##
**Model theory of difference fields.**
*(English)*
Zbl 0922.03054

A difference field is a field \(K\) with a distinguished automorphism \(\sigma\). For instance, given a prime \(p\) and a power \(q\) of \(p\), consider \(K = {\mathbb F}_p^{\text{alg}}\) = the algebraic closure of the finite field \({\mathbb F}_p\) with \(p\) elements, and \(\sigma = \sigma_q : x \mapsto x^q\) a power of the Frobenius morphism \(x \mapsto x^p\). In the sequel, let \(K_q\) denote \(({\mathbb F}_p^{\text{alg}}, \sigma_q)\).

The interest in the model theory of difference fields goes back to the end of the eighties. It was awakened, for instance, by the hope of axiomatizing the theory of non-principal ultraproducts of the difference fields \(K_q\) (generalizing the work of Ax on finite fields). Another motivation is related to the failure of Zil’ber’s Conjecture, and to the consequent hope to find some new examples of strongly minimal structures by looking at a possible non-definable automorphism \(\sigma\) of \({\mathbb F}_p^{\text{alg}}\) and at the corresponding pair \(({\mathbb F}_p^{\text{alg}}, \sigma)\). By the way, the latter question still seems to be open; we will come back to the former at the end of this review.

So, almost ten years ago, Macintyre, van den Dries and Wood gave a first-order characterization of the model companion of the theory of difference fields. According to this axiomatization, the existentially closed difference fields \((K, \sigma)\) are exactly those satisfying the following conditions:

i) \(K\) is an algebraically closed field;

ii) \(\sigma\) is an automorphism of \(K\);

iii) if \(U\) and \(V\) are varieties defined over \(K\) and \(V \subseteq U \times \sigma (U)\) projects generically onto \(U\) and \(\sigma (U)\), then there is some tuple \(a\) in \(K\) such that \((a, \sigma(a))\) is in \(V\).

The corresponding theory is usually denoted ACFA. Macintyre, van den Dries and Wood showed several (unpublished) results about ACFA (decidability, a classification of the possible completions, and so on). It was also observed that, if \((K, \sigma)\) is a model of ACFA, then the subfield of \(K\) fixed by \(\sigma\), \(\text{Fix} (\sigma)\), is pseudofinite; as no pseudofinite field is stable, it follows that existentially closed difference fields fall into the class of unstable structures.

The present paper continues and deepens the study of ACFA in the light of Shelah’s stability theory and Zil’ber’s geometric stability theory, in addition to A. Robinson’s model companion theory, towards a description of the geometry of varieties defined by difference equations.

After a short but attractive historical account of the involved model theory (stability, dimension, one-dimensional structures, Zil’ber trichotomy), the authors present the basic analysis of existentially closed difference fields, by recalling the old results established by Macintyre, van den Dries and Wood. Then they propose an algebraic notion of independence for subsets \(A, B \supseteq C\) of a model \((K, \sigma)\) of ACFA: \(A\) and \(B\) are said to be independent over \(C\) if the algebraic closures of the difference fields generated by \(A\) and \(B\) are linearly disjoint over the algebraic closure of the difference field generated by \(C\). It is shown (Independence Theorem) that this notion of independence just coincides with nonforking, and that every model of ACFA is simple (although unstable, as said before). The Independence Theorem is also used to show that every existentially closed difference field \((K, \sigma)\) eliminates the imaginaries and its fixed subfield \(\text{Fix} (\sigma)\) is stably embedded (in other words, every definable subset of \(\text{Fix}(\sigma)^m\) can be defined with parameters in \(\text{Fix}(\sigma)\); moreover, it is definable in the pure language of fields). Elimination of imaginaries in \(\text{Fix}(\sigma)\) is also discussed.

