     # O-minimal structures-basics and some Examples of de¯¬¾nable groups in o-minimal structures 1.Semi-linear

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• O-minimal structures-basics and some applications

Kobi Peterzil

Department of Mathematics University of Haifa

11th Panhellenic Logic Symposium at Delphi

Kobi Peterzil 1

• Map of the universe (thanks to Gabriel Conant)

2

• Map of the universe (thanks to Gabriel Conant)

3

• The genrel plan

Talk I Definitions, examples and a variety of basic results about o-minimal structures. Including some proofs.

Talk II O-minimality and Diophantine geometry.

4

• O-minimal structures

Definition A linearly ordered structureM = 〈M;

• Main examples of o-minimal structures

For simplicity we restrict to dense linear orderings.

I 〈D;

• Basic properties of o-minimal structures

We assume thatM = 〈M, x ∧ φ(y , ā)).

Then M |= ∃x∀y(y > x → φ(y , ā)).

7

• Definable Choice (Skolem functions)

Assume thatM = 〈M,

• Structural properties of definable sets I

The Monotonicity-Continuity Theorem Let f : I = (a,b)→ M be a definable function (i.e. its graph is a definable set).Then there are a = a0 < a1 · · · < an = b such that on each interval (ai ,ai+1) the function is continuous and either strictly monotone or constant.

Idea of Proof We prove a much weaker statement: There is an open (nonempty) interval J ⊆ I such that f is either constant or strictly monotone on J.

9

• Lemma-an ordered Ramsey Theorem (no proof here)

Let I ⊆ M be an open interval, and assume I 2 ⊆ X1 ∪ · · · ∪ Xn, for definable X ′i s. Then there exists an open interval J ⊆ I and a fixed 1 6 j 6 n, such that for every a < b in J, (a,b) ∈ Xj .

X X

X

1

2

3

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• End of the proof of CMT

Back to our f : I → M. Let

X< = {(x , y) ∈ I 2 : f (x) < f (y)}.

X> = {(x , y) ∈ I 2 : f (x) > f (y)}.

X= = {(x , y) ∈ I 2 : f (x) = f (y)}.

Clearly I 2 = X< ∪ X> ∪ X=. By Ordered Ramsey, there is an open interval J ⊆ I and ♦ ∈ {>,

• Topological properties of definable sets

• The order topology on M and box-topology on Mn have definable basis. • Caution: In general, the space M is not locally connected, or locally compact (unless the underlying ordered set is R). BUT

Exercise I M is definably connected, i.e. it cannot be partitioned into two

definable open sets. I Mn is also definably connected.

12

• Cell decomposition

Definition of a k -cell • In M: A 0-cell is a point, a 1-cell is an open interval. • In Mn+1: Assume C ⊆ Mn is a k -cell. A cell C′ in Mn+1 is either (i) the graph Γ(f ) of a definable continuous function f : C → M, in which case C′ is a k -cell. OR (ii) C′ = {(x̄ , y) ∈ C ×M : f (x̄) < y < g(x̄)}, for f ,g : C → M definable continuous, with f < g. In which case dim C′ is a k + 1-cell.

Exercise: Each cell is definably connected.

13

• The fundamental theorem of o-minimality-Knight Pillay Steinhorn Every definable set in Mn can be partitioned into finitely many cells. Hence, every definable set has finitely many definably connected components. Moreover, if {Xt : t ∈ T} is a definable family of sets then there is a uniform bound on the number of these components.

Many important corollaries I Finer stratification results yield strong geometric tameness. I Dimension theory: dim X := the maximal dimension of a cell in

X . I Important!: For every definable set X ⊆ Mn,

dim(Cl(X ) \ X ) < dim X .

(easy to prove for cells).

14

• A model theoretic corollary

Theorem IfM is o-minimal and N ≡M, then N is also o-minimal. I.e. o-minimality is a property of complete theories.

Exercise: The above is equivalent to:

Uniform o-minimality Let {Xā : ā ∈ T} be a definable family of subsets of M. I.e. for some formula φ(x , t̄) and a definable T ⊆ Mn, for every ā ∈ T ,

Xā := {b ∈ M : φ(b,a)}.

Then there is a fixed K ∈ N such that every Xā can be written as a finite union of at most K -many open intervals and K -many points. And this follows from cell decomposition.

15

• O-minimality and algebra

Theorem Let 〈G;

• O-minimality and algebra-definable groups

Definition A group G is called definable inM if the set G and the group operation are definable inM.

