Derivations of the Frobenius Map

Piotr Kowalski^{∗} Wrocław University Univ. Illinois at Urbana-Champaign

Abstract

We prove that the theory of fields with a derivation of Frobenius has the model companion which is stable and admits elimination of quantifiers up to the level of the λ-functions. Along the way, we give new geometric axioms of DCF_{p}.

0 Introduction

Throughout this paper Ω is a big algebraically closed field of characteristic p > 0, all the other fields are embedded into Ω, n ≥ 0 and q = p^{n}. Let K be a field and δ : K ⟶ K. δ is a derivation of the n-th power of Frobenius if for any a, b ∈ K,

δ(a + b) = δ(a) + δ(b), δ(ab) = b^{q}δ(a) + a^{q}δ(b).

Let DF_{p,n} denotes the theory of fields with a derivation of the n-th power of Frobenius. For brevity, we call a derivation of the n-th power of Frobenius an n-derivation and a field with an n-derivation an n-differential field. Similarly, we talk about n-differential fields extensions, etc. Thus differential fields are 0-differential fields in our terminology. A natural example of an n-derivation is the composition of the n-th power of Frobenius with a derivation. However, not all the n-derivations can be obtained in such a way.

∗Supported by a postdoc under NSF Focused Research Grant DMS 01-00979 and the Polish KBN grant 2 P03A 018 24

In this paper we prove that DF_{p,n} has the model companion DCF_{p,n}, which behaves much as the theory of differentially closed fields in positive characteristic (see [Wo1]); in particular it is stable, not superstable and admits elimination of quantifiers in the language with the λ-functions. We do not know if adding just the p-th root function is enough, as in the case of DCF_{p}.

The axioms of DCF_{p,n} are in the style of Pierce-Pillay axioms of DCF_{0} [PP]. In particular, this gives new axioms of DCF_{p}. The proof that DCF_{p,n} does axiomatize the model companion of DF_{p,n} follows the lines from [PP], however some care needs to be taken while dealing with separability issues.

The main motivation for considering n-derivations is a theorem [Kow, 2.3.1] saying that the n-derivations constitute the remaining (besides derivations and endomorphisms) class of jet operators [Bu] (up to a natural notion of equivalence of operators [Bu]) on fields . Therefore, this paper shows that for any jet operator, its theory has the model companion.

At first I thought that the Shelah-Wood theory [Sh], [Wo1] can be extended smoothly to the case of n-derivations just by ”twisting arguments by a power of Frobenius”. However, not all the arguments from the differential algebra extend to the n-differential case. Some results trivially follow from functorial properties of the structure ring together with the existence of extensions to fields of fractions. For others some computations are needed, but there are always twisted versions of the proofs from differential algebra. For example, the role of the tangent space is played by the Frobenius tangent space. Finally, there are some results which do not hold for derivations of non-zero powers of Frobenius. Note that the real difference between derivations and n-derivations can be seen via looking at compositions. The composition of an n-derivation with an m-derivation is an n + m-derivation for m > 0 and is not an l-derivation for any l, if m = 0. The lack of higher order derivations is the main obstacle for proving the quantifier elimination up to the level of the (partial) inverse of Frobenius.

I would like to thank Zoé Chatzidakis and Ludomir Newelski for their comments and suggestions. I would like also to thank the referee for many valuable suggestions.

1 Some n-differential algebra

To prove certain properties of n-derivations, we need to go beyond the context of fields. All the rings considered in this section are of characteristic p commutative and with unity. One can associate with a ring R its structure ring of n-derivations R_{(1)} := (R^{2}, ⊕, ∗), where:

(a_{0}, a_{1}) ⊕ (b_{0}, b_{1}) = (a_{0} + a_{1}, b_{0} + b_{1}), (a_{0}, a_{1}) ∗ (b_{0}, b_{1}) = (a_{0}b_{0}, a_{0}^{q}b_{1} + b_{0}^{q}a_{1}).

One can check that R_{(1)} is really a ring (0 = (0, 0), 1 = (1, 0)). This is functorial: for f : R ⟶ S we define f_{(1)} : R_{(1)} ⟶ S_{(1)} as f × f.

Definition 1.1 Let R be a ring. A map δ : R → R is an n-derivation if the map R ∋ a ↦ (a, δ(a)) ∈ R_{(1)} is a ring homomorphism.

One easily checks that this definition coincides with the one given in the introduction. Thus n-derivations are jet operators in the sense of [Bu] (see also [Kow]), or D-operators from [Sc] .