Independence allows to define a rank notion for types, called \(SU\)-rank and sharing some significant properties with the usual Lascar \(U\)-rank. When \((K, \sigma)\) is a (large) model of ACFA, \(a\) is in \(K\) and \(E\) is a difference subfield of \(K\), then the \(SU\)-rank of the type of \(a\) over \(E\) is bounded by the transcendence degree of the difference field generated by \(a\) over \(E\). When this transcendence degree is infinite, i.e. \(a\) is transformally transcendental over \(E\), then the \(SU\)-rank is \(\omega\). On the other hand, it is shown that there are enough \(SU\)-rank 1 sets to control arbitrary types. For instance, in characteristic \(0\), if \(E\) is an elementary substructure of \((K, \sigma)\) and \(a \not \in E\) has a type of finite \(SU\)-rank over \(E\), then there is some \(b\) in the algebraic closure of the difference field generated by \(a\) over \(E\) such that the \(SU\)-rank of \(b\) over \(E\) is just \(1\).

So the authors concentrate their attention on the \(SU\)-rank 1 types. The main result here is a Trichotomy Theorem à la Zil’ber for these types in characteristic \(0\): For a type \(p\) of \(SU\)-rank 1 over a difference subfield \(E\) of \((K, \sigma)\), the (pre)geometry of realizations of \(p\) either is locally modular (and moreover is stable and stably embedded), or interprets and is interpreted in a pseudofinite field – actually the fixed subfield \(\text{Fix}(\sigma)\). The authors also announce a similar Trichotomy Theorem in prime characteristic, proved by the authors and Y. Peterzil in a subsequent paper; notice that, in arbitrary characteristic, various pseudofinite fields can arise in the latter case. The significance of the Trichotomy Theorem with respect to difference equations is also discussed.

Finally, the trichotomy result is applied to the description of types of finite \(SU\)-rank, as well as of groups definable in existentially closed difference fields (extending the work of Hrushovski and Pillay on groups definable in pseudofinite fields).

With respect to the question quoted at the beginning of this review and originating the interest in model theory of difference fields, it is conjectured that ACFA is just the theory of non-principal ultraproducts of the difference fields \(K_q\). A positive solution, due to Hrushovski and, independently, to Macintyre, is announced.

The interest in the model theory of difference fields goes back to the end of the eighties. It was awakened, for instance, by the hope of axiomatizing the theory of non-principal ultraproducts of the difference fields \(K_q\) (generalizing the work of Ax on finite fields). Another motivation is related to the failure of Zil’ber’s Conjecture, and to the consequent hope to find some new examples of strongly minimal structures by looking at a possible non-definable automorphism \(\sigma\) of \({\mathbb F}_p^{\text{alg}}\) and at the corresponding pair \(({\mathbb F}_p^{\text{alg}}, \sigma)\). By the way, the latter question still seems to be open; we will come back to the former at the end of this review.

So, almost ten years ago, Macintyre, van den Dries and Wood gave a first-order characterization of the model companion of the theory of difference fields. According to this axiomatization, the existentially closed difference fields \((K, \sigma)\) are exactly those satisfying the following conditions:

i) \(K\) is an algebraically closed field;

ii) \(\sigma\) is an automorphism of \(K\);

iii) if \(U\) and \(V\) are varieties defined over \(K\) and \(V \subseteq U \times \sigma (U)\) projects generically onto \(U\) and \(\sigma (U)\), then there is some tuple \(a\) in \(K\) such that \((a, \sigma(a))\) is in \(V\).

The corresponding theory is usually denoted ACFA. Macintyre, van den Dries and Wood showed several (unpublished) results about ACFA (decidability, a classification of the possible completions, and so on). It was also observed that, if \((K, \sigma)\) is a model of ACFA, then the subfield of \(K\) fixed by \(\sigma\), \(\text{Fix} (\sigma)\), is pseudofinite; as no pseudofinite field is stable, it follows that existentially closed difference fields fall into the class of unstable structures.

The present paper continues and deepens the study of ACFA in the light of Shelah’s stability theory and Zil’ber’s geometric stability theory, in addition to A. Robinson’s model companion theory, towards a description of the geometry of varieties defined by difference equations.

