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arXiv:math/0401303v1 [math.LO] 22 Jan 2004

Analytic and pseudo-analytic structures

B. Zilber

University of Oxford

2 April 2001

One of the questions frequently asked nowadays about model theory is whether it is

still logic. The reason for asking the question is mainly that more and more of model

theoretic research focuses on concrete mathematical fields, uses extensively their tools

and attacks their inner problems. Nevertheless the logical roots in the case of model

theoretic geometric stability theory are not only clear but also remain very important

in all its applications.

This line of research started with the notion of a κ-categorical first order theory, which

quite soon mutated into the more algebraic and less logical notion of a κ-categorical

structure.

A structure M in a first order language L is said to be categorical in cardinality

κ if there is exactly one, up to isomorphism, structure of cardinality κ satisfying the

L-theory of M.

In other words, if we add to Th(M) the (non first-order) statement that the cardi-

nality of the domain of the structure is κ, the description becomes categorical.

The principal breakthrough, in the mid-sixties, from which stability theory started

was the answer to J.Los’ problem

The Morley Theorem A countable theory which is categorical in one uncountable

cardinality is categorical in all uncountable cardinalities.

The basic examples of uncountably categorical structures in a countable language are:

(1) Trivial structures (the language allows only equality);

(2) Abelian divisible torsion-free groups; Abelian groups of prime exponent (the lan-

guage allows +,=); Vector spaces over a (countable) division ring

(3) Algebraically closed fields in language (+,·,=) .

Also, any structure definable in one of the above is uncountably categorical in the

language which witnesses the interpretation.

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The structures definable in algebraically closed fields, for example, are effectively ob-

jects of algebraic geometry.

As a matter of fact the main logical problem after answering the question of J.Los

was what properties of M make it κ-categorical for uncountable κ?

The answer is now reasonably clear:

The key factor is measurability by a dimension and high homogeneity of the structure.

This gave rise to (Geometric) Stability Theory, the theory studying structures with

good dimensional and geometric properties (see [Bu] and [P]). When applied to fields,

the stability theoretic approach in many respects is very close to Algebraic Geometry.

The abstract dimension notion for finite X ⊂ M mentioned above could be best un-

derstood by examples:

(1a) Trivial structures:

size of X;

(2a) Abelian divisible torsion-free groups; Abelian groups of prime exponent; Vector

spaces over a division ring: linear dimension of X;

(3a) Algebraically closed fields: transcendence degree tr.d.(X).

Dually, one can classically define another type of dimension using the initial one:

dimV = max{tr.d.(¯ x) | ¯ x ∈ V }

for V ⊆ Mnan algebraic variety. The latter type of dimension notion is called in

model theory the Morley rank.

The last example can serve also as a good illustration of the significance of homo-

geneity of the structures. So, in general, the transcendence degree makes good sense

in any field, and there is quite a reasonable dimension theory for algebraic varieties

over a field. But the dimension theory in arbitrary fields fails if we want to consider

it for wider classes of definable subsets, e.g. the images of varieties under algebraic

mappings. In algebraically closed fields any definable subset is a boolean combination

of varieties, by elimination of quantifiers, which eventually is the consequence of the

fact that algebraically closed fields are existentially closed in the class of fields. The

latter effectively means high homogeneity, as an existentially closed structure absorbs

any amalgam with another member of the class.

One of the achievements of stability theory is the establishing of some hierarchy of

types of structures that allows to say which ones are more ’analysable’ (see [Sh]).

The next natural question to ask is whether there are ’very good’ stable structures

which are not reducible to (1) - (3) above?

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The initial hope of the present author in [Z1], that any uncountably categorical struc-

ture comes from the classical context (the trichotomy conjecture), was based on the

general belief that logically perfect structures could not be overlooked in the natural

progression of mathematics. Allowing some philosophical licence here, this was also

a belief in a strong logical predetermination of basic mathematical structures.

As a matter of fact this turned out to be true in many cases. Specifically for Zariski

geometries, which are defined as the structures with a good dimension theory and

nice topological properties, similar to the Zariski topology on algebraic varieties (see

[HZ]).

Another situation where this principle works, is the context of o-minimal structures

(see [PS]).

Powerful applications of the result on Zariski geometries and of the underlying method-

ology were found by Hrushovski [H3],[H4]. This not only lead to new and indepen-

dent solutions to some Diophantine problems, Manin-Mumford and Mordell-Lang

(the functional case) conjectures, but also a new geometric vision of these.

Yet the trichotomy conjecture proved to be false in general as Hrushovski found a

source of a great variety of counterexamples.

We analyse below the Hrushovski construction, purporting to answer the question of

whether the counterexamples it provides dramatically overhaul the trichotomy con-

jecture or if there is a way to save at least the spirit of it. As the reader will find

below the author is inclined to stick to the second alternative.

1Hrushovski construction of new structures

The main steps:

Suppose we have a, usually elementary, class of structures H with a good dimension

notion d(X) for finite subsets of the structures. We want to introduce a new function

or relation on M ∈ H so that the new structure gets a good dimension notion.

