{'fraction_non_alphanumeric': 0.08747981150025443, 'fraction_numerical': 0.011659549990044028, 'mean_word_length': 3.3154477754439564, 'pattern_counts': {'":': 0, '<': 61, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 25, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Remarks around the non-existence of difference-closure

This paper shows that in general, difference fields do not have a difference closure. However, we introduce a stronger notion of closure (kappa-closure), and show that every algebraically closed difference field K of characteristic 0, with fixed field satisfying a certain natural condition, has a closure, and this closure is unique up to isomorphism over K.


Introduction
In this paper, a difference field is a field K with a distinguished automorphism σ. A difference field L is difference-closed if every finite system of difference equations with coefficients in L which has a solution in a difference field extending L, has already a solution in L. If K is a difference field, then a difference-closure of K is a difference-closed field containing K, and which K-embeds into every difference-closed field containing K.
The theory of difference-closed difference fields has been extensively studied, and is commonly denoted by ACFA. The algebra of difference fields was developped by Ritt, in analogy with the algebra of differential fields. It is well-known that any differential field of characteristic 0 has a differential closure, and that this differential closure is unique up to isomorphism. In 2016, Michael Singer asked whether this result generalises to the context of difference fields. One of the main results of this paper is that it does not, even after imposing some natural conditions on the difference field K. We will show by two examples (1.3 and 1.4) that even the existence of a difference closure can fail.
There are several natural strengthenings of the notion of difference-closure (originating from model-theory but which have a natural algebraic translation), and we will show that these notions do satisfy existence and uniqueness of closure, provided we work over an algebraically closed difference field of characteristic 0 whose fixed subfield is large enough. The proof uses the fact that in characteristic 0, SU-rank 1 types which are orthogonal to the fixed field are stable and stably embedded, which means that one can use techniques of stability theory. They provide examples of structures which are stable over a predicate, see [10] and [12] for definitions. The other main result of the paper is Theorem 3.14. Let κ be an uncountable cardinal or ℵ ε , and let K be an algebraically closed difference field of characteristic 0 such that F := Fix(σ)(K) is pseudo-finite and is κ-saturated.
The paper is organised as follows. In section 1 we discuss the problem, reformulate it in modeltheoretic terms, and describe the two examples. In section 2, we state the preliminary results we will need from difference algebra and model theory. Section 3 contains the proof of the main result.
1 Discussion of the problems and the examples 1.1. Notation and conventions. All difference fields will be inversive, i.e., the endomorphism σ will be onto. Let K be a difference field, contained in some large difference field U. If a is a tuple in U, we denote by K(a) σ the difference field generated by a over K, i.e., the subfield K(σ i (a)) i∈Z of U. The algebraic and separable closure of a field L are denoted by L alg and L s respectively, and G(K) denotes the absolute Galois group of K, i.e., Gal(K s /K). If A ⊂ U, then acl(A) denotes the smallest algebraically closed difference field containing A; it coincides with the model-theoretic closure of A for the theory ACFA. We denote by L the language {+, −, ·, 0, 1, σ}.

Translation into model-theoretic terms.
Let K be a difference field. Recall that any complete theory extending the theory ACFA of difference-closed difference fields is supersimple, unstable, of SU-rank ω, and does not eliminate quantifiers, but it eliminates imaginaries. It is extensively studied in [5]. The reason ACFA does not eliminate quantifiers is that given an automorphism σ of a field K, there may be several non-isomorphic ways of extending σ to K alg . So, the first obvious obstacle to the existence of a difference closure is that, and a natural condition to impose is to assume that K is algebraically closed. There is another natural condition one needs to impose: if L is difference closed, then its fixed field Fix(σ)(L) = {a ∈ L | σ(a) = a} is pseudo-finite. Moreover, every pseudo-finite field can occur as the fixed field of some difference closed field ( [1]). Thus if L is the difference-closure of a difference field K, then Fix(L) must be prime over Fix(K) (for its theory in the language of rings). Duret showed in [7] that any completion of the theory of pseudo-finite fields has the independence property. From his proof one extracts easily the fact that non-algebraic types are non-isolated, and this forces us to require in case K is countable that Fix(σ)(K) be pseudo-finite in order to hope to have a difference closure. The case when K is uncountable is a little more complicated, the question is adressed and solved in [3].
It is therefore reasonable to make the following two assumptions: K is algebraically closed, and K ∩ Fix(σ) is pseudo-finite.
But even this is not enough. What we need to do is the following: Exhibit a difference field K satisfying the above two conditions, and a finite system of difference equations which does not have a solution in K, and such that any finite strengthening of this system has several completions.
This looks easy, since even our stable types are only supersimple, not ω-stable. However, the first obvious examples . . . do not satisfy the first condition. Here is a more involved example, taken from [5] (example 6.7): Example 1.3. An example in characteristic 0. Let k be a pseudo-finite field of characteristic 0 containing Q alg , and consider K = (k alg , σ), where σ is a (topological) generator of Gal(K/k). We conside the elliptic curve J, with jinvariant a / ∈ K, and which is defined by We let A ′ be a cyclic subgroup of J of order p 2 , A = [p]A ′ and a 1 the j-invariant of the elliptic curve J/A, a 2 the j-invariant of the elliptic curve J/A ′ . Then the map Q alg (a, a 1 ) → Q alg (a 1 , a 2 ) which is the identity on Q alg and sends (a, a 1 ) to (a 1 , a 2 ) extends to a field automorphism of Q(a) alg , which in turns extends to an automorphism of K(a) alg which agrees with σ on K. Let Φ(x, x 1 , x 2 ) be the finite system of polynomial equations which describe the algebraic locus of (a, a 1 , a 2 ) over K, and consider the difference equation The formula Φ can be expressed, in the notation of Chapter 5 §3 of [11] (see in particular Note that necessarily, the kernel of the map J b → J σ n (b) for n > 0 is cyclic of order p n . As σ(b) = b, we know that b is transcendental. Hence the curve J b is not of CM-type, its endomorphism group is isomorphic to Z, and therefore J b is not isomorphic to any of its quotients by finite cyclic subgroups (see e.g. section C.11 in [13]). Therefore, the elements b, σ(b), σ 2 (b), . . . are all distinct, and b / ∈ K. Furthermore the isomorphism type of So any difference-closed field containing K must contain a solution of Φ(x, σ(x), σ 2 (x)) ∧σ(x) = x. However, Example 6.7 of [5] shows that if b is as above, and L is any finite extension of K(b) σ , then there are 2 ℵ 0 non-isomorphic ways of extending σ to L alg . Thus K does not have a difference closure.
