A theory of a generic function with derivations

Boris Zilber University of Oxford

6 April 2001

1 Introduction

There are two main motives for the research presented in this paper. The first one is Hrushovski’s discovery of new strongly minimal structures ([H1], [H2]) or rather his method of constructing new structures with stability conditions. The other one comes from the attempts of the author to find a model theoretic generalisation of nice analytic structures on complex manifolds [Z1]. The two lines intersected in the observation that the main ingredient of Hrushovski’s construction, the notion of a predimension, or rather a predimension inequality, is very close to one of the main common properties of classical transcendental functions formulated in the form of the Schanuel conjecture (and its generalisations):

tr.d.(X ∪ exp(X)) − l.d.(X) ≥ 0,

where l.d.(X) is the linear dimension over ℚ, the maximal number of additively independent elements of X. A fuller account of this can be found in [Z2] and [Z3]. The ultimate expression of the observed similarity is the conjecture that the field of complex numbers with function exp is isomorphic to an abstract structure constructed by Hrushovski method, the field with a pseudoexponentiation. This statement is stronger than the Schanuel conjecture, since Hrushovski’s predimension inequality for this structure exactly coincides with the Schanuel conjecture. So there is no hope for an immediate success with this conjecture. Instead, model theory provides us with a very general view on the situation and a very general form of the conjecture: Nice analytic structures over the complex numbers are isomorphic to corresponding abstract pseudo-analytic structures, constructed by Hrushovski’s method. This conjecture, we believe, can be verified at least in simple cases. One of the simplest cases is a structure on an algebraically closed field K of characteristic zero with an abstract function f satisfying the following Hrushovski’s predimension inequality for every finite X ⊆ K :

tr.d.(X ∪ f(X)) − size X ≥ 0.  (1)

Here, size (X) is the number of elements in X. We have written down a list of first order axioms, which form the complete ω-stable theory of the class of structures obtained by Hrushovski’s construction with the above inequality. Let us call this the theory of generic function. The general conjecture then presumes that there is an entire holomorphic function f on the complex numbers such that (ℂ, +, ·, f) satisfies the axioms. A.Wilkie in [W] constructed an entire function f (Liouville function) which satisfies (1) and a weaker version of the second group of the axioms, denoted EC in the present paper ([W] discusses the variant of (1) with extra condition that 0 ∉ X, which practically does not change anything). So the question of whether the complex numbers with Liouville function is elementarily equivalent to an abstract structure with generic function remains open, as well as the question of whether the Liouville theory is ω-stable. Under these circumstances it is reasonable to look for further properties of the abstract structure and compare them to the properties of its analytic prototype. This is our aim in this paper. We show that the structure of an abstract generic function f can be expanded to an ω-stable structure which contains analogues of nth derivatives of f. The structure is then given a Zariski-type topology, and we show that under this topology one can carry out usual differential calculations, including the Taylor’s formula. This makes the structure very close to satisfy the definition of a quasi-Riemann surface of [Z1].

2 Definitions and notation

This section introduces definitions which are just versions of the corresponding definitions in [Z2]. So sometimes we state the properties of the notions without proofs, which just would repeat with slight modifications the corresponding proofs from [Z2]. We work with structures K = (K; L), which are thought as being expansions of fields in relational language L whose basic relations are: n-ary predicates ⟨x_{1}, …, x_{n}⟩ ∈ V for each algebraic variety V ⊆ K^{n} defined over the prime subfield of K (V -predicates); a binary relation F_{i}(x, y) for each i ∈ ℤ.

Definition Fix a characteristic p. Denote E the class of structures K in language L axiomatized by A1 : K is a field of characteristic 0 with distinguished algebraic relations V, and F_{i}, graphs of surjective function f^{(i)} : K → K.

Denote for finite X ⊆_{fin} K size X, the cardinality of X; tr.d.(X), the transcendence degree of X; For any finite I ⊆ ℕ denote f^{I}(X) = ⋃_{i∈I} f^{(i)}(X) and call

δ^{I}(X) = tr.d.(X ∪ f^{I}(X)) − size X · size I

the I-predimension of X.

For Y ⊆ K finite δ^{I}(X/Y) = δ^{I}(X ∪ Y) − δ^{I}(Y);

For infinite A ⊆ K δ^{I}(X/A) ≥ k means that for any finite Y ⊆ A there is finite Y ⊆ Y′ ⊆ A such that δ^{I}(X/Y′) ≥ k. And δ^{I}(X/A) = k if δ^{I}(X/A) ≥ k and not δ^{I}(X/A) ≥ k + 1.

Remark δ^{I}(X/A) = δ^{I}(X/Y) for any Y ⊆ A such that tr.d.(X∪f^{I}(X)/Y) = tr.d.(X ∪ f^{I}(X)/A) and X ∪ f^{I}(X) ∩ A ⊆ Y.

Denote E_{0} the subclass of E consisting of all K satisfying the condition:

δ^{I}(X) ≥ 0 for all finite X ⊆ K, I ⊆ ℕ.

Proposition 1 E_{0} is an axiomatizable class.

Proof For any algebraic variety V in (k + 1)n-space defined over the prime field with dim V < kn and any I = {i_{1}, …, i_{k}} ⊆ ℤ consider the axiom:

A2_{V,I} ⟨v_{1} …, v_{n}, f^{i_{1}}(v_{1}) …, f^{i_{j}}(v_{m}) … f^{i_{k}}(v_{n})⟩ ∈ V → ⋁_{i<j≤n} v_{i} = v_{j}.