The main principle, which Hrushovski found will allow us to do this, is that of the

free fusion. That is, the new function should be related to the old structure in as a

free way as possible. At the same time we want the structure to be homogeneous.

He then found an effective way of writing down the condition: the number of ex-

plicit dependencies in X in the new structure must not be greater than the size (the

cardinality) of X.

The explicit L-dependencies on X can be counted as L-codimension, size(X)−d(X).

The explicit dependencies coming with a new relation or function are the ones given

by simplest ’equations’, basic formulas.

So, for example, if we want a new unary function f on a field (implicit in [H2]), the

condition should be

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tr.d.(X ∪ f(X)) − size(X) ≥ 0, (1)

since in the set Y = X ∪f(X) the number of explicit field dependencies is size(Y ) −

tr.d.(Y ), and the number of explicit dependencies in terms of f is size(X).

If we want, e.g., to put a new ternary relation R on a field, then the condition would

be

tr.d.(X) − r(x) ≥ 0, (2)

where r(X) is the number of triples in X satisfying R.

The very first of Hrushovski’s examples (see [H1]) introduces just a new structure of a

ternary relation, which effectively means putting new relation on the trivial structure.

So then we have

size(X) − r(X) ≥ 0. (3)

If we similarly introduce an automorphism σ on the field (difference fields, [CH]),

then we have to count

tr.d.(X ∪ σ(X)) − tr.d.(X) ≥ 0, (4)

and the inequality here always holds.

Similarly for differential fields with the differentiation operator D (see [Ma]), where

we always have

tr.d.(X ∪ D(X)) − tr.d.(X) ≥ 0. (5)

The left hand side in each of the inequalities (1) - (5), denote it δ(X), is a counting

function, which is called predimension, as it satisfies some of the basic properties

of the dimension notion.

At this point we have carried out the first step of the Hrushovski construction, that is:

(Dim) we introduced the class Hδof the structures with a new function or relation,

and the extra condition

(GS)δ(X) ≥ 0 for all finite X.

(GS) here stands for ’Generalised Schanuel’, the reason for which will be given below.

The condition (GS) allows us to introduce another counting function with respect to

a given structure M ∈ Hδ

∂M(X) = min{δ(Y ) : X ⊆ Y ⊆finM}.

We also need to adjust the notion of embedding in the class for further purposes.

This is the strong embedding, M ≤ L, meaning that ∂M(X) = ∂L(X) for every

X ⊆finM.

The next step is

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(EC) Using the inductiveness of the class construct an existentially closed structure

in (Hδ,≤).

If the class has the amalgamation property, then the existentially closed structures are

sufficiently homogeneous. Also ∂M(X) for existentially closed M becomes a dimension

notion.

So, if also the class EC of existentially closed structures is axiomatisable, one can

rather easily check that the existentially closed structures are ω-stable. This is the

case for examples (1) - (3) and (5) above.

In more general situations the e.c. structures may be unstable, but still with a

reasonably good model-theoretic properties.

Notice that though condition (GS) is trivial in examples (4) - (5), the derived dimen-

sion notion ∂ is non-trivial. In both examples ∂(x) > 0 iff the corresponding rank

of x is infinite (which is the SU-rank in algebraically closed difference fields and the

Morley rank, in differentially closed fields).

Notice that the dimension notion ∂ for finite subsets, similarly to the example (3a),

gives rise to a dual dimension notion for definable subsets S ⊆ Mnover a finite set

of parameters C :

dim(S) = max{∂({x1,...,xn}/C) : ?x1,...,xn? ∈ S}.

(Mu) This stage originally had been considered prior to (EC), but as one easily sees,

it can be equivalently introduced after.

We want to find now a finite Morley rank structure as a substructure (may be non-

elementary) of a structure M ∈ EC. In fact an existentially closed M would be of

finite Morley rank, if ’dim(S) = 0’ is equivalent to ’S is finite’. But in general dim(S)

may be zero for some infinite definable subsets S, e.g. the set S = {x ∈ M : f(x) = 0}

is one such in example (1) : ’some equations have too many solutions’.

To eliminate the redundant solutions Hrushovski introduces a counting function µ

for the maximal allowed size of potentially Morley rank 0 subsets. Then Hδ,µis the

subclass of structures of Hδ satisfying the bounds given by µ. Equivalently, since

existentially closed structures are universal for structures of Hδ, so Hδ,µis the class

of substructures of existentially closed structures M, satisfying the bound by µ.

Inside this class we can just as well carry out the construction of existentially closed

structures Mµ. Again, if the subclass has the amalgamation property and is first or-

der definable, then an existentially closed substructure Mµof this subclass is of finite

Morley rank, in fact strongly minimal in cases (1) - (3). It is also important for the

further discussion that Mµ⊆ M.

The infinite dimensional structures emerging after step (EC) in natural classes we

call natural Hrushovski structures. Some but not all of them lead after step (Mu) to

finite Morley rank structures.

It follows immediately from the construction, that the class of natural Hrushovski

structures is singled out in H by three properties: the generalised Schanuel property

(GS), the property of existentially closedness (EC) and the property (ID), stating the

existence of n-dimensional subsets for all n.

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