One can build other examples along the same lines, using moduli spaces of abelian varieties.
where k is a pseudo-finite field fixed by σ, σ restricts to a generator of Gal(k alg /k), and A is the set of solutions of the equation σ(x) p − σ(x) + x p = 0 (in some countable difference-closed overfield). So, any difference-closed field containing K will contain the set B of solutions of the equation σ(x) −x p + x = 0, an infinite-dimensional F p -vector space. However, as was shown in example 6.5 of [5], there are 2 |A| ways of extending σ from Kk(B) alg σ to K(B) alg σ : there is a definable non-degenerate bilinear map q : A × B → F p , which can be chosen totally arbitrarily.
In fact this example is part of a large family of examples: let f and g be additive polynomials with coefficients in a difference field K, and assume that the subgroup A of G a defined by f (x) = g(σ(x)) is locally modular. Then there is a definable subgroup B of G a , and a definable non-degenerate bilinear map A × B → F p . As above, there is no prime model over K(B) σ .
While we provided examples of difference fields not having a difference closure, we did not provide a procedure which, given a difference field which is not difference closed, exhibits a non-isolated type which needs to be realised. So, the following remains open: Question 1.5. Are there any difference fields which are not difference closed but admit a difference closure?

Preliminaries
Basic difference algebra.
2.1. Let K ⊂ U be difference fields. If X = (X 1 , . . . , X n ), the ring K[X] σ = K[σ i (X j )] 1≤j≤n,i∈N is called the n-fold difference polynomial ring. A difference equation is an equation of the form f (X) = 0 for some f (X) ∈ K[X] σ .
If a ia a finite tuple in U, and L is a difference subfield of K(a) σ containing K, then L = K(b) σ for some finite tuple b (5.23.18 in [6]).
An element a ∈ U is transformally algebraic over K if it satisfies some non-trivial difference equation with parameters in K. Otherwise, it is transformally transcendental over K. A tuple a is transformally algebraic over K if all its elements are. A (maybe infinite) tuple of elements of U is transformally independent over K if it does not satisfy any non-trivial difference equation with coefficients in K. A transformal transcendence basis of U over K is a subset B of U which is transformally independent over K and maximal such; every element of K will then be transformally algebraic over K(B) σ . We denote by ∆(K), the transformal transcendence degree of K, i.e., the cardinality of a transformal transcendence basis of K, and if L is difference field containing K, by ∆(L/K) the cardinality of a transformal transcendence basis of L over K.
2.2. The fixed field. The fixed field of U is the field Fix(σ)(U) := {a ∈ U | σ(a) = a}. Then Fix(σ)(U) and K are linearly disjoint over their intersection. (Choose n minimal such that there are c 1 , c 2 , . . . , c n ∈ Fix(σ) and d 1 = 1, d 2 , . . . , d n ∈ K such that i c i d i = 0; applying σ we get c i σ(d i ) = 0, and by minimality of n, that σ(d i ) = d i for all i.) This implies in particular that if E is a difference subfield of K, then EFix(σ)(U) and K are linearly disjoint over their intersection E(Fix(σ)(U) ∩ K). In positive characteristic, similar results hold for the other fixed fields Fix(σ n Frob m ).
Basic model-theoretic facts.
2.3. For references see [5]. The theory ACFA is supersimple, of SU-rank ω. It eliminates imaginaries, but does not eliminate quantifiers. The completions of ACFA are given by describing the isomorphism type of the automorphism σ of the algebraic closure of the prime field (1.4 in [5]). Let U be a model of ACFA, and K a difference subfield of U.
2.4. Types, algebraic closure, independence. If a is a tuple of elements of U, then tp(a/K) is determined by the isomorphism type of the difference field K(a) alg σ over K: a and b have the same type over K if and only if there is a K-isomorphism of difference fields K(a) σ → K(b) σ which send a to b and extends to the algebraic closure of K(a) σ . (1.5 in [5]). The SU-rank of a over K, denoted by SU(a/K), is bounded by tr.deg(K(a) σ /K), and is finite if and only if tr.deg(K(a) σ /K) is finite (if and only if a is transformally algebraic over K). If A ⊂ U, then acl(A) as defined above in 1.1 coincides with the model-theoretic closure of A (1.7 in [5]). Let A, B, C be subsets of U. Then A is independent from B over C, A | ⌣C B, if and only if the fields acl(AC) and acl(BC) are free over acl(C). Equivalently, if whenever a is a tuple of elements in A, then the prime σ-ideal I σ (a/acl(BC)) := {f (X) ∈ acl(BC)[X] σ | f (a) = 0} is generated (as a σ-ideal) by its intersection with acl(C)[X] σ . Then independence coincides with non-forking, and we will also say, in that case, that tp(A/B) does not fork over C. 2.6. Notions of canonical bases. If a is a tuple in U, then Cb(a/K) denotes the smallest difference field over which I σ (a/K) is defined. Then tp(a/K) does not fork over Cb(a/K). Also, Cb(a/K) is contained in the algebraic closure over K of n independent realisations of tp(a/K), for some n; if K(a) σ is a regular extension of K, then Cb(a/K) is contained in the difference field generated over K by finitely many independent realisations of tp(a/K) (see the proof of Lemma 2.13(4) in [5]). Cb(a/K) will denote Cb(a/K) alg . Note that a type does not fork over the algebraic closure of some finite set.
2.7. The generic type. The generic 1-type is the type of a transformally transcendental element. It is axiomatised by its quantifier-free part, is definable and stationary 1 . Similarly, if V is a variety defined over the algebraically closed difference field K, then the generic type of V (which is characterised by having a realisation a with ∆(K(a) σ /K) = dim(V )) is axiomatised by its quantifier-free part, is definable and stationary (see 2.11 in [5]).