Since R_{(1)} is a ring, we get that n-derivations not coming from derivations (i.e. not being the compositions of the n-th power of Frobenius with a derivation) exist. Consider the zero n-derivation on a field K. Since an nderivation on K[X] is equivalent to a homomorphism from K[X] to K[X]_{(1)}, for any F ∈ K[X], X ↦ F extends uniquely to an n-derivation of K[X]. It is enough to choose F not belonging to K[X]^{p}, to get an n-derivation not coming from a derivation. Using Lemma 1.4, we can produce such an n-derivation on a field.

If f : R ⟶ S is a homomorphism of rings, we call a map δ : R ⟶ S an n-derivation of f, if δ is additive and the map (f, δ) : R → S_{(1)} is a ring homomorphism. So the usual n-derivations can be considered as nderivations of the identity map. After Fact 1.2, f is always an inclusion map, and all the n-derivations are denoted by δ.

As was noted by Udi Hrushovski and David Pierce, n-derivations on a ring R correspond to derivations of the inclusion R^{q} ⊂ R (actually, we need also to assume that the Frobenius map is injective on R). In this correspondence, an n-derivation coming from a derivation correspond to a derivation of R^{q} ⊂ R whose image is contained in R^{q}.

Fact 1.2 If δ : R ⟶ S is an n-derivation of f : R ⟶ S, and I ⊲ R, J ⊲ S, are ideals such that δ(I), f(I) ⊂ J, then δ̄: R/I ⟶ S/J is an n-derivation of \overline{f} : R/I ⟶ S/J.

Proof The map (f, δ) : R ⟶ S_{(1)} is a ring homomorphism. Let g : S ⟶ S/J be the quotient map. Then g_{(1)} : S_{(1)} ⟶ (S/J)_{(1)} is a ring homomorphism, and (f, δ)(I) ⊂ ker(g_{(1)}) = J × J, hence the composition:

R ⟶ S_{(1)} ⟶ (S/J)_{(1)}

factors through the quotient map R ⟶ R/I, inducing the homomorphism:

(f̄, δ̄) : R/I ⟶ (S/J)_{(1)}.

This means that δ̄ is an n-derivation of f̄.

Fact 1.3 Suppose R ⊂ R_{1}, R_{2} are n-differential ring extensions. Then there is a unique n-derivation on R_{1} ⊗_{R} R_{2} such that the maps R_{1}, R_{2} → R_{1} ⊗_{R} R_{2} are n-differential. Moreover R_{1} ⊗_{R} R_{2} is the coproduct of R_{1} and R_{2} in the natural category of n-differential R-algebras.

Proof It is a general (obvious) fact about jet operators [Kow, p. 19]. The n-derivation on R_{1} ⊗_{R} R_{2} is given by the following formula:

δ(∑_{i} a_{i} ⊗ b_{i}) = ∑_{i} a_{i}^{q} ⊗ δ(b_{i}) + δ(a_{i}) ⊗ b_{i}^{q} .

Lemma 1.4 Suppose i : R ⊂ S is an extension of domains, and δ : R ⟶ S is an n-derivation of i. Then δ extends to an n-derivation of the embedding R_{0} ⟶ S_{0} of the fields of fractions.

Proof The map (i, δ) : R ⟶ S_{(1)} is a ring homomorphism. Denote by Φ the following composition:

R ⟶ S_{(1)} ⟶ (S_{0})_{(1)},

where the latter map comes from the embedding of S into its fraction field S_{0}. It is enough to check, whether Φ factors through R ⟶ R_{0}. It does if Φ(R \ {0}) is a subset of the multiplicative group of (S_{0})_{(1)}.

Take any r ∈ R \ {0}. It is easy to check that

Φ(r) ∗ (\frac{1}{i(r)}, −\frac{δ(r)}{i(r)^{2q}}) = (1, 0) = 1.

Hence Φ(r) is invertible in (S_{0})_{(1)}.

Now we turn our attention to jet spaces and their connection with extensions of δ. If L is a field, then an affine L-variety is the set of zeroes in a cartesian power of Ω of a finite set of polynomials over L (we do not demand Lirreducibility). For an L-variety V, let I_{L}(V) = {F ∈ L[X̄] : F (V) = 0}. By Hilbert Nullstellensatz and properties of polynomial ideals, L-varieties correspond to radical ideals in L[X̄], and L-irreducible L-varieties correspond to prime ideals in L[X̄].