Obviously, the set A of all the axioms A2 together with A1 defines the class E_{0}.□

Notation Denote SE the class of L-substructures A of structures from E, i.e. A satisfy the condition: A in the field language is embeddable in a field of characteristic p and F_{i} are graphs of partial functions f^{(i)} : A → A. For X ⊆ A denote

f_{A}^{(i)}(X) = {y ∈ A : ⟨x, y⟩ ∈ F_{i} for some x ∈ X}.

Denote

D^{i}A = {x ∈ A : ∃y ∈ A F_{i}(x, y)};

D^{I} = ⋂_{i∈I} D^{i}.

Definition δ_{A}^{I}(X), the predimension of X in A, is defined for X ⊆_{fin} D^{I}A as

δ_{A}^{I}(X) = tr.d.(X ∪ f^{I}(X)) − size I · size X,

and for arbitrary X ⊆_{fin} A

δ_{A}^{J}(X) = δ_{A}^{J}(X ∩ D^{J}A);
δ_{A}(X) = min{δ_{A}^{J}(X) : J ⊆_{fin} ℤ}.

We usually omit the subscript A, when A is fixed.

Notation Denote SE_{0} the subclass of SE consisting of A which satisfy δ_{A}^{I}(X) ≥ 0 for any I ⊆_{fin} ℤ and X ⊆_{fin} A.

Notation For A, B ∈ SE denote A ≤ B the fact that A ⊆ B as structures and δ^{I}(X/A) ≥ 0 for all finite I ⊆ ℤ and X ⊆ B.

It follows immediately from the definitions that

Lemma 2.1 If X ⊆_{fin} A and δ_{A}(X) = δ_{A}^{I}(X), then for any J ⊆ ℤ \ I the set f^{J}(X) is algebraically independent over X ∪ f^{I}(X).

Lemma 2.2 For A, B, C ∈ SE

(i) if A ≤ B and B ≤ C, then A ≤ C; (ii) if A ≤ B, Y ⊆ B, δ(Y/A) = 0, then AY ≤ B. (ii) if A ≤ B, X ⊆ A and f^{(}i)(X) ⊆ B, then Af^{(}i)(X) ≤ B.

Notation Let A ∈ SE, X ⊆_{fin} A and A_{0} ≤ A.. Denote

∂_{A}(X) = min{δ_{A}(X′) : X ⊆ X′ ⊆_{fin} A},
∂_{A}(X/A_{0}) = min{δ_{A}(X′/A_{0}) : X ⊆ X′ ⊆_{fin} A}.

Lemma 2.3 Let A, B ∈ SE, A ≤ B and X ⊆_{fin} B. Choose X ⊆ X′ ⊆_{fin} A such that

δ_{B}(X′/A) = ∂_{B}(X/A).

Then A ∪ X′ ≤ B.

Proof Immediately from the definition.□

Lemma 2.4 Let A, B ∈ SE, A ≤ B and X a finite subset of A. Then

∂_{A}(X) = ∂_{B}(X).

Proposition 2 For any A ∈ SE_{0} there is K ∈ E_{0} and an embedding of A into K such that A ≤ K. In particular, E_{0} ≠ ∅.

Proof Let K be an algebraically closed field of characteristic p such that A ⊆ K as a substructure in the V -language and tr.d.(K/A) > card A. Claim 1. Given i ∈ ℕ and a ∈ K we can extend the definitions of F_{i} from A to A′ by adding at most two elements so that A ≤ A′ and a ∈ D^{i}A′. Claim 2. Given i ∈ ℕ and a ∈ K we can extend the definitions of F_{i} from A to A′ by adding at most two elements so that A ≤ A′ and a ∈ f^{(i)}(A′). Applying the claims inductively we can extend A to K so that A ≤ K, D^{i}(K) = K and f^{(i)}(K) = K for all i ∈ ℕ. □

3 Existentially closed structures

Definition Given N ⊆_{fin} ℕ and I ⊆_{fin} ℤ, size I = k, size N = n, we call a variety V ⊆ K^{n(k+1)} an (I, N)-variety, if it is given as a zero-set of polynomials in variables {x_{m}, y_{m}^{i} : i ∈ I, m ∈ N}. Usually N = {1, …, n}.

We say for an irreducible (I, N)-variety V over C and an integer s that

δ(V) = s

if a generic point {x_{m} = a_{m}, y_{m}^{i} = b_{m}^{i} : i ∈ I, m ∈ N} of V satisfies

tr.d.({a_{m}, b_{m}^{i} : m ∈ N}/C) − kn = s.

For M ⊆ N we call the M-projection V_{M} of V the variety which is the locus over C of {a_{m}, b_{m}^{i} : i ∈ I, m ∈ M}, the (I, M)-part of the generic solution of V.

We say that V is normal if

δ(V_{M}) ≥ 0

for any M ⊆ N.

Definition An (I, N)-variety V over C is said to be free over C if no irreducible component of V is contained in a variety given by an equation of the form x_{i} = x_{j} or x_{i} = c for some i ≠ j ∈ N and c ∈ C.

The variety is said to be absolutely free if it is free over acl(C).

Lemma 3.1 Let C ≤ A ∈ SE_{0}, A \ C finite and ā = ⟨a_{1}, …, a_{n}⟩ be a string of length n of all distinct elements of A \ C. Let I ⊆_{fin} ℤ be such that

δ^{I}(X/C) = ∂(X/C)

for any X ⊆ A \ C. Define b_{m}^{i} to be f^{(}i)(a_{m}) if the latter is defined in A, and some element of K, otherwise, so that

{b_{j}^{i} : i ∈ I, 1 ≤ j ≤ n} \ f^{I}(A \ C)

is algebraically independent over A. Let V be the (I, N) variety (N = {1…. , n}), over C determined by its generic point {x_{m} = a_{m}, y_{m}^{i} = b_{m}^{i} : i ∈ I, m ∈ N}. Then V is free and normal over C.