2.8. Orthogonality of types. Let p and q be (partial) types over A and B respectively. If A = B, we say that p and q are almost orthogonal (or weakly orthogonal), denoted by p ⊥ a q, if whenever a realises p and b realises q, then a | ⌣A b. We say that p and q are orthogonal, denoted by p ⊥ q, if whenever C contains A ∪ B, and a realises p, b realises q, and a | ⌣A C, b | ⌣B C, then a | ⌣C b.
2.9. The dichotomy in characteristic 0. Recall that a partial type π over a set A is called one-based 2 if whenever a 1 , . . . , a n realise π and B ⊃ A, then (a 1 . . . a n ) | ⌣C B, where C = acl(Aa 1 , . . . , a n ) ∩ acl(B) 3 . Types of finite SU-rank are analysable in terms of types of SU-rank 1. The main result of [5] says that in characteristic 0, a type q of SU-rank 1 is either one-based, or non-orthogonal to the fixed field. Moreover, if q is one-based, then it is stable stably embedded, and definable. See Theorem 4.10 in [5].
2.10. Stable embeddability of the fixed field. Recall that a subset S of U n , which is definable or ∞-definable, is stably embedded if whenever D ⊂ U nm is definable with parameters from U, then D∩S m is definable with parameters from S. An important result of [5] (Proposition 1.11) says that the fixed field F := Fix(σ) of U is stably embedded: if D ⊂ F n is definable in the difference field U (with parameters from U), then it is definable in the pure field language in F (with parameters from F ). In fact, one has more: let C = acl(C) ⊂ U, and b a tuple in F ; then [4]), and therefore any (C ∩ F )-automorphism of the field F extends to a Cautomorphism of the difference field acl(CF ), since it obviously extends to a C-automorphism of CF , and the automorphism σ of CF alg extends uniquely to acl(CF ) (Babbitt's Theorem. See e.g. Lemma 2.8 in [5]). For more properties of stably embedded sets or types, see the appendix of [5].
2.11. More on stable stably embedded types. For a definition of a (partial) type being stable stably embedded, see Lemma 2 of the appendix of [5]. Here we will use the following consequence: let A = acl(A) be algebraically closed, and suppose that tp(a/A) is stable stably embedded. Then tp(a/A) is definable (over A; see Lemma 1 in the Appendix of [5]). Also, if B = Cb(a/A) and tp(a/B) ⊥ a tp(A/B), then tp(a/B) ⊢ tp(a/A): this is because tp(a/B) has a unique non-forking extension to any superset of the algebraically closed set B. (1) p is qf-internal to F if there is some finitely generated over A difference field C such that whenever a realises p, there is a tuple b in F such that a ∈ C(b). I.e., a ∈ CF .
(2) p is almost-internal to F if there is some some finitely generated over A difference field C such that whenever a realises p, there is a tuple b in F such that a ∈ acl(Cb).
Remarks 2.13. Clearly qf-internality implies almost-internality. Moreover, to show qf-internality or almost-internality of p, it is enough to do it for a particular realisation a of the type p, i.e., to find C independent from a over A such that a ∈ CF or a ∈ acl(CF ). See Lemma 5.2 in [5]. Internality or almost internality (to F ) of a type is in fact a property of its quantifier-free part.
Recall that a difference field E is linearly disjoint from F over F ∩ E. It follows that in (1) above, the tuple b can be taken so that Lemma 2.14. Let A = acl(A), and assume that tp , and a ∈ acl(Aa ′ ).
Proof. By assumption there is some tuple c independent from a over A and such that a ∈ acl(F Ac).
, and therefore that AF (c, b) σ and A(c, a) σ are linearly disjoint over A(c, b) σ , whence a ∈ acl(Acb). As c is independent from a over A = acl(A), it follows that A(c, a) σ = A(c, a, b) σ is a regular extension of A(a) σ , and therefore that Cb(b, c/A(a) σ ) is contained in the difference field generated by finitely many realisations of tp(b, c/A(a) σ ) (see 2.6). Again, as c is independent from a over A and b is in F , it follows that if a ′ is such that Cb(c, b/A(a) σ ) = A(a ′ ) σ , then tp(a ′ /A) is qf-internal to F . As b ∈ A(a ′ , c) σ and c is independent from a over A, it follows that a ∈ acl(Aa ′ ) as desired. As A(c, a ′ ) σ = A(c, b) σ and b ∈ F , it follows that A(c, a ′ ) σ is finitely generated as a field extension of A(c) σ . But as a ′ and c are independent over A, the same holds of the field extension A(a ′ ) σ /A, i.e., for some n, σ n (a ′ ) ∈ A(a ′ , σ(a ′ ) . . . , σ n−1 (a ′ )). We then replace a ′ by (a ′ , σ(a ′ ) . . . , σ n−1 (a ′ )) 2.15. The semi-minimal analysis. Let a be a tuple which is transformally algebraic over K. Thus SU(a/K) < ω. As Th(U) is supersimple, there is a sequence a 1 , . . . , a n ∈ acl(Ka), such that a ∈ acl(Ka 1 , . . . , a n ), and for every 0 < i ≤ n, tp(a i /acl(K, a 1 , . . . , a i−1 )) is either one-based of rank 1, or almost-internal to a non-one-based type of rank 1. This is a classical result in supersimple theories, for a proof in our case in characteristic 0, see Theorem 5.5 in [5]. Note that in characteristic 0 by the above dichotomy 2.9, all non-one-based types of rank 1 are non-orthogonal to σ(x) = x, and by Lemma 2.14, almost-internality to F may be replaced by qf-internality to F . (1) We say that M is ℵ ε -saturated if whenever A ⊆ M is finite, then every strong 1-type over A is realised in M. Equivalently, as our theory eliminates imaginaries, if every 1-type over acl(A) is realised in M.
(2) Let κ be an infinite cardinal or ℵ ε , and A ⊆ M. We say that M is κ-prime over A, if M is κ-saturated, and A-embeds elementarily into every κ-saturated model of Th(M, a) a∈A . When κ = ℵ ε , one also says that M is a-prime.