For a field L and a finite tuple a ⊂ Ω, the locus of a over L, denoted by locus_{L}(a), is the smallest affine L-variety which contains a. It is the set of zeroes of the ideal I_{L}(a) = {f ∈ L[X̄] : f(a) = 0}. A locus over a field L is an L-irreducible L-variety and any L-irreducible L-variety is a locus over L of a certain tuple a ⊂ Ω. An affine variety V is defined over L, if I_{Ω}(V) can be generated over L[X̄]. Note that locus_{L}(a) need not be defined over L; e.g. if a ∈ L − L^{p}, then locus_{L^{p}}(a) is an L^{p}-variety defined over L and not defined over L^{p} (but still definable over L^{p}).

We fix an n-differential field (K, δ), and denote by C the kernel of δ. One checks that C is a field, called the field of constants of K. Note some obvious consequences of the twisted Leibnitz rule (a ∈ K, c ∈ C, m > 0):

δ(a^{m}) = ma^{(m−1)q}δ(a), δ(ca) = c^{q}δ(a)

Hence, as in the case of derivations K^{p} ⊆ C. We call K strict, if K^{p} = C.

If X̄ = (X_{1}, ..., X_{n}), X̄′ = (X′_{1}, ..., X′_{n}) are tuples of variables, then X̄ ↦ X̄′ extends to an n-derivation of the inclusion K[X̄] ⊂ K[X̄, X̄′] extending δ on K. For F ∈ K[X̄] we have a similar formula as in the (0-)differential case:

δ(F)(X̄, X̄′) = F^{δ}(X̄^{q}) + ∑_{i} \frac{∂F}{∂X_{i}} (X̄)^{q}X′_{i},

where F^{δ} is the polynomial obtained from F by applying δ to its coefficients, and X̄^{q} = (X_{1}^{q}, ..., X_{n}^{q}).

Definition 1.5 Let V be an affine K-variety. The jet space of V, denoted by V^{(1)}, is the set of zeroes of (I_{K}(V), δ(I_{K}(V))) ⊲ K[X̄, X̄′].

One can check that V^{(1)} is a torsor of the n-th Frobenius tangent space to V, which is the pull-back of the tangent space to F r^{n}(V) by the morphism F r^{n} : V → F r^{n}(V). Also, V ↦ V^{(1)} is a functor, and we have always a projection map V^{(1)} → V. For c ∈ V, we denote by V_{c}^{(1)}, the fibre of V^{(1)} → V over c.

Fact 1.6 If a, b ∈ Ω^{m} (m > 0) and V = locus_{K}(a), then (a, b) ∈ V^{(1)} if and only if a ↦ b extends to an n-derivation of the inclusion K[a] ⊂ K[a, b].

Proof Obviously, K[a] ≅ K[X̄]/I_{K}(a) and (a, b) ∈ V^{(1)} if and only if δ(I_{K}(a)) vanishes at (a, b). The latter condition is equivalent to δ(I_{K}(a)) ⊂ I_{K}(a, b). By 1.2, δ(I_{K}(a)) ⊂ I_{K}(a, b) if and only if a ↦ b extends to an n-derivation of the inclusion K[a] ⊂ K[a, b].

Remark Note that an n-derivation of K[a] ⊂ K[a, b] not necessarily extends to an n-derivation of K(a, b). For example we have a derivation of K[X] ⊂ K[X^{1/p}] induced by δ(X) = X^{1/p}, which clearly does not extend to K(X^{1/p}). In the next fact, we give conditions equivalent to the existence of an extension to the fraction field.

Fact 1.7 Let a, b ∈ Ω^{m} (m > 0), V = locus_{K}(a), W = locus_{K}(a, b). The following are equivalent:

i) There exists an n-derivation on K(a, b) extending δ and taking a to b. ii) There exists an n-differential field extension K ⊂ M such that a, b ⊂ M and δ(a) = b. iii) W ⊂ V^{(1)} and (a, b) ∈ Im(W_{(a,b)}^{(1)} → V_{a}^{(1)}). iv) W ⊂ V^{(1)} and W′ := {(x, y) ∈ W : (x, y) ∈ Im(W_{(x,y)}^{(1)} → V_{x}^{(1)})} is Zariski dense in W.