Proof Immediate from definitions.□

Definition A structure K in E_{0} is said to be f-closed (f.c.) if for any K′ ∈ E_{0}, such that K ≤ K′, any finite quantifier-free type over K which is realized in K′ has a realization in K.

Proposition 3 For any K in E_{0} there exists an f.c. structure containing K.

Proof The class E_{0} is inductive with respect to normal embedding, so the standard construction of existentially closed structures applies.□

Notation The class of f-closed structures is denoted EC.

Below K is an f-closed structure.

Lemma 3.2 Assume I ⊆ ℤ of size k. Let C be a finite subset of K and V ⊆ K^{n(k+1)} a normal free irreducible (I, N)-variety over C. Let V′ ⊆ V be its proper subvariety over C. Then there is ā in K realizing the type v̄^{⌢}f^{I}(v̄) ∈ (V \ V′). Moreover in some extension K′ ≥ K ā^{⌢}f^{I}(ā) can be chosen generic in V over C.

Proof Choose in some field K′ extending K a point on V generic over K : {x_{m} = a_{m}, y_{m}^{i} = b_{m}^{i} : m ∈ N, i ∈ I}. Consider the subset

A = K ∪ {a_{m}, b_{m}^{i} : m ∈ N, i ∈ I}

with V -predicates induced on A from K′, F^{i} ∩ A^{2} = F^{i} ∩ K^{2} for all i ∈ ℤ \ I, and for i ∈ I f^{i} extends that of K to the domain K ∪ {a_{1}, …, a_{n}} on A as f^{i}(a_{m}) = b_{m}^{i}. This is a consistent definition of partial functions since V is free. Thus we obviously defined a structure on A belonging to SE. Moreover, it follows immediately from the normality of V that K ≤ A. Hence by Proposition 2 there is a normal extension A ≤ K′ ∈ E_{0}. By the construction in K′ the type V \ V′ is realized. And since K is f.c. the type is also realized in K.□

Corollary 1 A structure K in E_{0} is f.c. iff given any C ⊆ K, a normal free V over C and an algebraic subvariety V′ ⊂ V over C, there is ā^{⌢}f^{I}(ā) ∈ V \V′.

By the standard trick of algebraic geometry we can represent an open affine variety V \V′ as a biregular projection of a closed affine variety V^{∗} in a higher dimensional space. This allows us to strengthen the previous statement to the following

Theorem 1 A structure K ∈ E_{0} is in EC iff for any normal free (I, N)- variety V over K there is an ā ∈ K^{n} such that āf^{I}(ā) ∈ V.

Lemma 3.3 Given V (c̄), V′(c̄) ⊆ K^{2n} the statement about parameters c̄:

V (c̄) is free and normal and V′(c̄) is a proper subset of V (c̄) is quantifier-free definable in L.

Corollary 2 The subclass EC of f.c.-structures of E_{0} is axiomatizable by the set of axioms (an axiom for any pair of (I, N)-varieties defined over parameters c̄) EC(V, V′): If V (c̄) is free and normal and V′(c̄) is a proper subset of V (c̄), then there is x̄ such that x̄^{⌢}f^{I}(x̄) ∈ V (c̄) \ V′(c̄).

Remark Modulo the theorem above we can simplify EC by omitting V′.

Lemma 3.4 For K ∈ EC, given finite X ⊆ K and K′ ∈ E such that K ≤ K′,

∂_{K}(X) = ∂_{K′}(X).

Proof Immediate from the definitions.□

By Lemma above we may omit K when writing ∂_{K} for K f.closed. Also in this case we call, for X ⊆ K, ∂(X) the dimension of subset X.

Lemma 3.5 For any finite A ⊆ K and any a, b ∈ K

(i) ∂(A) ≤ ∂(aA) ≤ ∂(A) + 1; (ii) ∂(abA) = ∂(aA) = ∂(A) implies ∂(bA) = ∂(A); (iii) ∂(abA) = ∂(aA)&∂(A) < ∂(bA) implies ∂(abA) = ∂(bA); (iv) ∂(aA) = ∂(A) = ∂(bA) implies ∂(abA) = ∂(A);

(v) ∂(aA) = ∂(A) implies ∂(bA) = ∂(abA).

Lemma 3.6 Given n ∈ ℕ and m ∈ ℤ

(i) there is a quantifier-free type q_{m} in variables v_{1}, …, v_{n} such that x̄ in K realizes q_{m} iff δ(x̄) ≥ m; (ii) there is an ∀-type r_{m} in variables v_{1}, …, v_{n} such that x̄ in K realizes r_{m} iff ∂(x̄) ≥ m.

Proof (i) Obviously, we can take for q_{m} the set of formulas of the form A2_{V,I}(v̄) (see axioms in the proof of Proposition 1) where V ranges the varieties over ℚ of dimension less than kn + m.

(ii) Type r_{m} should contain all statements

∀w̄A2_{V,I}(v̄w̄)

for w̄ a string of variables of length l, all l ≥ 0,size I = n and dim V < k(n + l) + m.□

Corollary 3 There is a set of axioms ID which holds in an f.c.-structure K iff K is of infinite ∂-dimension, i.e. for every m there is a subset X ⊆ K with ∂(X) > m.

Notation Denote EC^{∗} the class of all infinitely dimensional f.c.-structures.

Definition The extension of the initial language L by predicates

E_{P} (v̄) ≡ ∃w̄P (v̄w̄),

where P is a quantifier-free formula, is denoted L^{E} and the predicates E_{P} are called E-predicates.

