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 Macintyre, van den Dries and Wood showed several (unpublished) results about ACFA (decidability, a classification of the possible completions, and so on).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.


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
Full Text: DOI


[1] James Ax, The elementary theory of finite fields, Ann. of Math. (2) 88 (1968), 239 – 271. · Zbl 0195.05701
[2] Steven Buechler, Locally modular theories of finite rank, Ann. Pure Appl. Logic 30 (1986), no. 1, 83 – 94. Stability in model theory (Trento, 1984). · Zbl 0627.03016
[3] Zoé Chatzidakis, Lou van den Dries, and Angus Macintyre, Definable sets over finite fields, J. Reine Angew. Math. 427 (1992), 107 – 135. · Zbl 0759.11045
[4] G. Cherlin, E. Hrushovski, Large finite structures with few \(4\)-types, preprint 1998 (earlier version: Smoothly approximable structures, 1994).
[5] Richard M. Cohn, Difference algebra, Interscience Publishers John Wiley & Sons, New York-London-Sydeny, 1965. · Zbl 0127.26402
[6] Lou van den Dries, Dimension of definable sets, algebraic boundedness and Henselian fields, Ann. Pure Appl. Logic 45 (1989), no. 2, 189 – 209. Stability in model theory, II (Trento, 1987). · Zbl 0704.03017
[7] L. van den Dries and K. Schmidt, Bounds in the theory of polynomial rings over fields. A nonstandard approach, Invent. Math. 76 (1984), no. 1, 77 – 91. · Zbl 0539.13011
[8] Jean-Louis Duret, Les corps faiblement algébriquement clos non séparablement clos ont la propriété d’indépendence, Model theory of algebra and arithmetic (Proc. Conf., Karpacz, 1979), Lecture Notes in Math., vol. 834, Springer, Berlin-New York, 1980, pp. 136 – 162 (French).
[9] David M. Evans and Ehud Hrushovski, On the automorphism groups of finite covers, Ann. Pure Appl. Logic 62 (1993), no. 2, 83 – 112. Stability in model theory, III (Trento, 1991). · Zbl 0788.03043
[10] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001
[11] E. Hrushovski, Contributions to stable model theory, Ph. D. Thesis, Berkeley 1985.
[12] Ehud Hrushovski, Unimodular minimal structures, J. London Math. Soc. (2) 46 (1992), no. 3, 385 – 396. · Zbl 0804.03023
[13] Ehud Hrushovski, Finitely axiomatizable ℵ\(_{1}\) categorical theories, J. Symbolic Logic 59 (1994), no. 3, 838 – 844. · Zbl 0808.03015
[14] E. Hrushovski, Pseudo-finite fields and related structures, preprint (1991). · Zbl 1082.03035
[15] Ehud Hrushovski, Finite structures with few types, Finite and infinite combinatorics in sets and logic (Banff, AB, 1991) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 411, Kluwer Acad. Publ., Dordrecht, 1993, pp. 175 – 187. · Zbl 0845.03013
[16] E. Hrushovski, The Manin-Mumford conjecture and the model theory of difference fields, preprint (1995). · Zbl 0987.03036
[17] E. Hrushovski, The first-order theory of the Frobenius, preprint (1996). · Zbl 0864.03026
[18] U. Hrushovski and A. Pillay, Weakly normal groups, Logic colloquium ’85 (Orsay, 1985) Stud. Logic Found. Math., vol. 122, North-Holland, Amsterdam, 1987, pp. 233 – 244.
[19] Ehud Hrushovski and Anand Pillay, Groups definable in local fields and pseudo-finite fields, Israel J. Math. 85 (1994), no. 1-3, 203 – 262. · Zbl 0804.03024
[20] E. Hrushovski and A. Pillay, Definable subgroups of algebraic groups over finite fields, J. Reine Angew. Math. 462 (1995), 69 – 91. · Zbl 0823.12005
[21] B. Kim, Forking in simple unstable theories, J. London Math. Soc. (2) 57 (1998), 257-267. CMP 99:01
[22] Byunghan Kim and Anand Pillay, Simple theories, Ann. Pure Appl. Logic 88 (1997), no. 2-3, 149 – 164. Joint AILA-KGS Model Theory Meeting (Florence, 1995). · Zbl 0897.03036
[23] A. Macintyre, Generic automorphisms of fields, in: Proc. AILA-KGS conference (Florence, 1995), A. Lachlan, D. Mundici editors, Ann. Pure Appl. Logic 88 (1997), 165 - 180. CMP 98:07
[24] A. Macintyre, Nonstandard Frobenius, in preparation.
[25] Anand Pillay, An introduction to stability theory, Oxford Logic Guides, vol. 8, The Clarendon Press, Oxford University Press, New York, 1983. · Zbl 0526.03014
[26] Anand Pillay, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. · Zbl 0871.03023
[27] Bruno Poizat, Cours de théorie des modèles, Bruno Poizat, Lyon, 1985 (French). Une introduction à la logique mathématique contemporaine. [An introduction to contemporary mathematical logic]. · Zbl 0583.03001
[28] Derek J. S. Robinson, A course in the theory of groups, 2nd ed., Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1996. · Zbl 0483.20001
[29] Gary Cornell and Joseph H. Silverman , Arithmetic geometry, Springer-Verlag, New York, 1986. Papers from the conference held at the University of Connecticut, Storrs, Connecticut, July 30 – August 10, 1984. · Zbl 0596.00007
[30] Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. · Zbl 0423.12016
[31] Jean-Pierre Serre, Topics in Galois theory, Research Notes in Mathematics, vol. 1, Jones and Bartlett Publishers, Boston, MA, 1992. Lecture notes prepared by Henri Damon [Henri Darmon]; With a foreword by Darmon and the author. · Zbl 0746.12001
[32] Igor R. Shafarevich, Basic algebraic geometry. 1, 2nd ed., Springer-Verlag, Berlin, 1994. Varieties in projective space; Translated from the 1988 Russian edition and with notes by Miles Reid. Igor R. Shafarevich, Basic algebraic geometry. 2, 2nd ed., Springer-Verlag, Berlin, 1994. Schemes and complex manifolds; Translated from the 1988 Russian edition by Miles Reid.
[33] Saharon Shelah, Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam-New York, 1978. · Zbl 0389.03014
[34] Saharon Shelah, Simple unstable theories, Ann. Math. Logic 19 (1980), no. 3, 177 – 203. · Zbl 0489.03008
[35] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. · Zbl 0585.14026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.