Proof i)⟶ii) Obvious. ii)⟶iii) By 1.6, (a, b) ∈ V^{(1)} and (a, b, b, δ(b)) ∈ W^{(1)}, since b = δ(a). Therefore W ⊂ V^{(1)}, and as the map W^{(1)} → V^{(1)} is the restriction of the projection on the first and third coordinate, we get:

W_{(a,b)}^{(1)} ∋ (a, b, b, δ(b)) ↦ (a, b) ∈ V_{a}^{(1)}.

iii) ⟶ i) By iii), we know that the set

Z := {c ∈ Ω^{m} : (a, b, b, c) ∈ W^{(1)}}

is non-empty. By the form of δ(F) (for F ∈ K[X̄]), Z is also the set of zeroes of linear polynomials over K(a, b), so we can find c ∈ Z(K(a, b)). By 1.6, there is an n-derivations of the inclusion: K[a, b] ⊂ K[a, b, b, c] taking a to b. By 1.4, it extends to an n-derivation of the inclusion K(a, b) ⊂ K(a, b, b, c). But the latter is identity, since c ⊂ K(a, b). iii)⟷iv) Let W′′ be the Zariski closure of W′. W′ is K-definable, hence W′′ is a K-variety. By quantifier elimination for ACF_{p}, W′ is K-constructible, therefore (a, b) ∈ W′ if and only if W′ contains a K-definable Zariski open subset of W. Since W is K-irreducible and W′′ is a K-variety containing W′, the former condition is equivalent to W′ being Zariski dense in W.

We investigate now algebraic extensions of n-differential fields and show that the results from differential algebra are still true in this context.

Lemma 1.8 Suppose (K, δ) is an n-differential field. Then δ extends in a unique way to an n-derivation on the separable closure of K. Moreover, if K ⊂ L is any n-differential field extension, then the derivation on L preserves the relative separable closure of K in L.

Proof Let b ∈ Ω be separable over K, and F ∈ K[X] be the minimal polynomial of b over K. For G ∈ K[X], we define δ on K[X] extending the n-derivation on K and such that δ(X) = G. Then:

δ(F)(X) = F^{δ}(X^{q}) + (F^{′q}G)(X)

By 1.2, we need to find a polynomial G such that I, the ideal generated by F, is preserved by δ. By the twisted Leibnitz rule, it is enough to find G such that δ(F) ∈ I.

Since K[X]/I is a field, it is enough to show that F^{′q} ∉ I. Since F is separable, F′ ≠ 0 and deg(F′) < deg(F). Hence F^{′q} can not belong to the maximal ideal I.

Therefore, we can extend δ to K(b), for any b separable over K. Since DF_{p,n} is inductive, we can extend δ to the separable closure of K.

Uniqueness and the moreover claim follow in the same way: for b ∈ L separable over K and F a minimal polynomial of b over K, δ(b) = −\frac{F^{δ}(b^{q})}{F^{′q}(b)}, so δ(b) is uniquely determined and separable over K (even δ(b) ∈ K(b)).

In a similar manner we prove:

Lemma 1.9 Any n-differential field has a separably closed strict extension.

Let L_{λ} be the language of (n-)differential fields expanded by the symbols for the λ-functions, see [Ch, 1.8] for the definition of the λ-functions. Since the λ-functions are definable in the pure field language, any n-differential field has naturally the L_{λ}-structure. We will use thoroughly the fact that the extensions of fields preserving λ-functions are exactly the separable extensions [Ch, 1.9].

Fact 1.10 The theory of n-differential fields has the amalgamation property and is universal in the language L_{λ}.

Proof The proof of the amalgamation property goes similarly as in [Zi]. Let K ⊂ K_{1}, K_{2} be extensions of L_{λ}-structures (in particular, separable extensions). Using 1.8, we can assume that K_{1}, K_{2} are separably closed. Hence K^{s}, the separable closure of K is a subfield of K_{1}, K_{2}. By the uniqueness and moreover claim in 1.8, K^{s} is an n-differential subfield of K_{1}, K_{2} and the induced n-differential structures coincide. The extensions K^{s} ⊂ K_{1}, K_{2} are still separable, because if we take a p-basis of K, it remains a p-basis of K^{s} and is still p-independent in K_{1}, K_{2}. Therefore, we can assume (replacing K by K^{s}) that the extensions K ⊂ K_{1}, K_{2} are regular. We can also assume that K_{1} is a lgebraically disjoint from K_{2} over K. Hence, by the regularity of the extensions, K_{1} is linearly disjoint from K_{2} over K [FJ, 9.9]. This means that K_{1}K_{2} ≅_{K} (K_{1} ⊗_{K} K_{2})_{0}, the field of fractions of K_{1} ⊗_{K} K_{2}, and the extensions K_{1}, K_{2} ⊂ K_{1}K_{2} are separable (even regular [FJ, 9.10]). By 1.3 and 1.4, δ extends to K_{1}K_{2}, so the extensions K_{1}, K_{2} ⊂ K_{1}K_{2} are L_{λ}-extensions.