It follows from Lemma 3.6 that

Corollary 4 If K ⊆ K′ in language L^{E}, then K ≤ K′.

Lemma 3.7 Assume K_{1}, K_{2} ∈ E_{0} satisfy EC. Suppose also A ⊆ K_{1}, A ⊆ K_{2} and b̄^{(i)} ∈ K_{i}^{n} be such that Ab̄^{(i)} ≤ K_{i} for i = 1, 2 and the L-quantifier-free types of b̄_{1} and b̄_{2} over A coincide. Then the L^{E}-quantifier-free types of b̄_{1} and b̄_{2} over A coincide.

Proof Let ∃x̄P_{b_{1}}(x̄) be an E-predicate with parameters in Ab̄_{1} which holds in K_{1}, P_{b_{1}} quantifier-free. Let d̄ be a string in K_{1} for which P_{b_{1}}(d̄) holds. Then P_{b_{1}}(x̄) is a consequence of a formula P_{b_{1}}^{0}(x̄) of the form

x̄āb̄_{1}^{⌢}f^{I}(x̄āb̄_{1}) ∈ (V \ V′),

where ā is the string of elements of A, V is the minimal (I, N)-variety over the prime field containing d̄āb̄_{1}^{⌢}f^{I}(d̄āb̄_{1}), and V′ ⊂ V is an algebraic subvariety over the prime field. By Lemma 3.1 it follows that V (āb̄_{1}) is normal. Moreover, since the normality is expressible quantifier-free in L, V (āb̄_{1}) is normal. It follows from axioms EC that ∃x̄P̄_{b_{2}}^{0}(x̄) holds in K_{2} and hence ∃x̄P_{b_{2}}(x̄) holds. Thus b̄_{1} and b̄_{2} satisfy the same E-predicates over A.□

Proposition 4 For every K ∈ EC^{∗} any finite L^{E}-quantifier-free type which is realized in an E_{0}-extension of K is realized in K itself.

Proof Consider an elementary extension K^{∗} of K. We prove that for any finite A ⊆ K^{∗} and c̄ in an ≤-extension K′ the L^{E}-quantifier-free type q of c̄ over A is realized in K^{∗}. W.l.o.g. we assume that A ≤ K^{∗}.

Consider first the case ∂(c̄/A) = 0. Choose c̄′ in K′ extending c̄ such that ∂(c̄/A) = δ(c̄′/A) = 0. Since A ≤ K′ by Lemma 3.1 and axioms EC the L-quantifier-free type of c̄′ over A is realized in K^{∗}. Let b̄′ be the realization. Since ∂(c̄′/A) = δ(c̄′/A) we have Ac̄′ ≤ K′. Since δ(b̄′/A) = 0 and A ≤ K^{∗} we have Ab̄′ ≤ K^{∗}. Hence by Lemma 3.7 the L^{E}-quantifier-free types of b̄′ and c̄′ over A coincide. Hence the corresponding substring of b̄′ is a realization we sought for. Let now ∂(c̄/A) = k > 0 and {c_{1}, …, c_{k}} be a ∂-base of c̄ over A. It follows from ID and the saturatedness that there are {b_{1}, …, b_{k}} in K^{∗} which are ∂-independent over A. Evidently the L-quantifier-free types of ⟨b_{1}, …, b_{k}⟩ and ⟨c_{1}, …, c_{k}⟩ over A coincide. Also A ∪ {b_{1}, …, b_{k}} ≤ K^{∗} and A ∪ {c_{1}, …, c_{k}} ≤ K′. Hence the L^{E}-quantifier-free types of the strings coincide too. Thus we may identify the strings and since ∂(c̄/A∪{c_{1}, …, c_{k}}) = 0 by the case considered above the L-quantifier-free type of c̄ over A is realized in K^{∗}. □

Lemma 3.8 Let K_{1} and K_{2} be structures in EC^{∗} and A_{1} ≤ K_{1}, A_{2} ≤ K_{2} such that there is an L^{E}-isomorphism

φ : A_{1} → A_{2}.

Let K_{1}^{A} and K_{2}^{A} be the expansions of K_{1}, K_{2} by the set of constants naming elements of A_{1} and A_{2} correspondingly with φ. Then

K_{1}^{A} ≡ K_{2}^{A}.

Proof We prove that given ω-saturated elementary extensions K_{1}^{∗} of K_{1} and K_{2}^{∗} of K_{2}, given finite B ⊆ K_{1}^{∗}, c ∈ K_{1}^{∗}\(A_{1}∪B) and a partial L^{E}-isomorphism φ of A_{1} ∪ B into K_{2}^{∗} one can extend the isomorphism to c. By the symmetry it would allow to play Ehrenfeucht game and we are done. We may assume that φ is the identity and A_{1} ∪ B = A = φ(A). It is enough to show that under the assumption for any c ∈ K_{1}^{∗} \ A we can extend φ to some A′ ⊇ Ac as an L^{E}-isomorphism and A′ ≤ K_{1}^{∗}, φ(A′) ≤ K_{2}^{∗}. If ∂(c/A) = 1 then define A′ = Ac and φ(c) to be any element from K_{2}^{∗} which is not in the ∂-closure of A in K_{2}^{∗}. Then A′ and φ(A′) are as required. If ∂(c/A) = 0 then extend c to a finite string c̄ from K_{1}^{∗} so that δ(c̄/A) = 0. Again, as in the proof of Proposition 4, there is b̄ in K_{2}^{∗} which realizes the L-quantifier-free type of c̄ over A. Since δ(b̄/A) = 0, Ab̄ ≤ K_{2}^{∗} holds. Thus by Lemma 3.7 the L^{E}-quantifier-free types of b̄ and c̄ over A coincide. Put φ(c̄) = b̄, A′ = Ac̄.□

Lemma 3.9 For any finite subset A of a structure K ∈ EC^{∗} there is a subset Ã such that

(i) Ã ≤ K; (ii) if B is a subset of an ω-saturated structure K′ ∈ EC and there is a partial L^{E}-isomorphism φ : A → B (onto) in language, then φ can be extended to Ã and

φ(Ã) = B̃ ≤ K′.

