δ-dominantly onto V and V_{i} is a K-irreducible component of V, then a K-irreducible component of W ∩τV_{i} projects Δ-dominantly onto V_{i}.

The second issue, that of Δ-dominant projections, is more difficult to deal with. Let us note here that when Δ = ∅, that is, in the case of DCF_{0}, one can just replace dominant projections by surjective projections in the Pierce-Pillay axiomatization. Indeed this reformulation is stated in [15]. We will not give a proof here as it will follow from Theorem 4.3 below. However, what makes this work is the fact that if a is D-algebraic over K, then D^{k+1}a ∈ K(a, Da, …, D^{k}a) for some k. In several derivations it is not necessarily the case that if a is Δ ∪ {D}-algebraic over K, then D^{k+1}a is in the Δ-field generated by a, Da, …, D^{k}a over K, for some k. However, by a theorem of Kolchin (Fact 4.2 below), this can always be achieved if we allow ℚ-linear transformations of the derivations. Our modification of Theorem 3.5 will therefore need to refer to such transformations.

For every M = (c_{i,j}) ∈ GL_{m+1}(ℚ), let Δ′ = {δ′_{1}, …, δ′_{m}} and D′ be the derivations on K defined by δ′_{i} = c_{i,1}δ_{1} + ⋯ + c_{i,m}δ_{m} + c_{i,m+1}D and D′ = c_{m+1,1}δ_{1} +⋯+c_{m+1,m}δ_{m} +c_{m+1,m+1}D. In this case we write (Δ′, D′) = M(Δ, D). Clearly, the elements of Δ′ ∪ {D′} are also commuting derivations on K.

Fact 4.2 ([4], Chap. II, §11). Let (K, Δ ∪ {D}) ⊧ DF_{0,m+1}. Let ā = (a_{1}, …, a_{n}) be a tuple of a Δ ∪ {D}-field extension of K. Suppose all the a_{i}’s are Δ ∪ {D}- algebraic over K, then there exists k > 0 and a matrix M ∈ GL_{m+1}(ℚ) such that, writing (Δ′, D′) = M(Δ, D), we have that D^{ℓ}ā is in the Δ′-field generated by ā, D′ā …, D^{′k}ā over K, for all ℓ > k.

Theorem 3.5 characterizes DCF_{0,m+1} in terms of the geometry of DCF_{0,m}. The idea, of course, was that DCF_{0,m} has a similar characterization relative to DCF_{0,m−1}, and so on. In our final formulation (Theorem 4.3 below) we will implement this recursion and give, once and for all, a geometric first-order axiomatization of DCF_{0,m+1} for all m ≥ 0, that refers only to the base theory ACF_{0}.

Theorem 4.3. Suppose (K, Δ∪{D}) ⊧ DF_{0,m+1}. Then (K, Δ∪{D}) ⊧ DCF_{0,m+1} if and only if

(1) K ⊧ ACF_{0}

(2) Suppose M ∈ GL_{m+1}(ℚ), (Δ′, D′) := M(Δ, D), V = V(f_{1}, …, f_{s}) ⊆ K^{n} is a nonempty Δ′-closed set, and

W ⊆ V(f_{1}, …, f_{s}, τ_{D′/Δ′}f_{1}, …, τ_{D′/Δ′}f_{s}) ⊆ K^{2n}

is a Δ′-closed set that projects onto V. Then there is ā ∈ V such that (ā, D′ā) ∈ W.

Proof. Suppose (K, Δ ∪ {D}) ⊧ DCF_{0,m+1}. Clearly K ⊧ ACF_{0}. Suppose M, Δ′, V and W are as in condition (2). Clearly (K, Δ′ ∪ {D′}) ⊧ DCF_{0,m+1}, so by Proposition 3.3 we have that V(f_{i}, τ_{D′/Δ′}f_{i} : i = 1, …, s) = τ_{D′/Δ′}V. Let V_{i} be an irreducible component of V and W′ = W ∩ τ_{D′/Δ′}V_{i}. By Lemma 4.1, we can find an irreducible component of W′ projecting Δ′-dominantly onto V_{i}. Now just apply Theorem 3.5 (with Δ′ ∪ {D′} rather than Δ ∪ {D}) to get the desired point.

For the converse, we assume conditions (1) and (2) and prove that (K, Δ ∪ {D}) ⊧ DCF_{0,m+1}. Given r = 1, …, m + 1 and N ∈ GL_{m+1}(ℚ), let K_{r,N} = (K, Δ̄_{r−1} ∪ {D̄}) where (Δ̄, D̄) = N(Δ, D) and Δ̄_{r−1} = {δ̄_{1}, …, δ̄_{r−1}}. Set K_{0,N} to be the pure algebraic field K. We show by induction that for each r = 0, …, m+1, K_{r,N} ⊧ DCF_{0,r} for all N ∈ GL_{m+1}(ℚ). The result will then follow by setting r = m+1 and N = id. The case of r = 0 is just assumption (1). We assume 0 ≤ r ≤ m, N ∈ GL_{m+1}(ℚ), and we show that K_{r+1,N} = (K, Δ̄_{r} ∪ {D̄}) ⊧ DCF_{0,r+1}.