It remains to show that DF_{p,n} is universal in L_{λ}. The sentences expressing properties of λ-functions are actually existential, but any L_{λ}-substructure of an L_{λ}-model of DF_{p,n} is still an L_{λ}-model of DF_{p,n}, since the λ-functions are also the Skolem functions for the sentences expressing p-independence. Namely, we can write an universal sentence expressing that a tuple is pindependent, as well as a universal sentence expressing that the p-th powers of λ-functions are p-coordinates of a p-dependent tuple.

Remark 1.11 The above fact was proved in [Kol] in the case of n = 0 (i.e. for differential fields).

Remark 1.12 DF_{p,n} does not have the amalgamation property in the language of n-differential fields.

Proof An example from [Wo1] works also for n-differential fields. Namely, if K is an n-differential field and a ∈ K − K^{p} such that δ(a) = 0, then δ extends to K(a^{1/p}) with an arbitrary value for δ(a^{1/p}) (since the ideal (X^{p} −a)K[X] is closed under δ, no matter how δ(X) is defined). Hence there are extensions of K with e.g. δ(a^{1/p}) = 0 and δ(a^{1/p}) = 1 and they can not be amalgamated.

2 The model companion of DF_{p,n}

Now we are ready to give axioms of the model companion of DF_{p,n}, which we call DCF_{p,n}. The axioms actually consist of a scheme of axioms.

Axioms of DCF_{p,n}

Suppose V, W are K-irreducible K-varieties, and X is a proper Ksubvariety of W. If W ⊂ V^{(1)}, W projects generically onto V, and the set

{(x, y) ∈ W : (x, y) ∈ Im(W_{(x,y)}^{(1)} → V_{x}^{(1)})}

is Zariski dense in W, then there is a ∈ V (K) such that (a, δ(a)) ∈ W − X.

We need to say a word whether this scheme of axioms is first-order. It is rather standard (see the axiomatization of ACFA in [CH], or SCFA in [Ch]). The main point is that the theory ACF_{p} is strongly minimal, eliminates quantifiers and is complete. In particular, the most important density assumption translates as a quantifier-free formula in the language of fields about the parameters defining W and V saying that the Morley rank of a subset of W coincides with the Morley rank of W. This formula is true in Ω if and only if it is true in K. The definability of being K-irreducible (even uniformly in K) is classical, see the introduction to Chapter IV in [vdD] (van den Dries also gives a new proof there). As usual, we can remove X from the assumptions by replacing W with some higher-dimensional K-variety.

Theorem 2.1 K is an existentially closed model of DF_{p,n} if and only if K ⊧ DCF_{p,n}.

Proof ⟹ Take V, W and X satisfying the axiom assumptions. Since V and W are irreducible K-varieties and W projects generically onto V, there are a, b ⊂ Ω such that V = locus_{K}(a), W = locus_{K}(a, b).

By the density assumption and 1.7, there exists an n-derivation δ on K(a, b) such that δ(a) = b. Obviously (a, b) ∈ W −X. Since K is existentially closed, we can find a′ ∈ V (K) such that (a′, δ(a′)) ∈ W − X. ⟸ By the standard trick, it is enough to show that for any quantifier-free ζ(x, y) in the language of fields, if M ⊧ (∃x)ζ(x, δ(x)) for some n-differential field M extending K, then K ⊧ (∃x)ζ(x, δ(x)).

Take a ⊂ M such that M ⊧ ζ(a, δ(a)) and let V = locus_{K}(a), W = locus_{K}(a, δ(a)). We take X such that the axiom assumptions concerning it are satisfied and ζ(x, y) is implied by (x, y) ∈ W − X.

By 1.7, the density assumption holds. Hence we get a′ ∈ V (K) such that (a′, δ(a′)) ∈ W − X. Therefore K ⊧ (∃x)ζ(x, δ(x)).

Now, we discuss the model-theoretical properties of DCF_{p,n}. We follow the lines from [Wo1]. Let L denotes the language of n-differential fields and L_{λ} the language of n-differential fields with λ-functions. In the next theorem we list some algebraic and model-theoretic properties of DCF_{p,n}. We say that an n-differential field K is n-differentially perfect if any n-differential extension of K is separable.