Proof Let ā be the string of all elements of A and c̄ in K such that δ(āc̄) = ∂(ā) = m. From general properties of ≤ Ac̄ ≤ K. Let q^{0}(x̄ȳ) be the Lquantifier-free type of āc̄. Let b̄ be a string in K′ which is L^{E}-isomorphic to ā. Then E-predicates guarantee that q^{0}(b̄ȳ) is consistent and thus ∂(b̄) ≤ m = ∂(ā). By the symmetry ∂(b̄) = m = ∂(ā). Let ȳ = d̄ be the realization of q^{0}(b̄ȳ) in K′. Since δ(b̄d̄) = ∂(b̄), b̄d̄≤ K′. Now Lemma 3.7 says that b̄d̄ is of the same L^{E}-quantifier-free type as āc̄. □

Theorem 2 The theory A+EC+ID of class EC^{∗} is a model completion of E_{0}. It is complete and has quantifier elimination in L^{E}.

Proof The completeness is proved in Lemma 3.8. It follows from Lemma 3.9 and Lemma 3.8 that the theory is submodel complete. Thus it has elimination of quantifiers [S].□

Theorem 3 The theory A+EC+ID is ω-stable.

Proof Let K ∈ EC^{∗} and card K = λ. We want to determine the cardinality of the set S(K) of complete 1-types over K. Let K^{∗} be an elementary extension of K which realizes all n-types over K for all natural n. Let S^{#}(K) be the set of all complete n-types over K which are realized in K^{∗} by n-tuples b̄ = ⟨b_{1}, …, b_{n}⟩ such that δ(b̄/K) = ∂(b_{1}/K). Since any element a in an elementary extension K′ of K can be can be included in a finite subset ā such that Kā ≤ K′, it follows that card S(K) ≤ card S^{#}(K). From general properties of ≤ Kb̄ ≤ K^{∗}, and by Lemma 3.7 the L^{E}-quantifier-free type of b̄ over D is determined by the L-quantifier-free type of that. By the quantifier elemination the complete type of b̄ over K is determined by the L-quantifier-free subtype. Thus the card S(K) is less or equal to the cardinality of the set of all L-quantifier-free complete types over K, which is of power λ + ℵ_{0} since each such type is uniquely determined by V for some algebraic (I, N)-variety V. □

Consider now in a big f.c.-structure K a ∂-independent set B_{κ} of power κ and then, using ω-stability of the theory construct a prime model K(B_{κ}) over B_{κ}. By Shelah’s Theorem it is unique over B_{κ}. Since the isomorphism type of B_{κ} is determined just by κ, the isomorphism type of the structure K(B_{κ}) is determined by κ alone. Call such a structure canonical f.c. of cardinality κ.

We discussed in the introduction of this paper the class of fields with one generic function. We now show how this class is related to the above class EC. Let

L^{0} = L \ {F_{i} : i ≠ 0}

be the reduction of language L, which contains only one binary predicate F_{0} of the form F_{i}. We denote further F_{0} as F and correspondingly f^{(0)} as f. A^{0} denote the set of axioms A1, reduced to L^{0} together with A2_{V}^{0} which is the subset of A2_{V,J} with J = {0} Correspondingly we define δ^{0}(X) and adjust the notions of normality and freeness to the case of one function. EC^{0} is the subset of the set of axioms EC with I = {0} and correspondingly ID^{0}. One can easily extract from the above construction and proofs

Theorem 4 The theory A^{0}+EC^{0}+ID^{0} is complete, ω-stable and allows the elimination of quantifiers in the ∃-definable expansion of L^{0}. The reduction of a model of A+EC+ID to L^{0} is a model of A^{0}+EC^{0}+ID^{0}.

Theory A^{0}+EC^{0}+ID^{0} is what we called the theory of generic function in the introduction. Correspondingly A+EC+ID is the theory of generic function with derivations.

4 Specialisation and topology on K

We now slightly extend K by adding one element, the infinity point ∞, to its domain. Algebraic geometrically we just switched from the affine line over the field K to a projective line P(K), which is usually represented as quotient of the set

{⟨v, t⟩ : v, t ∈ K, v ≠ 0 ∨ t ≠ 0}

by the equivalence relation

⟨v_{1}, t_{1}⟩ ∽ ⟨v_{2}, t_{2}⟩ iff v_{1}t_{2} = v_{2}t_{1}.

In this representation x ∈ K corresponds to the equivalence class given by any pair ⟨tx, t⟩ with t ≠ 0, and ∞ corresponds to ⟨v, 0⟩ with v ≠ 0.

Any polynomial equation p(v_{1}, …, v_{n}) = 0 on K can be transformed to a homogeneous polynomial equation q(v_{1}, …, v_{n}, t_{1}, …, t_{n}) = 0, which can be interpreted on P(K) by replacing occurrences of each v_{i} by v_{i}/t_{i} and eliminating then the denominator (the homogenisation). We denote K̂ = K ∪ {∞} = P(K), the completion of K. Correspondingly, a usual affine variety V on K^{m} can be completed to a variety V̂ ⊆ K̂^{m} by transforming the equations defining it to the homogeneous equations as described above.