A 0 is finite if we are dealing with ℵ ε -saturation, and has cardinality < κ if we are dealing with κ-saturation. We will also say that (4) Let κ be an infinite cardinal or ℵ ε , and A ⊆ M. A type p over A is κ-isolated if it is implied by its restriction to some small subset of acl(A).
Recall also that M is atomic over A if every finite tuple realises an isolated type over A.
with B κ-atomic over A and C κ-atomic over B, then C is κ-atomic over A. This is however not necessarily true when κ is singular. However, this will hold if B = acl(Ab) for some finite tuple b (since every finite tuple in B realises an isolated type over Ab), or if C is atomic over B. (There are stronger statements involving cardinals λ < cf(κ).) (2) If M is a κ-saturated model of T containing A and M is κ-constructed over A, then M is κ-prime over A.
(3) The property of being κ-constructed is preserved under towers and union of chains indexed by ordinals.

Algebraic translation of the model-theoretic notions
Let us translate what the notions of saturation mean in our case. We will be dealing with either uncountable cardinals or ℵ ε . Recall that tp(a/A) is entirely determined by the isomorphism type over the difference field generated by A of the difference field acl(Aa) (2.4). So, for κ an uncountable cardinal, the κ-saturation of a model M of ACFA simply means: every system of < κ difference equations with coefficients in M, which has a solution in a difference field extending M, already has a solution in M. This is what was called κ-closed in the introduction. The notion of κ-prime over a difference subfield corresponds to being a κ-closure of that difference field.
In the case of ℵ ε -saturation, the algebraic description is a little more complicated, and is better expressed in terms of embedding problems: work inside a large model U, and consider a submodel M of U. Then M is ℵ ε -saturated if whenever a is a finite tuple of elements of M and b an element of U, there is an acl(a)-embedding of acl(a, b) inside M.
A similar description holds for κ-saturated, with the base set a of cardinality < κ.

The results
Results of Hrushovski ([9]) show that if F is a pseudo-finite field and C ⊂ F , then Th(F, c) c∈C eliminates imaginaries if and only if the absolute Galois group of the relative algebraic closure inside F of the field generated by C is isomorphic toẐ. It may therefore happen that Th(F ) eliminates imaginaries in the ring language, but it may also happen that extra elements are needed, for instance if F contains Q alg . The following lemma will therefore be useful when dealing with ℵ ε -saturation.
Lemma 3.1. Let F be an ℵ ε -saturated pseudo-finite field and a a finite tuple in F . Then there is a finitely generated subfield A of F containing a and such that G(A alg ∩ F ) ≃Ẑ.
Proof. Let k be the relative algebraic closure inside F of the subfield generated by a, and consider k(t), where t is transcendental over k. Let Q 0 be the set consisting of all integers n which are either prime numbers or 4 and such that G(k) does not have a quotient isomorphic to Z/nZ. If char(k) = 0, we let Q = Q 0 \ {4}, and if char(k) = 0, we let Q = Q 0 \ {2}. If Q is empty, then G(k) ≃Ẑ, and we are done. So, we assume that Q is non-empty.
By Proposition 16.3.5 of [8], for each n, k(t) has a Galois extension L n which is regular over k and with Gal(L n /k(t)) = Z/nZ. Let L be the field composite of all L n , n ∈ Q. Then Gal(L/k(t)) ≃ n∈Q Z/nZ. Observe that L ∩ k alg = k, because all L n 's are regular extensions of k, and Galois over k(t) of relatively prime order. Take a topological generator σ 0 of Gal(L/k 0 ), and a topological generator σ 1 of G(k). Let σ ∈ G(k(t)) extend (σ 0 , σ 1 ) ∈ Gal(Lk alg /k(t)) (≃ Gal(L/k(t)) × G(k)); then the subfield A of k(t) alg fixed by σ is a regular extension of k, with Galois group isomorphic toẐ, since its Galois group is procyclic, projects onto G(k), onto all Z/pZ with p a prime, and onto Z/4Z if char(k) = 0. By general properties of pseudo-finite fields and by ℵ ε -saturation of F , there is a k-embedding ϕ of A inside F , in such a way that ϕ(A) alg ∩F = A. This is classical, and follows from instance from Lemma 20.2.2 in [8].
Lemma 3.2. Let κ be an uncountable cardinal or ℵ ε , let K be a difference field, with Fix(σ)(K) pseudo-finite and κ-saturated. Then there is a model U of ACFA containing K, which is κsaturated, and with Fix(σ)(U) = Fix(σ)(K).
Proof. (Compare with Afschordel's result [1]). Let U 1 be a κ-saturated model of ACFA containing K, and let U ⊆ U 1 be maximal such that F := Fix(σ)(U) = Fix(σ)(K). We will show that U satisfies our conclusion. First observe that U is algebraically closed. Let A = acl(A) ⊂ U be small, let p ∈ S 1 (A). Then p is realised in U 1 , and we take some a ∈ U 1 realising p, with SU(a/U) minimal. Let B ⊃ A be small and such that a | ⌣B U and replace p by tp(a/B). If tp(a/U) ⊥ a Fix(σ), then U(a) alg σ has the same fixed field as U: indeed, U(a) alg σ and Fix(σ)(U 1 ) are linearly disjoint over their intersection, which is contained in U and therefore in K. So by maximality of U, a ∈ U.
Assume now that tp(a/U) ⊥ a Fix(σ). Then there is some small C ⊂ U containing B, and a realisation a ′ of tp(a/B) such that C(a ′ ) σ ∩ Fix(U 1 ) contains some element b not in U. We may and will assume that Fix(σ)(C) has absolute Galois group isomorphic toẐ (by Lemma 3.1). But as F is κ-saturated, tp F (b/C ∩ F ) is realised in F , by some b 1 . Then b 1 realises tp(b/C) (see the first paragraph of 2.10). Thus, by κ-saturation of U 1 , there is some a 1 ∈ U 1 such that tp(a 1 , b 1 /C) = tp(a ′ , b/C). But then a 1 realises p, and SU(a 1 /U) ≤ SU(a ′ /B) − SU(b/C) < SU(a/B), which gives us the desired contradiction.