After a short but attractive historical account of the involved model theory (stability, dimension, one-dimensional structures, Zil’ber trichotomy), the authors present the basic analysis of existentially closed difference fields, by recalling the old results established by Macintyre, van den Dries and Wood. Then they propose an algebraic notion of independence for subsets \(A, B \supseteq C\) of a model \((K, \sigma)\) of ACFA: \(A\) and \(B\) are said to be independent over \(C\) if the algebraic closures of the difference fields generated by \(A\) and \(B\) are linearly disjoint over the algebraic closure of the difference field generated by \(C\). It is shown (Independence Theorem) that this notion of independence just coincides with nonforking, and that every model of ACFA is simple (although unstable, as said before). The Independence Theorem is also used to show that every existentially closed difference field \((K, \sigma)\) eliminates the imaginaries and its fixed subfield \(\text{Fix} (\sigma)\) is stably embedded (in other words, every definable subset of \(\text{Fix}(\sigma)^m\) can be defined with parameters in \(\text{Fix}(\sigma)\); moreover, it is definable in the pure language of fields). Elimination of imaginaries in \(\text{Fix}(\sigma)\) is also discussed.

Independence allows to define a rank notion for types, called \(SU\)-rank and sharing some significant properties with the usual Lascar \(U\)-rank. When \((K, \sigma)\) is a (large) model of ACFA, \(a\) is in \(K\) and \(E\) is a difference subfield of \(K\), then the \(SU\)-rank of the type of \(a\) over \(E\) is bounded by the transcendence degree of the difference field generated by \(a\) over \(E\). When this transcendence degree is infinite, i.e. \(a\) is transformally transcendental over \(E\), then the \(SU\)-rank is \(\omega\). On the other hand, it is shown that there are enough \(SU\)-rank 1 sets to control arbitrary types. For instance, in characteristic \(0\), if \(E\) is an elementary substructure of \((K, \sigma)\) and \(a \not \in E\) has a type of finite \(SU\)-rank over \(E\), then there is some \(b\) in the algebraic closure of the difference field generated by \(a\) over \(E\) such that the \(SU\)-rank of \(b\) over \(E\) is just \(1\).

So the authors concentrate their attention on the \(SU\)-rank 1 types. The main result here is a Trichotomy Theorem à la Zil’ber for these types in characteristic \(0\): For a type \(p\) of \(SU\)-rank 1 over a difference subfield \(E\) of \((K, \sigma)\), the (pre)geometry of realizations of \(p\) either is locally modular (and moreover is stable and stably embedded), or interprets and is interpreted in a pseudofinite field – actually the fixed subfield \(\text{Fix}(\sigma)\). The authors also announce a similar Trichotomy Theorem in prime characteristic, proved by the authors and Y. Peterzil in a subsequent paper; notice that, in arbitrary characteristic, various pseudofinite fields can arise in the latter case. The significance of the Trichotomy Theorem with respect to difference equations is also discussed.

Finally, the trichotomy result is applied to the description of types of finite \(SU\)-rank, as well as of groups definable in existentially closed difference fields (extending the work of Hrushovski and Pillay on groups definable in pseudofinite fields).

With respect to the question quoted at the beginning of this review and originating the interest in model theory of difference fields, it is conjectured that ACFA is just the theory of non-principal ultraproducts of the difference fields \(K_q\). A positive solution, due to Hrushovski and, independently, to Macintyre, is announced.

Reviewer: C.Toffalori (Camerino)

### MSC:

03C60 | Model-theoretic algebra |

12L12 | Model theory of fields |

08A35 | Automorphisms and endomorphisms of algebraic structures |

12H10 | Difference algebra |

03C45 | Classification theory, stability, and related concepts in model theory |

### Keywords:

difference field; simple theory; Zil’ber trichotomy; existentially closed fields; geometric stability theory; geometry of varieties defined by difference equations; independence; nonforking; elimination of imaginaries; rank notion for types; \(SU\)-rank 1 types; pseudofinite field; types of finite \(SU\)-rank
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\textit{Z. Chatzidakis} and \textit{E. Hrushovski}, Trans. Am. Math. Soc. 351, No. 8, 2997--3071 (1999; Zbl 0922.03054)

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