It is classically known then that the varieties in K̂^{m}, all m, can be taken as closed sets of Zariski topology on K̂^{m} with the following properties:

(i) Closed sets satisfy the descending chain condition; (ii) Under the projection pr : K̂^{m+1} → K̂^{m} the image of a closed set is closed. Also, it is well known that (iii) Given closed V ⊆ K̂^{m+1} and an s ∈ ℕ the set

{w ∈ K̂^{m} : dim(pr^{−1}(w) ∩ V) ≥ s}

is closed.

Now we are ready to define the completion K̂ of structure K, which is a structure in the same language, with the interpretation of L-symbols on K̂ as follows:

(i) the V -symbols are interpreted by completions of the varieties on K; (ii) F_{i}(x, y) has the previous meaning if x, y ∈ K and F_{i}(∞, y) holds for any y ∈ K̂.

We now consider structure K̂ in the extended Zariski language L̂ ⊇ L, which is obtained by adding to L new ternary relations G_{i}, i ∈ ℤ. We introduce on K functions

g_{i}(x_{1}, x_{2}) = \{ \begin{matrix}\frac{f^{(i)}(x_{1})−f^{(i)}(x_{2})}{x_{1}−x_{2}} if x_{1} ≠ x_{2} \\ f^{(i+1)}(x_{1}) if x_{1} = x_{2} \end{matrix}

and define the interpretation in K̂ : (iii) G_{i}(x_{1}, x_{2}, , y) ≡ (g_{i}(x_{1}, x_{2}) = y & x_{1}, x_{2} ∈ K) ∨ x_{1} = ∞ ∨ x_{2} = ∞.

Obviously, K̂ in the language L̂ is definably equivalent to K in language L, but our aim here is to view K̂ as a topological structure, and we consider agreeing topologies on K̂^{n}, all n, the closed bases for which are positively definable (in language L̂) subsets of K̂^{n}.

Definition Suppose M is a first order structure, M^{∗} ≻ M be an elementary extension of M, M ⊆ A ⊆ M^{∗}. A map π : A → M will be called a (partial) specialisation, if for for every a ∈ A^{n} and an n-ary atomic formula S, if M^{∗} ⊧ S(a) then M ⊧ S(π(a)), and π is an identity on M. So, given K̂′ ≻ K̂ a specialisation π : K̂′ → K̂ preserves all Zariski-closed relations as well as F_{i} and G_{i}.

It is rather immediate from the definition, and also proved in [Ho] and [Z2], that

Lemma 4.1 A specialisation

π : M′ → M

preserves all positively definable relations, thus, in the case M = K̂, it maps closed subsets of K̂^{′n} to closed subsets of K̂.

Proposition 5 (B.Weglorz) For any M′ ≻ M there exists a specialisation π : M′ → M iff any finitely satisfiable collection of positive formulas has a realisation in M.

The latter property for M = K̂ means that the topology on K̂ is quasicompact in the sense that any family of closed subsets with finite intersection property has a non-empty intersection, and the projection K̂^{n+1} → K̂^{n} maps closed sets onto closed sets.

Theorem 5 For any K′ ≻ K there is a specialisation π : K̂′ → K̂.

Proof of the theorem follows from lemmas below.

Assume I ⊆ ℤ is finite. Let C be a finite subset of K and V_{C} ⊆ K̄^{n(k+1)}, normal free irreducible I-variety over C in variables x_{k}, y_{k}^{i}, k ∈ {1, …, n} and i ∈ I. We also consider a variety V_{C}^{∗} in bigger space in variables x_{k}, y_{k}^{i}, z_{k_{1},k_{2}}^{i}, z_{k,c}^{i} ( k, k_{1}, , k_{2} ∈ {1, …, n}, k_{1} < k_{2}, i ∈ I and c ∈ C) determined by the homogenisations of the following equations over C, f^{I}(C) :

(i) the equations for V; (ii) the equations r_{k_{1},k_{2}}^{i} = 0, where r_{k_{1},k_{2}}^{i} ≡ y_{k_{1}}^{i} − y_{k_{2}}^{i} − z_{k_{1},k_{2}}^{i}(x_{k_{1}} − x_{k_{2}}), k_{1} < k_{2} ≤ n; (iii) the equations r_{k,c}^{i} = 0 where r_{k,c}^{i} ≡ y_{k}^{i} − f^{i}(c) − z_{k,c}^{i}(x_{k} − c), c ∈ C, k ∈ {1, …, n}.

Lemma 4.2 (i) The projection of V^{∗} on (x, y)-coordinates is just V; (ii) Suppose x_{n−1} = x_{n} holds identically on V. Then V^{∗} ∩ {y_{n−1}^{i} = y_{n}^{i} : i ∈ I} is equivalent to ((V ∩ {y_{n−1}^{i} = y_{n}^{i} : i ∈ I})_{1,...,n−1})^{∗}, (obtained by projecting to x_{k}, y_{k}^{i}-coordinates with k < n, and then taking ^{∗}).