So in both cases, p is realised in U. Corollary 3.3. Let κ be as above, and K an algebraically closed difference field with F ix(σ)(K) κ-saturated. If U is a κ-prime model of ACFA over K then Fix(σ)(U) = Fix(σ)(K).   Proof. Let k ⊂ K be small such that a | ⌣k K and Gal(Fix(σ)(k) alg /Fix(σ)(k)) is isomorphic tô Z. Then σ(a) ∈ k(a) and kF contains Fix(σ ℓ )(U) for all ℓ ≥ 1. By assumption, there is some small B (in U, by ℵ ε -saturation of U) independent from a over k, such that a ∈ BF . Hence, there is a tuple c in B(a) σ ∩ F = B(a) ∩ F such that B(a) = B(c). Let D = Cb(a, c/B). Then D(c) = D(a), and D ⊂ k(c 1 , a 1 , . . . , c n , a n ) for some independent realisations (c i , a i ) of qf tp(c, a/B) (in some elementary extension of U). By ℵ ε -saturation of U, we may assume that (c 1 , a 1 , . . . , c n , a n ) is in U, and is independent from (c, a) over D. We let b = (a 1 , . . . , a n ), A = Cb(k, c 1 , a 1 , . . . , c n , a n /K); then D ⊂ kF (b), and A is small. As A contains c 1 , . . . , c n (∈ F ⊂ K) and k, we also have D ⊂ A(b), whence a ∈ F A(b). Note that a | ⌣k A since A ⊂ K. If a ′ ∈ U realises qf tp(a/A(b)), then the difference fields D(a) and D(a ′ ) are isomorphic; hence there is some Let a ′ be an arbitrary realisation of qf tp(a/A), and let b ′ be a realisation of qf tp(b/A), which is independent from (b, a ′ ) over A. By the previous paragraph (as b ′ consists of n realisations of qf tp(a/A(b))) we know that b ′ ∈ F A(b). The difference fields A(b) and A(b ′ ) are A-isomorphic, and this isomorphism extends to an isomorphism of difference fields A(b, a) → A(b ′ , a ′ ). Hence, a ′ ∈ F A(b ′ ) ⊆ F A(b), as desired. If a ′ realises qf tp(a/A)[ℓ] and is independent from D over k, then the σ ℓ -difference fields D(a ′ ) and D(a) are isomorphic over D. Let f (x) be the tuple of rational functions over D such that f (a) = c; then σ ℓ (f (a ′ )) = f (a ′ ) and D(a ′ ) = D(f (a ′ )). Hence a ′ belongs to F A(b). An argument similar to the one given in the first case shows it for arbitrary realisation of qf tp(a/A)[ℓ] and shows (1).
Note that we have in fact shown that F A(b ′ ) = F A(b), and so the conclusion of (1) also holds for b ′ . An easy argument allows to remove the assumption that b ′ is independent from b over A: let b ′′ realise qf tp(b/A), independent from (b, b ′ ) over A; then by the proof of the first part: , and noting that if m divides ℓ, then the realisations of qf tp(a/A)[m] also realise qf tp(a/A)[ℓ], part (1) gives (2).
For the proof of (3), we will first show that every realisation b ′ of qf tp(b/A)[ℓ] (in U) is independent from K over A. Indeed, by (2), we know that F A(b) = F A(b ′ ), and therefore F K(b) = F K(b ′ ). This implies that tr.deg(b/K) = tr.deg(b ′ /K), and therefore that b ′ | ⌣A K. As U is ℵ ε -saturated, this shows that if d ∈ K, then By Proposition 4.9 of [5], if tp(b/A) ⊢ tp(b/K), then there would be some tuple d ∈ K and integer ℓ ≥ 1 such that tp(d/A)[ℓ] ⊥ a tp(b/A) [ℓ]. But as we just saw, this is impossible, and this gives us (3). (This is where the characteristic 0 assumption is crucial).
Lemma 3.6. Let K, A, b, U be as in Lemma 3.4, and let L be a difference subfield of U containing K. Then there is a small A ′ containing A such that tp(b/A ′ ) ⊢ tp(b/L). In particular, tp(a/A ′ ) ⊢ tp(a/L).
Proof. Let A ′ ⊂ L be small, containing A and such that b | ⌣A ′ L. Then the proof of (3) works.
Corollary 3.7. Let K and U be as in Lemma 3.4, and p be a type which is almost internal to Fix(σ). Then any K-indiscernible sequence (a i ) of realisations of p in U is finite.
Proof. Let (a i ) i<ω be a sequence of realisations of p in U which is K-indiscernible. Then either a 0 ∈ K, or tp(a 0 /K) is almost orthogonal to Fix(σ) (since K contains F := Fix(σ)(U)). By Lemma 2.14 there is a ′ 0 ∈ K(a 0 ) σ such that σ(a ′ 0 ) ∈ K(a ′ 0 ), a 0 ∈ K(a ′ 0 ) alg and tp(a ′ 0 /K) is qf-internal to Fix(σ). It suffices to show the result for p = tp(a ′ 0 /K). Let b be the finite tuple of realisations of tp(a ′ 0 /K) given by Lemma 3.4. If n > d = tr.deg(K(b)/K) and tp(a ′ i , a i /K) = tp(a ′ 0 , a 0 /K), then we know that a ′ n ∈ K(a ′ 0 , . . . , a ′ d−1 ) alg (because K ⊃ F ). Hence the sequence is finite. Definition 3.8. We call a type p over a set A acceptable (in K ⊃ A) if A is the algebraic closure of a finite tuple, and either SU(p) = 1 and p is one-based, or p is qf-internal to Fix(σ), almost orthogonal to Fix(σ), and if b realises p then σ(b) ∈ A(b), tp(b/A) ⊢ tp(b/K), p ⊥ a Fix(σ), and the set of realisations of qf tp(b/A)[ℓ] for ℓ ≥ 1, in some model U of ACFA containing K, is contained in A(b)Fix(σ)(U). Notation 3.9. Let p be a one-based type of SU-rank 1 over the very small set A. If A ⊂ B ⊂ C, we denote by p|B the unique non-forking extension of p to B, and by dim B p(C) the cardinality of a maximal B-independent subset of realisations of p|B in C. Lemma 3.10. Let p be an acceptable one-based type over the very small A, let K be an algebraically closed difference field containing A. We work in a sufficiently saturated model U of ACFA. Let κ be an infinite cardinal.