Theorem 2.2 (i) Models of DCF_{p,n} (as pure fields) are separably closed and of infinite imperfection degree. (ii) Models of DCF_{p,n} are n-differentially perfect. (iii) DCF_{p,n} has the amalgamation property and the join embedding property in L and L_{λ}. (iv) DCF_{p,n} is the model companion of DF_{p,n} in L. DF_{p,n} does not have the model completion in L. DCF_{p,n} is complete.

(v) DCF_{p,n} is the model completion of DF_{p,n} in L_{λ}. DCF_{p,n} has quantifier elimination in L_{λ}.

Proof

(i) By 1.9 and 2.1, models of DCF_{p,n} are separably closed. Take K ⊧ DCF_{p,n} and assume that the imperfection degree of K is finite. Let a_{1}, … a_{m} be a basis of K over K^{p}. For convenience, we take the actual basis as opposed to a p-basis, so the imperfection degree of K equals log_{p} m.

We are looking for a contradiction. For n = 0, the result was proved by Wood [Wo1]. Let us assume n > 0. By the universal property of the ring of polynomials, equivalence of nderivations with certain homomorphisms of rings and 1.4, we can extend an n-derivation δ on K to K(X) (the field of rational functions) such that δ(X) = X. Since DF_{p,n} is inductive, there exists K̄, an existentially closed extension of K. Since K ≺ K̄, a_{1}, … a_{m} is a basis of K̄ over K̄^{p}. Therefore we can write

X = ∑_{i=1}^{m} α_{i}a_{i}

for certain α_{1}, …, α_{m} ∈ K̄^{p}. Let B = (b_{ij}) ∈ M_{m}(K^{p}) such that for each i ≤ m, δ(a_{i}) = ∑_{j} b_{ij}a_{j}. Then

X = δ(X) = ∑_{i=1}^{m} α_{i}^{q}δ(a_{i}) = ∑_{i,j}^{m} α_{i}^{q}b_{ij}a_{i}.

Let ᾱ = (α_{1}, …, α_{m}). By the previous computation, Bᾱ^{q} = ᾱ. Let X̄ = (X_{1}, …, X_{m}) be a tuple of variables and consider BX̄^{q} − X̄, the tuple of polynomials over K. Then BX̄^{q} −X̄ vanishes on α and the Jacobian of BX̄^{q} − X̄ is the identity matrix (we use n > 0 here!). By [La, Chapter VIII, 5.3], the extension K ⊆ K(ᾱ) is separable algebraic. Since K is separably closed, α_{i} ∈ K for each i ≤ m. Therefore X ∈ K, a contradiction. (ii) Take K ⊂ M, an extension of n-differential fields with K ⊧ DCF_{p,n}. Take a finite tuple a from K. Since K is existentially closed, and the sentence expressing that a is linearly dependent over the field of p-th powers is existential, a is linearly dependent over K^{p} if and only if a is linearly dependent over M^{p}. (iii) DCF_{p,n} has the amalgamation property in L_{λ} by 1.10 and the inductiveness of DF_{p,n}. By (ii), any L-extension of models of DCF_{p,n} is also an L_{λ}-extension, hence DCF_{p,n} has also the amalgamation property in L. For the joint embedding property, take K, M ⊧ DCF_{p,n}. Let F denotes the algebraic closure of the prime field. By (i), we can assume that F (together with the 0-function as the n-derivation) is an n-differential subfield of both K and M. Since F is perfect, the extensions F ⊂ K, M are separable, hence they are L_{λ}-extensions. By 1.10 and the inductiveness of DF_{p,n}, K and M jointly L-embed into a model of DCF_{p,n}. By (ii), DCF_{p,n} has also the joint embedding property in L_{λ}. (iv) Since DF_{p,n} is inductive, the model companion of DF_{p,n} exists if and only if the class of existentially closed models of DF_{p,n} is elementary, and is given by an axiomatization of this class [Ho, 8.3.6]. By (iii) and 2.1, DCF_{p,n} is complete. Since DF_{p,n} does not have the amalgamation property in L (1.12), it does not have the model completion in L.

(v) By (ii), any L-extension of a model of DCF_{p,n} is also an L_{λ}-extension, hence (by (iii)) DCF_{p,n} is the model companion of DF_{p,n} in L_{λ} as well. Since DF_{p,n} has the amalgamation property and is universal in L_{λ} (1.10), DCF_{p,n} is the model completion of DF_{p,n} in L_{λ} and DCF_{p,n} has quantifier elimination in L_{λ}.