Proof Use the definitions.□

Lemma 4.3 If V above is a normal irreducible projective I-variety, not necessarily free, then there is ā in K̄ such that

ā^{⌢}f^{I}(ā) ∈ V.  (2)

Moreover, ā can be extended to a point in V^{∗} over Cā satisfying extra conditions:

x_{j} = a ≠ ∞ ⇒ z_{j,a}^{i} = f^{(i+1)}(a)  (3)

and

x_{j} = x_{k} = a ≠ ∞ ⇒ z_{j,k} = f^{(i+1)}(a)  (4)

Proof By induction on n. If n = 1 then either V is free, and thus there is a trivial solution with x = ∞ negating the assumptions in (3) and (4), or V implies x ∈ acl(C). Put x = a for a satisfying the corresponding algebraic type over C and notice that by normality dim V (a) = k, i.e. {y^{i} : i ∈ I} are allowed to take any values in K. Thus x_{1} = a and y_{1}^{i} = f^{(i)}(a) is a solution for V. By definition the ideal Ann(V^{∗}) (of polynomials vanishing on V^{∗}) is generated by Ann(V) and polynomials r_{1,c}^{i}. Put

z_{1,c}^{i} = \frac{f^{(i)}(a) − f^{(i)}(c)}{a − c}

for c ≠ a and z_{1,a}^{i} = f^{(i+1)}(a). This will vanish all r_{1,c}^{i}.

For arbitrary n > 1 suppose for distinct m < n indices dim V_{i_{1},...,i_{m}} = m. Then by induction on n there is a solution ā_{|i_{1},...,i_{m}}f(ā_{|i_{1},...,i_{m}}) in K̂. Moreover, the solution extends to that of V_{i_{1},...,i_{m}}^{∗} satisfying (3) and (4). It follows, using the fact that dimension of fibers is minimal on an open subset of parameters, that V (ā_{|i_{1},...,i_{m}}, f^{I}(ā_{|i_{1},...,i_{m}})) is a normal variety of variables {x_{j}, y_{j}^{i} : i ∈ I, j ≠ i_{l}, l = 1, … m}. So by induction this has a solution, satisfying the extra conditions, too. Which completes the solutions to that of V. The condition (3) holds for the solution because it holds on each of the two steps of the construction. The condition (4) holds when both j, k belong to {i_{1}, …, i_{m}} or to the complement of this set, by the same reason. And if j ∈ {i_{1}, …, i_{m}} and k ∉ {i_{1}, …, i_{m}} and x_{j} = x_{k} = a ≠ ∞, we get that a belongs to the set of parameters defining V (ā_{|i_{1},...,i_{m}}, f^{I}(ā_{|i_{1},...,i_{m}})). So z_{k,a}^{i} = f^{(i+1)}(a) as defined on the second step. But z_{j,k}^{i} = z_{k,a}^{i} modulo x_{j} = a. Thus the requirements of Lemma are satisfied in this case. So we assume

dim V_{i_{1},...,i_{m}} > mk, when m < n.  (5)

Notice that, in particular, it then excludes the case that V forces x_{i} ∈ acl(C). We are going to show that there is a solution with x_{n} = ∞. It follows from normality and assumtion (5) that for a generic point {x_{j}, y_{j}^{i} : i ∈ I, 1 ≤ j ≤ n} of V

tr.d.({y_{n}^{i} : i ∈ I}/C∪{x_{j}, y_{j}^{i} : i ∈ I, 1 ≤ j ≤ n−1}∪{x_{n}}) < k or dim V > nk.

(6)

Consider i_{1} < … < i_{m} < n. If m < n − 1, then dim V_{i_{1},...,i_{m},n} > (m + 1)k by (5). Hence

dim(V_{i_{1},...,i_{m},n} ∩ {x_{n} = ∞}) ≥ (m + 1)k.

It follows

dim ∃{y_{n−1}^{i} : i ∈ I}(V_{i_{1},...,i_{m},n−1} ∩ {x_{n−1} = ∞}) ≥ mk,  (7)

since the fibers of the projection are at most of dimension k. If m = n − 1, then using (6) we again get (7). Observe that

∃{x_{n}, y_{n}^{i} : i ∈ I}(V_{i_{1},...,i_{m},n} ∩ {x_{n} = ∞}) = (V ∩ {x_{n} = ∞})_{i_{1},...,i_{m}},

the variety of {i_{1}, …, i_{m}}-variables. We have thus checked above that

(V ∩ {x_{n} = ∞})_{1,...,n−1}

is a normal I-variety. Thus there is a point x_{k} = a_{k}, y_{k}^{i} = f^{(i)}(a_{k}) (k = 1, …, n − 1) on the variety and we also can satisfy the extra conditions (3) and (4). We can then add x_{n} = ∞ and y_{n}^{i} some value, so that we get a point of the form āf^{I}(ā) on V with the extra conditions (3) and (4) satisfied.□

Given a variety V_{C} ⊆ K^{n(k+1)} and a Zariski-specialisation π : K → K̂, obviously π(V_{C}) = V_{π(C)} ⊆ K̂^{n(k+1)}, the variety obtained by replacing parameters C by π(C).

Lemma 4.4 If V_{C} is a normal variety, then V_{π(C)} is too.

Proof For any variety

dim W_{C} ≤ dim W_{π(C)},

which follows from the well-known fact that for any d the condition dim W_{C} < d is a Zariski-open condition in the space of parameters.□

Proof of the theorem. Suppose C ≤ K′ (e.g. C = ∅) and there exists a partial specialisation

π : C → K̂.