(1) If K contains κ many A-independent realisations of p, then the non-forking extension of p to K is not κ-isolated, and conversely.
(b) Not case (a). If B is a set of K-independent realisations of p|K of size λ < κ, then dim A p(acl(KB)) < κ. Proof.
(1) If C = acl(C) ⊂ K is small, then C contains < κ A-independent realisations of p, so that the non-forking extension of p to C will be realised in K, and p|K is not κ-isolated. The converse is clear: the non-forking extension of p to K is implied by its restriction to acl(A, p(K)).
(2) (a) is clear by (1) and because dim is additive. So, assume that there is no such n, and let B be as in (b), and (a i ) i<λ ⊂ B a sequence of independent over K realisations of p, and assume that λ < dim A p(acl(KB)) = µ ≥ κ. So acl(K, a i | i < λ) contains a set C consisting of µ many A-independent realisations of p. Then for each c ∈ C, there is some finite I c ⊂ λ such that c ∈ acl(Ka i | i ∈ I c ). As λ < µ, some set I c appears µ times. Thus dim A p(acl(Ka i | i ∈ I c )) = µ ≥ κ, which contradicts our assumption.
Remark 3.11. Let p be the generic 1-type over K, and κ an infinite cardinal. Then p is κ-isolated if and only if ∆(K) < κ. This follows easily from the description and properties of the generic types, see 2.7.
Definition 3.12. Let K = acl(K) ⊂ L = acl(L) ⊂ U. We say that L is normal over K (in U) is whenever a is a tuple in L, then L contains all realisations of tp(a/K) in U.
Lemma 3.13. Let κ be an uncountable cardinal or ℵ ε , let K ⊆ L be algebraically closed difference subfields of U, where U is κ-saturated, and Fix(σ)(U) ⊂ K. Assume that U is κ-atomic over K.
(1) Let B ⊂ U be transformally independent over K, and assume that either |B| < κ, or that B is a tranformal transcendence basis of U over K. Then U is κ-atomic over acl(KB).
(2) If L is normal over K then U is κ-atomic over L. Proof.
(1) If B ⊂ K there is nothing to prove, so suppose it is not. Then ∆(K) < κ by 3.11. Let a be a finite tuple in U, and let b ⊂ B be a finite tuple such that a | ⌣Kb B. Let c ⊂ a be a tranformal transcendence basis of K(a, b) σ over K(b) σ (and therefore also over K(B) σ ).
But tp(a, b, c/K) is κ-isolated, and this gives the result.
(2) Let a be a finite tuple in U, and consider tp(a/L). Let d ⊂ a be maximal transformally independent over L. If d = ∅, then d is transformally independent over K, which implies that ∆(L/K) = 0 (by normality of L/K), and that ∆(K) = ∆(L) < κ (by κ-isolation of tp(d/K)). Therefore tp(d/L) is κ-isolated. If ∆(L/K) = 0, note that by normality of L over K in U, every element of the tuple a which is not in L is transformally algebraic over K. So, replacing a by a \ L, we may assume they are all transformally algebraic over K, i.e., that SU(a/K) < ω. We then let d = ∅.
In both cases, by (1), U is κ-atomic over acl(Kd), and the normality of L over K implies the normality of acl(Ld) over acl(Kd). Working over acl(Kd), we may therefore assume that a and D := Cb(a/L) are transformally algebraic over K. We use induction on SU(a/L), and using the semi-minimal analysis, we find b ∈ acl(Da) such that tp(a/acl(Db)) is either one-based of SU-rank 1, or almost-internal to Fix(σ). If tp(a/acl(Db)) is almost-internal to Fix(σ), then so is tp(a/acl(Lb)). By Lemma 2.14, there is a ′ ∈ acl(Lba) such that a ∈ acl(Lba ′ ) and tp(a ′ /acl(Lb)) is qf-internal to Fix(σ). By Lemma 3.4, there is a very small D ′ ⊇ D such that tp(a ′ /acl(D ′ b)) ⊢ tp(a ′ /acl(Lb)), and we may choose it so that a ∈ acl(D ′ ba ′ ). This shows that tp(a/acl(Lb)) is κ-isolated, and therefore so is tp(a/L). So assume that p := tp(a/acl(Db)) is one-based of SU-rank 1, and let c be a tuple containing b such that acl(Db) = acl(c) =: C. We need to show that dim C p(acl(Lc)) < κ. As U is κ-atomic over K, we know that tp(a, c/K) is κ-isolated, and therefore dim C p(acl(Kc)) < κ. So, if dim C p(acl(Lc)) ≥ κ, then there is some a ′ ∈ acl(Lc) \ acl(Kc) realising p. Recall that by our earlier step, c, a ′ are transformally algebraic over K, and therefore so is e = Cb(c, a ′ /L). Consider now acl(Kca ′ )∩acl(Ke) =: E ⊂ L; by Proposition 3.1 of [2], tp(e/E) is almost-internal to Fix(σ), and therefore orthogonal to all one-based types. As tp(a ′ /Kc) is one-based, and a ′ ∈ acl(Kce) \ acl(Kc), it follows that e ∈ E, since internality to Fix(σ) and non-orthogonality to a one-based type imply algebraicity. Thas is, e ∈ acl(Kca ′ ) ∩ L, and as a ′ / ∈ acl(Kc), the tuples a ′ and e are equi-algebraic over acl(Kc). Hence acl(Kca) contains a realisation of tp(e/acl(Kc)), because tp(a/acl(Kc)) = tp(a ′ /acl(Kc)). But this contradicts the normality of acl(Lc) over acl(Kc). So, dim C (p(acl(Lc)) < κ, and tp(a/Lb) is κ-isolated.
Theorem 3.14. Let κ be an uncountable cardinal or ℵ ε , and let K be an algebraically closed difference field of characteristic 0 such that F := Fix(σ)(K) is pseudo-finite and is κ-saturated.
(1) Then there is a κ-prime model U over K.
We now define U = n∈ω K n .
