Differentially closed fields

David Pierce October 13, 2006

[Added, June 6, 2007: I wrote these notes in preparation for the talk I gave at the MODNET meeting in Antalya.] Notation.

(1) 𝔐 is a structure;

(2) T is a first-order theory;

(3) 𝔐_{M} is 𝔐, with all elements named. Definition 1.

(1) diag 𝔐 is the quantifier-free theory of 𝔐_{M}.

(2) T_{∀} comprises the universal consequences of T. Lemma 2.

(1) The class of structures in which 𝔐_{M} embeds is elementary, and its theory is diag 𝔐.

(2) The class of substructures of models of T is elementary, and its theory is T_{∀}.

Definition 3. T is model-complete [10] if T ∪ diag 𝔐 is complete whenever 𝔐 ⊨ T. Theorem 4 (Robinson [10]). The following are equivalent:

(1) T is model-complete.

(2) ⊆ is ≼ in Mod(T).

(3) ⊆ is ≼_{1} in Mod(T) (Robinson’s test).

Definition 5.

(1) A model 𝔐 of T is an existentially closed model of T if 𝔐 ≼_{1} 𝔑 whenever 𝔐 ⊆ 𝔑 and 𝔑 ⊨ T.

(2) Suppose T_{∀} = T^{∗}_{∀}.

(a) T^{∗} is a model-completion [9] of T if T ⊆ T^{∗}, and T^{∗} ∪ diag 𝔐 is complete whenever 𝔐 ⊨ T.

(b) T^{∗} is a model-companion [1] of T if T^{∗} is model-complete. Model-completions are model-companions.

Definition 6.

(1) T_{∀∃} comprises the ∀∃ consequences of T.

(2) T is inductive if the class of its models contains all unions of chains of its elements.

Lemma 7 (Chang–Łoś–Suszko [9]). T is inductive if and only if T_{∀∃} ⊢ T.

Theorem 8 (Eklof & Sabbagh [3]).

(1) 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.

(2) If T has a model-companion, then its models are just the existentially closed models of T.

Example. The theory of one-dimensional vector-spaces over algebraically closed fields is the model-completion of the theory of one-dimensional vector-spaces, and the model-companion of the theory of vector-spaces [7]. (The signature is that of fields for the scalars, that of groups for the vectors, with a symbol for multiplication of vectors by scalars.)

Lemma 9 (Robinson).

(1) Model-completions are unique.

(2) Assume T is inductive. Then T has a model-completion if and only if, for every open formula φ(x, y), there is a formula φ̂(x, y) such that, for all models 𝔐 of T, for all a from M,

𝔐 ⊨ ∃x φ̂(x, a) ⟺ T ∪ diag 𝔐 ∪ {∃x φ(x, a)} is consistent.

If it exists, the model-completion is axiomatized by

T ∪ {∀y (∃x φ̂(x, y) → ∃x φ(x, y)) : φ(x, y) open}.

Definition 10.

(1) ACF is the theory of algebraically closed fields.

(2) SCF is the theory of separably closed fields.

(3) DF is the theory of differential fields, that is, structures (K, D), where K is a field, and D is a derivation of K, that is, an additive endomorphism obeying the Leibniz rule, D(x · y) = Dx · y + x · Dy.

(4) DPF is the theory of differentially perfect fields, that is, models of DF satisfying ∀x ∃y (p = 0 ∧ Dx = 0 → y^{p} = x) for every prime p.

(5) If T is one of these theories, then T_{p} is the theory of models of T whose underlying fields have characteristic p.

Theorem 11 (Seidenberg [11]). The function φ ↦ φ̂ in Lemma 9 exists when T is DF_{p}. Corollary 12 (Robinson). DF_{0} has a model-completion, DCF_{0}. Theorem 13 (Wood [13]). If p > 0, then DF_{p} has a model-companion, DCF_{p}, which is the model-completion of DPF_{p}. Lemma 14 (Blum [2]).

(1) Say T ⊆ T^{∗}, and T_{∀} ⊆ T^{∗}_{∀}. Then T^{∗} is the model-completion of T if

and only if

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

when 𝔄, 𝔅 ⊨ T, and 𝔐 ⊨ T^{∗} and is |B|^{+}-saturated.

(2) If T = T^{∗}_{∀}, it is enough to assume that 𝔅 is generated over 𝔄 by a single element (Blum’s criterion).

The second part holds by the analysis

\xymatrix{𝔐 \\ 𝔄 \ar[u] \ar[r] & 𝔄(a_{0}) \ar@{-->}[ul] \ar[r] & 𝔄(a_{0}, a_{1}) \ar@{-->}[ull] \ar[r] & … \ar[r] & 𝔅 \ar@{-->}[ullll]}

Theorem 15 (Blum [2]). The models of DCF_{0} are those models (K, D) of DF_{0} that contain solutions to every system

f(x, Dx, …, D^{n}x) = 0 ∧ g(x, Dx, …, D^{n−1}x) ≠ 0,

where f(x_{0}, …, x_{n}) and g(x_{0}, …, x_{n−1}) are polynomials over K, and g ≠ 0, and f is non-constant in x_{n}.

Theorem 16 (Wood [14]). The models of DCF_{p} (where p is prime) are those models (K, D) of DPF_{p} that contain solutions to every system

f(x, Dx, …, D^{n}x) = 0 ∧ g(x, Dx, …, D^{n−1}x) ≠ 0,

where f(x_{0}, …, x_{n}) and g(x_{0}, …, x_{n−1}) are polynomials over K, and g ≠ 0, and f is non-constant in x_{n} and separable.