In the theory DCF_{p}, we get the elimination of quantifiers just after adding the λ_{0}-function (the inverse of Frobenius on K^{p}, zero elsewhere). It is implied by the fact that the strict differential fields are differentially perfect. We do not know, if this is true in the case of n-differential fields, however (as in [Wo1]), if a field is n-differentially perfect, then it is strict (see the proof of 1.12). What we really need is the property that for any extension K ⊂ L of ndifferential fields, the constants of L are linearly disjoint from K over the constants of K. To prove this the Wronskian technique is needed. This technique does not work in the n-differential case (for n > 0). For instance the solutions of the equation X′′ = 0 can have infinite dimension over constants: e.g. if δ = F r^{n} ∘ δ′ (n > 0), where δ′ is a derivation on K such that K has the infinite dimension over constants of δ′. The usage of Wronskian for n = 0 gives that the dimension of the solution set of the equation X′′ = 0 is not bigger than 2. Using Wronskian, one can also easily show that models of DCF_{p} have infinite imperfection degree: dependence over constants becomes a quantifier-free condition, so having big imperfection degree becomes an existential condition. Note also that our proof of 2.1(i) does not work in the (0-)differential case. The reason of the non-existence of Wronskian for n > 0 is the following: the composition of two n-derivations is a 2n-derivation, and to use Wronskian effectively we need higher order operators. But maybe some other techniques may be used, or the higher order operators may be obtained in an indirect way, so we find it natural to ask the following:

Question 1 Let K ⊂ L be an n-differential field extension. Are the constants of L linearly disjoint from K over the constants of K?

Question 2 Is any strict n-differential field n-differentially perfect?

Question 3 Does DCF_{p,n} has elimination of quantifiers in the language with the inverse of Frobenius?

As it was explained above, the positive answer to Question 1 implies the positive answer to Question 2 and the positive answer to Question 2 implies the positive answer to Question 3.

Our axioms of DCF_{p,n} include the Wood axiomatization [Wo1] of DCF_{p} in the case of n = 0. Let us recall these axioms:

• K is strict.

• If F, G are differential polynomials over K in one variable, ord(G) = m, ord(F) < m, and \frac{∂G}{∂X^{(m)}} ≠ 0, then there exists a ∈ K such that G(a) = 0, F (a) ≠ 0.

Note that these axioms make perfect sense also for n-derivations.

If we set V = A^{m}, W = Z(X′_{1} = X_{2}, ..., X′_{m−1} = X_{m}, G(X_{1}, ..., X_{m}, X′_{m})), X = W ∩Z(F (X_{1}, ..., X_{m})), then finding a ∈ K such that G(a) = 0, F (a) ≠ 0 is equivalent to finding a ∈ V (K) such that (a, δ(a)) ∈ W − X (for δ, an nderivation on K). The condition \frac{∂G}{∂X^{(m)}} ≠ 0 means exactly that the morphism W → V is separable. This morphism is separable if and only if the tangent morphism T_{w}W → T_{v}V is surjective for a generic w ∈ W if and only if the nth Frobenius tangent morphism F^{n}T_{w}W → F^{n}T_{v}V is surjective for a generic w ∈ W. As the jet space is a torsor of the n-th Frobenius tangent space (see remark after 1.5), the separability of W → V implies the density assumption axioms of DCF_{p,n}. So the Wood axioms are a part of our axioms in the case of n = 0, similarly as in the characteristic zero case, the Blum axioms are a part of Pierce-Pillay axioms [PP]. For p = 0, the density assumption is superfluous, since a dominant morphism is always separable. Removing this assumption from our axioms, we get exactly the Pierce-Pillay axioms of DCF_{0}. Pierce [Pi] has recently given new geometric axioms of DCF (including the positive characteristic case) and ACFA of somewhat dual nature to the original axioms. It is natural to ask the following:

Question 4 Suppose that the density assumption in Axiom 2 is replaced by the separability of the morphism W → V together with the strictness of K. Does it still axiomatize DCF_{p,n}?

Question 5 Can DCF_{p,n} be axiomatized by the Wood axioms?

The positive answer to Question 5 implies the positive answer to Question 4, since the axioms from Question 4 still include the Wood axioms.

The last question concerns the behavior of compositions of n-derivations:

Question 6 Suppose n > 0 and (K, δ) is a model of DCF_{p,n}. Is (K, δ ∘ δ) a model of DCF_{p,2n}?