Step 2. Show that U is κ-saturated. Let C ⊂ U be small, and p a 1-type over C, realised by a in U 1 . If SU(p) = ω, then a is transformally transcendental over C; as C is small, K 0 will contain a realisation of p. So we may assume that SU(p) < ω, and the proof is by induction on SU(p): we assume that for any small D, any 1-type q over D of smaller SU-rank than p is realised in U. If SU(p) = 0 there is nothing to prove, as p is realised in C. If there is some b ∈ C(a) alg σ such that 0 < SU(b/C) < SU(a/C), then we get the result by induction: tp(b/C) is realised by some b ′ ∈ U, and there is a ′ ∈ U such that tp(a ′ , b ′ /C) = tp(a, b/C), since acl(Cb ′ ) is small and SU(a/Cb) < SU(p).
Hence we may assume that there is no such b, whence p is either one-based of SU-rank 1, or almost-internal to Fix(σ) (by the semi-minimal analysis 2.15). We need to distinguish three cases. Case 1. p is one-based of SU-rank 1.
Let A ⊂ C be very small such that p does not fork over A. Let n < ω be such that A ⊂ K n ; then p, being acceptable, occurs as a p β , and is therefore realised in K n+1 .
Case 2. p is realised in Fix(σ). If a ∈ Fix(σ), we saw in 2.10 that tp F (a/C ∩ F ) ⊢ tp(a/C). The saturation hypothesis on F then gives the result: p is realised in F . Case 3. Assume now that p ⊥ a Fix(σ), p almost-internal to Fix(σ). By Lemma 2.14, there is a 1 ∈ C(a) σ such that tp(a 1 /C) is qf-internal to Fix(σ), σ(a 1 ) ∈ C(a 1 ), and a ∈ C(a 1 ) alg . We may replace p by tp(a 1 /C), i.e., assume that p is qf-internal to Fix(σ). Let C 0 ⊂ C be very small such that p does not fork over C 0 . By Lemma 3.4 there is a tuple b of realisations of p and a very small D containing C 0 , contained in acl(CF ), such that F D(b) contains all realisations of qf tp(a/D), and tp(b/D) ⊢ tp(b/acl(CF )). Thus, tp(b/D) is acceptable, and if n is such that D ⊂ K n , then p in realised in K n+1 .
Step 4. U is κ-atomic over K. When κ is regular or ℵ ε , then this folllows from U being κ-constructed over K. The proof in the singular case is a little more delicate, and is done by induction. We already saw that K 0 is κ-atomic over K. Let a be a finite tuple in U, and (in the notation of Step 1), choose n minimal such that a ∈ K n+1 , and β minimal such that a ∈ K ′ β+1 . If n = −1, there is nothing to prove (by Step 0), so assume n ≥ 0. By definition of K ′ β+1 , there are a tuple b in K ′ β and a tuple c of realisations of p β such that a ∈ acl(Kbc). We may assume that acl(Kb) contains A β , and that c | ⌣Kb K ′ β . By induction hypothesis, tp(b/K) is κ-isolated, and it therefore suffices to show that tp(c/acl(Kb)) is κ-isolated (by 2.17 (1)). If p β is qf-internal to Fix(σ) then we know by Lemma 3.4 that there is some very small D ⊂ acl(Kb) such that tp(c/D) ⊢ tp(c/acl(Kb)), and we are done. If p β is one-based, then we may assume that the elements of the tuple c are independent over K ′ β , maybe at the cost of increasing b ∈ K ′ β . Then, by the construction of K ′ β+1 in Step 1, we know that tp(c/K ′ β ) is κ-isolated, so that if c ′ is a proper subtuple of c (consisting of realisations of p β ), then dim A β p β (acl(K ′ β c ′ )) < κ. In particular, dim A β p(acl(Kbc ′ )) < κ, and tp(c/acl(Kb)) is κ-isolated (by Lemma 3.10).
Remarks. (Notation as in step 1 and above) The same proof shows that U is κ-atomic over each K n , and over each K ′ β . Moreover, the fact that U is κ-atomic over By supersimplicity, for some n < ω the elements b i , n < i < λ, are independent over K(b 0 , . . . , b n ) σ . If SU(b n+1 /Kb 0 , . . . , b n ) ≥ ω, then the tuple b n+1 contains an element which is transformally transcendental over K, and as the transformal transcendence degree of U over K is ≤ κ, we get λ ≤ κ. So we may assume SU(b n+1 /acl(Lb 0 . . . , b n )) < ω. Let L = acl(Kb 0 , . . . , b n ). Then the sequence (b i ) n<i<λ is indiscernible over L. Note that the sequence acl(Lb i ), n < i < λ, is also indiscernible over L under a suitable enumeration of each acl(Lb i ). Hence, if c n+1 ∈ acl(Lb n+1 ), there are c i ∈ acl(Lb i ), n + 1 < i < λ, such that the sequence (c i ) n<i<λ is indiscernible over L. Using the semi-minimal analysis 2.15 we may therefore assume that either tp(c i /L) is one-based of SU-rank 1, or that tp(c i /L) is almost-internal to Fix(σ). If tp(c i /L) is almost-internal to Fix(σ), then the result follows by Corollary 3.7. The one-based case is a little more complicated.
Towards a contradiction, assume that λ > κ, that tp(c n+1 /L) is one-based of SU-rank 1, and let C ⊂ L be a very small set such that tp(c n+1 /L) does not fork over C, and set p = tp(c n+1 /C). Then the tuples c i , n < i < λ, form a Morley sequence over C and over L. Let N be κ-prime over M := acl(L, c i | n < i < κ). We may assume that N ≺ U.
Claim. U is κ-prime over L. It suffices to show that U is κ-constructed over L. To do that it is enough to show that each LK m is κ-constructed over LK m−1 . If m = 0 and K 0 = K, let B 0 be a finite subset of B (the transformal transcendence basis of U over K) such that b := (b 0 , . . . , b n ) is independent from K 0 over acl(KB 0 ). In particular, b is transformally algebraic over acl(KB 0 ), and therefore tp(B/acl(KB 0 )) ⊢ tp(B/acl(LB 0 )) (reason as in the proof of Lemma 3.13(1)), and as tp(B 0 /L) is κ-isolated, it follows that K 0 is κ-constructed over L.