In the proof, Blum’s criterion is not used directly, all though in principle it can be, if the operation for extracting pth roots (where they exist) is given a symbol. The case n = 0 ensures that models of DCF_{p} are separably closed. If n > 0, then the condition on f is that ∂_{n}f(x_{0}, …, x_{n}) ≠ 0 has a solution. Hence DF has a model-companion, axiomatized by:

(1) DPF,

(2) SCF,

(3) ∀z ∀y ∃x ((∂_{n}(y_{0}, …, y_{n}, z) ≠ 0 ∧ g(y_{0}, …, y_{n−1}, z) ≠ 0) → (f(x, Dx, …, D^{n}x, z) = 0 ∧ g(x, Dx, …, D^{n−1}x, z) ≠ 0)).

In characteristic 0, (1) and (2) become DF_{0} and ACF_{0}. In this case, these theories can be retained, while (3) varies, and new model-complete theories of differential fields can be obtained [4]. An alternative is to vary (2).

Theorem 17 (Singer [12]). The theory of ordered differential fields (that is, ordered fields with a derivation) has a model-completion, whose models are those ordered differential fields (K, <, D) that are real-closed, and that contain a solution to every system

f(x, Dx, …, D^{n}x) = 0 ∧ ⋀_{j<m} g_{j}(x, Dx, …, D^{n}x) > 0,

provided that they contain a solution to

f(x_{0}, …, x_{n}) = 0 ∧ ∂_{n}f(x_{0}, …, x_{n}) ≠ 0 ∧ ⋀_{j<m} g_{j}(x_{0}, …, x_{n}) > 0.

The following can be isolated from Singer’s proof:

Lemma 18. In the category of real-closed fields, if the system

f(x_{0}, …, x_{n}) = 0 ∧ ∂_{n}f(x_{0}, …, x_{n}) ≠ 0

has a solution at all, then it has a generic solution.

By Blum’s criterion, to axiomatize the model-completion of a universal theory, it is enough to understand solutions to systems in one variable. For differential fields, an alternative is the following. For every differential field (K, D), every primitive sentence

∃x ( ⋀_{f} f(x, Dx, …, D^{n}x) = 0 ∧ g(x, Dx, …, D^{n}x)),

where g and the f are ordinary polynomials over K, is equivalent, modulo DF ∪ diag (K, D), to

∃(x_{0}, …, x_{n−1}, y_{0}, …, y_{n−1}) ( ⋀_{f} f(x_{0}, …, x_{n−1}, y_{n−1}) = 0 ∧

∧ g(x_{0}, …, x_{n−1}, y_{n−1}) ≠ 0 ∧ ⋀_{i<n−1} x_{i+1} = y_{i} ∧ ⋀_{i<n} Dx_{i} = y_{i}).

This sentence is a special case of the existentialization of the formula

( ⋀_{f} f(x, y) = 0 ∧ g(x, y) ≠ 0 ∧ Dx = y).  (1)

If this has a solution (a, b) in some extension of (K, D), let that solution be a generic point of the variety W over K, and let U be the open subset of this defined by g(x, y) ≠ 0. Let V have generic point a. Since Da = b, we have that (a, b) is in the prolongation τV; so U ⊆ W ⊆ τV. Any point (c, Dc) of U satisfies the formula (1); conversely, if (c, d) is a generic point of W, then D extends to K(c, d) so that Dc = d. Hence, ‘geometric’ axioms for DCF_{0} [8], for DCF_{p} [5], and for CODF [6].

References

[1] Jon Barwise and Abraham Robinson. Completing theories by forcing. Ann. Math. Logic, 2(2):119–142, 1970.

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

[3] Paul Eklof and Gabriel Sabbagh. Model-completions and modules. Ann. Math. Logic, 2(3):251–295, 1970/1971.

[4] E. Hrushovski and M. Itai. On model complete differential fields. Trans. Amer. Math. Soc., 355(11):4267–4296 (electronic), 2003.

[5] Piotr Kowalski. Derivations of the Frobenius map. J. Symbolic Logic, 70(1):99–110, 2005.

[6] Christian Michaux and Cédric Rivière. Quelques remarques concernant la théorie des corps ordonnés différentiellement clos. Bull. Belg. Math. Soc. Simon Stevin, 12(3):341–348, 2005.

[7] David Pierce. Model-theory of vector-spaces over unspecified fields. http://www.math.metu.edu.tr/~dpierce/papers/Vector-spaces/.

[8] David Pierce and Anand Pillay. A note on the axioms for differentially closed fields of characteristic zero. J. Algebra, 204(1):108–115, 1998.

[9] Abraham Robinson. Introduction to model theory and to the metamathematics of algebra. North-Holland Publishing Co., Amsterdam, 1963.

[10] Abraham Robinson. Complete theories. North-Holland Publishing Co., Amsterdam, second edition, 1977. With a preface by H. J. Keisler, Studies in Logic and the Foundations of Mathematics, first published 1956.

[11] A. Seidenberg. An elimination theory for differential algebra. Univ. California Publ. Math. (N.S.), 3:31–65, 1956.

[12] Michael F. Singer. The model theory of ordered differential fields. J. Symbolic Logic, 43(1):82–91, 1978.

[13] Carol Wood. The model theory of differential fields of characteristic p ≠ 0. Proc. Amer. Math. Soc., 40:577–584, 1973.

[14] Carol Wood. Prime model extensions for differential fields of characteristic p ≠ 0. J. Symbolic Logic, 39:469–477, 1974.