Given an element a′ ∈ K′ we want to find a finite A′ ⊆ K′ such that a′ ∈ A′ and CA′ ≤ K′, and to find an extension π′ ⊇ π of the specialisation defined on A′. We can find A′ with CA′ ≤ K′ using Lemma 2.3. Then there is I ⊆_{fin} ℤ such that δ(A′/C) = δ^{I}(A′/C). Let ā′ be a string of all elements of A′ and V_{C} the I-variety which is defined in Lemma 3.1 (with A = CA′). By Lemma 3.1 V_{C} is free and normal. Then V_{π(C)} is normal and we can apply Lemma 4.3 to the variety and so find ā such that ā^{⌢}f^{I}(ā) is in V_{π(C)} and the extra conditions are satisfied too. Define π′(ā′) = ā and π′(f^{(i)}(a′_{m})) = f^{(i)}(a_{m}) for all m ≤ n and i ∈ ℤ, when f^{(i)}(a′_{m}) is in CA′. It follows from (2) and Lemma 2.1 that π′ preserves all the V - relations and all relations of the form F_{i}. It follows from (3) and (4) that π′ also preserves all relations G_{i}. So, we can inductively extend the partial specialisation to get the required full specialisation. □

Given a fixed specialisation π : K̂′ → K̂, for K′ reasonably saturated one can introduce a π-topology and develop a non-standard analysis on K, as shown in [Z2]. The main notion then is

Definition Let a ∈ K̂. The infinitesimal neighbourhood of a is a subset

V_{a} = {a′ ∈ K̂′ : π(a′) = a}.

Elements of V_{0} are called infinitesimals.

A type-definable subset S ⊆ K̂^{n} is said to be closed in π-topology if π(S(K̂′)) = S(K̂).

Remark Since addition is a continuous operation in the topology on K̂, for a ∈ K V_{a} is the set of elements of the form a+α, where α is an infinitesimal.

Remark The π-topology constructed in Theorem 5 is not Noetherian, i.e. it does not satisfy the descending chain condition as the following shows: Consider π-closed set

S = {x ∈ K̂ : f^{(0)}(x) = 0}.

Let a ∈ S. Then

S \ {a} = {x ∈ S : g_{0}(x, a) = 0},

because g_{0}(a, a) = f^{(1)}(a) ≠ 0 by axiom A2, and so these sets are closed.

It follows immediately from Theorem 5

Corollary 5 (Differentiation formula for f^{(i)}’s) For any a ∈ K, an infinitesimal α and i ∈ ℤ there is an infinitesimal β such that

f^{(i)}(a + α) − f^{(i)}(a) = α · (f^{(i+1)}(a) + β).

Let now p(x_{1}, …, x_{n}) be a rational function over K and \frac{∂p}{∂x_{i}} its formal partial derivatives. It follows easily from the definition of infinitesimals and the fact that π preserves the addition and multiplication, that for any a_{1}, …, a_{n} ∈ K, such that p(a_{1}, …, a_{n}) is defined, for any infinitesimals α_{1}, …, α_{n} there is an infinitesimal β such that

p(a_{1} + α_{1}, …, a_{n} + α_{n}) = p(a_{1}, …, a_{n}) + ∑_{i≤n} α_{i} \frac{∂p}{∂x_{i}} (a_{1}, …, a_{n}) + β.

Combining this with the differentiation formula for f^{(i)}’s we get

Corollary 6 Let h(x_{1}, …, x_{n}) be a function obtained by compositions of rational functions and f^{(i)}’s. Let \frac{∂h}{∂x_{i}} be its formal partial derivatives. Then for any a_{1}, …, a_{n} ∈ K, such that h(a_{1}, …, a_{n}) is defined, for any infinitesimals α_{1}, …, α_{n} there is an infinitesimal β such that

h(a_{1} + α_{1}, …, a_{n} + α_{n}) = h(a_{1}, …, a_{n}) + ∑_{i≤n} α_{i} \frac{∂h}{∂x_{i}} (a_{1}, …, a_{n}) + β.

In the case K is a field of characteristic zero we can extend language L̂ by symbols of definable functions g_{i,n}, (i ∈ ℤ, n ∈ ℕ) defined as

g_{i,n}(x_{1}, x_{2}) = \{ \begin{matrix}\frac{n!}{(x_{1}−x_{2})^{n}} [f^{(i)}(x_{1}) − ∑_{0≤k<n} \frac{1}{k!}f^{(i+k)}(x_{2})(x_{1} − x_{2})^{k}], if x_{1} ≠ x_{2} \\ f^{(i+n)}(x_{1}), if x_{1} = x_{2}, \end{matrix}

and strengthen Theorem 5 to this language. Then we correspondingly get

Corollary 7 (Taylor formula for f^{(i)}’s) In the case K is a field of characteristic zero, for any a ∈ K, an infinitesimal α, a positive integer n and i ∈ ℤ there is an infinitesimal β such that

f^{(i)}(a + α) = ∑_{0≤k≤n} \frac{f^{(i+k)}(a)}{k!} · α^{k} + α^{n}β.

We then correspondingly get the Taylor formula for definable functions h as above.

REFERENCES

[Ho] W.A.Hodges, Model Theory, Cambridge University Press. Cambridge, 1993

[H1] E.Hrushovski, A New Strongly Minimal Set, Annals of Pure and Applied Logic, 62 (1993), 147-166

[H2] ————, Strongly Minimal Expansions of Algebraically Closed Fields, Israel Journal of Mathematics 79 (1992), 129-151 [S] G.E.Sacks, Saturated Model Theory, Benjamin Inc., Reading. Massachusets, 1972

[W] A.Wilkie, Liouville functions, Submitted

[Z1]——–, Quasi-Riemann surfaces, In: Logic: from Foundations to Applications, European Logic Colloquium, eds W.Hodges, M.Hyland, C.Steinhorn and J.Truss, Clarendon Press, Oxford, 1996, 515-536

[Z2] B.Zilber, Fields with pseudo-exponentiation, Submitted for publication. Available on www.maths.ox.ac.uk/˜ zilber

[Z3]——–, Analytic and pseudo-analytic structures, Submitted, Available on www.maths.ox.ac.uk/˜ zilber