Hence, U being κ-prime over L, there is an L-embedding f of U into N. So we have L ⊂ f (U) ≺ N ≺ U. As λ > κ and the c i 's are independent over L, there is some n < j < λ such that f (c j ) / ∈ M. But dim M (p) ≥ κ, and by Lemma 3.10, p|M is not isolated. But N is κ-atomic over M, and f (c j ) realises p and is not in M, which gives us the desired contradiction. This finishes the proof of (2) and of the theorem.
Proposition 3.15. Let κ be an uncountable cardinal or ℵ ε , let U and U ′ be κ-saturated models of ACFA of characteristic 0. Assume that U (resp. U ′ ) contains an algebraically closed difference field K (resp. K ′ ), over which it is κ-atomic and over which every sequence of indiscernibles has length ≤ κ. Assume moreover that F := Fix(σ)(K) = Fix(σ)(U), Fix(σ)(K ′ ) = Fix(σ)(U ′ ), and that we have an isomorphism f : K → K ′ . Let p be an acceptable type over some very small A ⊂ K, and p ′ = f (p). If L = acl(Kp(U)) and L ′ = acl(Kp ′ (U ′ )), then f extends to an isomorphism between L and L ′ .
Proof. Note that p ′ is also acceptable, with very small basis A ′ = f (A). If p is not one-based, then this is clear by Lemma 3.4: L = acl(Kb), L ′ = acl(K ′ b ′ ) for some tuples b realising p and b ′ realising p ′ . We extend f |A to an isomorphism g 0 : acl(Ab) → acl(A ′ b ′ ) which sends b to b ′ ; as tp(b/A) ⊢ tp(b/K) and tp(b ′ /A ′ ) ⊢ tp(b ′ /K ′ ), g 0 ∪ f extends to an isomorphism acl(Kb) → acl(K ′ b ′ ). Assume now that p is one-based. Any κ-saturated model of ACFA containing A will contain (at least) κ realisations of p which are independent over A; hence so do U and U ′ . Let (a i ) i<λ ⊂ U be a set of realisations of p which is maximal independent over K, with λ a cardinal, and let (a ′ i ) i<µ ⊂ U ′ be defined analogously over K ′ . By Lemma 3.10 and our hypothesis on the length of K-indiscernible sequences, either λ is finite, or λ = κ. If λ = n < ω, then as tp(a ′ 0 , . . . , a ′ n−1 /K ′ ) = f (tp(a 0 , . . . , a n−1 /K)), it follows that acl(K ′ a ′ 0 , . . . , a ′ n−1 ) contains κmany independent realisations of f (p), so that µ ≤ n. The symmetric argument gives µ = λ. Define g on K(a i | i < λ) σ by g(a i ) = a ′ i , and extend to L = acl(Ka i | i < λ).
Theorem 3.16. Let κ be an uncountable cardinal or ℵ ε , let U and U ′ be κ-saturated models of ACFA of characteristic 0 containing an algebraically closed difference field K, with F := Fix(σ)(K) = Fix(σ)(U) = Fix(σ)(U ′ ). Assume that U and U ′ are κ-atomic over K, and that any sequence of K-indiscernibles in U or in U ′ has length ≤ κ. Then U ≃ K U ′ .
Proof. We start with the generic type: if the transformal transcendence degree of K is ≥ κ, then U and U ′ are transformally algebraic over K. If not, then let D be a transformal transcendence basis of U over K, D ′ a transformal transcendence basis of U ′ over K. They have the same cardinality κ, and there is a K-isomorphism K(D) alg σ → K(D ′ ) alg σ . By Lemma 3.13, U and U ′ still satisfy the hypotheses over K(D) alg σ and K(D ′ ) alg σ . Hence we may assume that both U and U ′ are transformally algebraic over K. We define by induction on n an increasing sequence K n of algebraically closed subfields of U such that for each n, if p is an acceptable type over some (very small) A ⊂ K n−1 , then K n contains all realisations of p in U, and furthermore, K n = acl(K n−1 P ) for the set P of all realisations (in U) of acceptable types over some subset of K n−1 . Then each K n is normal over K n−1 (and in fact over K), and so by Lemma 3.13, U satisfies the hypotheses over K n . Note also that U = n<ω K n . We let L n ⊂ U ′ be defined analogously. It then suffices to build a sequence g n of K-isomorphisms K n → L n . Assume g n−1 already built. Let p β , β < λ, be an enumeration of all acceptable types over a subset of K n−1 , with associated small basis A β . Note that f (p β ), β < λ, enumerates all acceptable types over subsets of L n−1 , since if q is an acceptable type over the very small C ⊂ L n−1 , so is g −1 n−1 (q) (over g −1 n−1 (C) ⊂ K n−1 ). We build by induction on β < λ an increasing sequence K ′ β of algebraically closed difference subfields of U such that K ′ β contains all realisations in U of p γ for all γ < β. Assume we have extended g n−1 to an isomorphism f β : K ′ β → L ′ β , where L ′ β contains all realisations in U ′ of g n−1 (p γ ) for all γ < β. As U is κ-atomic over K n−1 , it is also κ-atomic over K ′ β (by Lemma 3.13), and similarly, U ′ is κ-atomic over L ′ β = f β (K ′ β ). Extending f β to an isomorphism f β+1 : K ′ β+1 → L ′ β+1 is given by Lemma 3.15. As remarked before, if q is an acceptable type over some A ′ ⊂ L ′ n−1 , then g −1 n−1 (q) = p β for some β < λ, and so L ′ n contains q(U ′ ), and K ′ n contains g −1 n−1 (q)(U). This finishes the induction step. Then g = n<ω g n is a K-isomorphism between U and U ′ . Theorem 3.17. Let κ be an uncountable cardinal or ℵ ε , let K be an algebraically closed difference field of characteristic 0, with Fix(σ)(K) pseudo-finite and κ-saturated. Then ACFA has a κ-prime model over K, and it is unique up to K-isomorphism.
Proof. This follows immediately from Theorem 3.16, as the properties are preserved by elementary substructures.