To prove stability of DCF_{p,n} we need the fact below. For necessary definitions see [De2]. We say that for K ⊂ L ⊂ F, K ⊂ M ⊂ F, separable extensions of fields, L is p-disjoint from M over K in F, if any subset of L which is p-free over K in F, remains p-free over M in F. This is equivalent to saying that there exist p-bases B_{K}, B_{L}, B_{M} of K, L, M respectively such that B_{K} ⊂ B_{L}, B_{M} and B_{L} ∪ B_{M} is p-independent in F. By [De1, p. 63], if F above is an enough saturated model of SCF_{p,∞} (the theory of separably closed fields of infinite imperfection degree), then L is forking independent from M over K if and only if L is p-disjoint and algebraically disjoint (in this case equivalently linearly disjoint) from M over K in F.

Fact 2.3 Suppose K ⊂ L ⊂ F, K ⊂ M ⊂ F are separable extensions of ndifferential fields , F (as a pure field) is an enough saturated model of SCF_{p,∞} and a ∈ F. Then:

i) If L is p-disjoint from M over K in F, then the extension LM ⊂ F is separable. ii) There exists K_{a}, a countable L_{λ}-substructure of F containing a, such that K_{a} is SCF_{p,∞}-independent from K over K_{a} ∩ K.

Proof

i) This is a fact about pure separable fields, and its proof appears in [Wo2]. It is enough to notice that B_{L} ∪ B_{M} (B_{L}, B_{M} as above) is a p-basis of LM and is still p-independent in F. ii) By stability of SCF_{p,∞}, for any countable a ⊂ F, there exists a countable field k ⊂ K such that a is SCF_{p,∞}-independent from K over k. Then K_{(a)}, the smallest λ-closed subfield of F containing k(a) satisfies the statement of (ii) except it is not n-differential subfield. But we can take a_{1}, the ndifferential field generated by K_{(a)}, and apply to this countable tuple the above construction again. Then we finish using the usual chain procedure.

Theorem 2.4 DCF_{p,n} is stable and not superstable.

Proof Since models of DCF_{p,n} are not perfect (as fields), DCF_{p,n} is not superstable. To prove stability we follow the Shelah’s proof of stability of DCF_{p} [Sh]. We need to prove that for any pair K ⊂ L of models of DCF_{p,n}, there at most |K|^{ℵ_{0}} 1-types over K of elements of L. We can assume that L (as a pure field) is an enough saturated model of SCF_{p,∞}. By 2.5(ii), to any a ∈ L we can associate K_{a}, a countable L_{λ}-substructure of L such that a ∈ K_{a} and K_{a} is linearly disjoint and p-independent from K over K ∩ K_{a}.

For a, b ∈ L, we let a ∼ b, if K ∩ K_{a} = K ∩ K_{b} and K_{a} is n-differentially isomorphic to K_{b} over K ∩ K_{a} by a map sending a to b. There are obviously ≤ |K|^{ℵ_{0}} equivalence classes of ∼. Suppose a ∼ b and let k := K ∩K_{a}. By 1.3, we can extend the n-differential isomorphism between K_{a} and K_{b} to an ndifferential isomorphism K_{a}⊗_{k} K ≅_{K} K_{b}⊗_{k} K. By the linear disjointness, we get a field isomorphism (K_{a}⊗_{k}K)_{0} ≅_{K} K_{a}K and (K_{b}⊗_{k}K)_{0} ≅_{K} K_{b}K. These isomorphisms are n-differential by the uniqueness of the n-derivations on the fields in question. Hence K_{a}K and K_{b}K are n-differentially isomorphic over K. The extensions KK_{a}, KK_{b} ⊂ L are separable by 2.5(i), so KK_{a}, KK_{b} are L_{λ}-substructures of L. By 2.4, DCF_{p,n} has elimination of quantifiers in L_{λ}, thus a and b have the same types over K.

Therefore, the equality of types is coarser than ∼, so there are ≤ |K|^{ℵ_{0}} types over K.

The proof of 2.6 also shows that the non-forking independence relation in DCF_{p,n} is (as usual) the natural one: A is independent from B over C, if < A >_{λ} is SCF_{p,∞}-independent from < B >_{λ} over < C >_{λ}, where for a set D, < D >_{λ} is the L_{λ}-substructure generated by D.

The theory DCF_{p,n} is an expansion of SCF_{p,∞}, which behaves to a certain degree similarly as DCF_{p}. It seems interesting to go to the limit with n, i.e. to consider the theory DCF_{p,∞}, the limit theory (ultraproduct) of DCF_{p,n} as n goes to infinity. This will be done in the forthcoming paper.

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