Suppose ϕ(x̄) is a conjunction of atomic L_{r+1}-formulas over K, with a realisation ā = (a_{1}, …, a_{n}) in some Δ̄_{r} ∪ {D̄}-field F extending K_{r+1,N}. We need to find a realisation of ϕ in K_{r+1,N}. We may assume that each a_{i} is Δ̄_{r} ∪ {D̄}-algebraic over K (this can be seen algebraically or one can use the existence of prime models of DCF_{0,r+1} over K, see §3.2 of [11]).

Let M′ ∈ GL_{r+1}(ℚ) and k > 0 be the matrix and natural number given by Fact 4.2. Let M ∈ GL_{m+1}(ℚ) be

M = E ( \begin{matrix}M′ & 0 \\ 0 & I \end{matrix} ) EN

where E is the elementary matrix of size (m + 1) that interchanges row (r + 1) with row (m + 1) and I is the identity matrix of size (m − r). Then, setting (Δ′, D′) = M(Δ, D), we get

(4.1)  D^{′k+1}ā = \frac{f(ā, D′ā, …, D^{′k}ā)}{g(ā, D′ā …, D^{′k}ā)}

for some f, g ∈ (K{x̄_{0}, …, x̄_{k}}_{Δ′_{r}})^{n}. Here Δ′_{r} = {δ′_{1}, …, δ′_{r}} and K{x̄}_{Δ′_{r}} denotes the Δ′_{r}-ring of Δ′_{r}-polynomials over K. Let

c̄ = (ā, D′ā, …, D^{′k}ā, \frac{1}{g(ā, D′ā …, D^{′k}ā)}) .

Let X ⊆ F^{n(k+2)} be the Δ′_{r}-locus of c̄ over K and Y ⊆ F^{2n(k+2)} the Δ′_{r}-locus of (c̄, D′c̄) over K. Claim. Y projects onto X. Consider the Δ′_{r}-polynomial map s(x̄_{0}, …, x̄_{k+1}) : X → F^{n(k+2)} given by

s = (x̄_{1}, …, x̄_{k}, f x̄_{k+1}, −x̄_{k+1}^{2}τ_{D′/Δ′_{r}}g(x̄_{0}, …, x̄_{k}, x̄_{1}, …, x̄_{k}, f x̄_{k+1}))

where any product between tuples is computed coordinatewise. Using (4.1), an easy computation shows s(c̄) = D′c̄. Given b̄ ∈ X, we note that (b̄, s(b̄)) ∈ Y. Indeed, if h is a Δ′_{r}-polynomial over K vanishing at (c̄, D′c̄), then h(·, s(·)) vanishes at c̄ and hence on all of X. So (b̄, s(b̄)) is in the Δ′_{r}-locus of (c̄, D′c̄) over K. That is, (b̄, s(b̄)) ∈ Y. As this point projects onto b̄ we have proven the claim.

Now, by induction, (K, Δ′_{r}) ⊧ DCF_{0,r}. Indeed, (K, Δ′_{r}) = K_{r,N′} where N′ is obtained from M by interchanging rows r and (m + 1). Hence, the claim implies that Y (K) projects onto X(K). Also, if X(K) = V(f_{1}, …, f_{s}) where each f_{i} is a Δ′_{r}-polynomial, then clearly Y (K) ⊆ V(f_{i}, τ_{D′/Δ′_{r}}f_{i} : i = 1, …, s). Hence, by condition (2), there is d̄∈ X(K) such that (d̄, D′d̄) ∈ Y (K).

Now, let ρ(x̄) be the L_{r+1}-formula over K obtained from ϕ by replacing each δ_{1}, …, δ_{r+1} for d_{i,1}δ_{1} + ⋯ + d_{i,r+1}δ_{r+1}, where (d_{i,j}) ∈ GL_{r+1}(ℚ) is the inverse matrix of M′. By construction, ϕ^{(K,Δ̄_{r}∪{D̄})} = ρ^{(K,Δ′_{r}∪{D′})}. Thus it suffices to find a realisation of ρ in (K, Δ′_{r} ∪{D′}). We may assume that the k of (4.1) is large enough so that we can write

ρ(x̄) = ψ(x̄, δ_{r+1}x̄, …, δ_{r+1}^{k}x̄)

where ψ is a conjunction of atomic L_{r}-formulas over K. Let

χ(x̄_{0}, …, x̄_{k+1}, ȳ_{0}, …, ȳ_{k+1}) := ψ(x̄_{0}, …, x̄_{k}) ∧ (∧_{i=1}^{k}x_{i} = y_{i−1}) .

Then (F, Δ′_{r}) ⊧ χ(c̄, D′c̄), and so, as (d̄, D′d̄) is in the Δ′_{r}-locus of (c̄, D′c̄) over K, we have that (F, Δ′_{r}) ⊧ χ(d̄, D′d̄). But since d̄ is a K-point, we get (K, Δ′_{r}) ⊧ χ(d̄, D′d̄). Writing the tuple d̄ as (d̄_{0}, …, d̄_{r+1}), we see that d̄_{0} is a realisation of ρ in (K, Δ′_{r} ∪ {D′}). This completes the proof. □

Remark 4.4.

(1) Condition (2) of Theorem 4.3 is indeed first-order; expressible by an infinite collection of L_{m+1}-sentences, one for each fixed choice of M, f_{1}, …, f_{s} and “shape” of W.

(2) In condition (2) we can strengthen the conclusion to ask for {ā ∈ V : (ā, D′ā) ∈ W } to be Δ′-dense in V.

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