Examples of definable groups in o-minimal structures 1. Semi-linear groups (groups definable in ordered vector spaces). 2. All real algebraic groups H(R), all semi-algebraic groups (e.g. 〈R>0, ·〉), are definable in the real field.

3. Groups definable in Rexp: E.g.  et tet u0 et v

0 0 1

 : t ,u, v ∈ R 

Solvable, not isomorphic to a semi-algebraic group.

17

• Definable groups, rich theory, many authors

Some general principles Definable groups resemble: I Lie groups (e.g. admit a manifold-like, definable topology), I Algebraic groups (e.g. A descending Chain Condition for definable

subgroups. Definable simple groups are isomorphic to algebraic groups).

I Compact groups (e.g. finite number of connected components).

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• O-minimality and complex analysis

Note: The complex field (C,+, ·) is definable in (R; +, ·): The field operations on C ∼ R2 are polynomial in the real and imaginary coordinates.

Question Which holomorphic functions are definable in o-minimal structures?

Fact If z0 is an isolated singularity of a definable holomorphic function f then z0 is not an essential singularity.

Proof Consider F = {(z, f (z)) ∈ C2 : z 6= z0}. If z0 is an essential singularity then z0 × C ⊆ Cl(F ) \ F . But then dim(Cl(F ) \ F ) = dim F = 2. Contradicting topological fact.

19

• O-minimality and complex analysis II

Based on this simple observation, there is a rich theory of complex analytic sets definable in o-minimal structures.

Positive (and negative) results 1. (positive) If f : Cn → C is definable and holomorphic then f is

necessarily a polynomial map. 2. Every definable complex analytic subset of Cn is algebraic. 3. (Negative) As a result, all classical periodic, modular and

automorphic functions are not definable on their entire domain. 4. Still, as we shall see next talk, some of these are definable on

partial domain, which is enough for applications.

20

• Some references

1. Book: Tame topology and o-minimal structures, L. van den Dries, London Lecture Notes 248, 1998.

2. Notes on o-minimality and variations, D. Macpherson, in Model Theory, Algebra and geometry, 2000.

3. Introduction to o-minimal geometry, M. Coste, 1998. 4. A self-guide to o-minimality, Y. Peterzil, 2005, on

http://math.haifa.ac.il/kobi 5. A survey on groups definable in o-minimal structures, M. Otero,

2006.

21

• O-minimal structures II

Definition A linearly ordered structureM = 〈M;

• Talk II-Diophantine applications of o-minimality

We fix an o-minimal expansion R̄ of the real field R.

Let X ⊆ Rn be a definable set. What can we say about the number of rational points in X?, namely about X ∩Qn?

A general principle A definable subset of Rn should not have “many” integer or rational points unless for a “a good reason”.

Examples: I The set {(x , x2) : x ∈ R}, has infinitely many points in Q2

(because of algebraicity). I But also the set {(x ,2x ) : x ∈ R} has infinitely many points in Q2

(in fact in Z2). Not algebraic. I And the (non definable) set {(x , sin(x/π))} has infinitely many

points in Z2 (because of “periodicity”). 23

• What is “many”? The height function

Definition The height of a reduced fraction m/k ∈ Q is max{|m|, |k |}. For q̄ = (q1, . . . ,qn) ∈ Qn, ht(q̄) = max{ht(qi) : i = 1, . . . ,n}.

• There are between H and H2 points in Q of height 6 H (to be precise (H + 1) · φ(H), where φ(H) is Euler’s totient function).

Definition For X ⊆ Rn and H ∈ N, we let #X (Q,H) = |{q̄ ∈ X ∩Qn : ht(q̄) 6 H}| (so always #X (Q,H) 6 H2n).

• For X = {(x , x2) : x ∈ R}, #X (Q,H) > H1/2 (definable-algebraic). • For X = {(x ,2x ) : x ∈ R}, #X (Q,H) 6 log(H2) (definable in Rexp-nonalgebraic). • For X = {(x , sin(πx)) : x ∈ R}, #X (Q,H) > H. (non-definable, non-algebraic).

24

• The Pila-Wilkie Theorem

Let X ⊆ Rn be definable in an o-minimal structure. Assume that X contains “polynomially many” rational points. Namely, there exists � > 0 such that,

lim sup H

#X (Q,H) H�

= +∞.

Then X contains a semialgebraic set (defined over Q), of positive dimension. Moreover, if we let X trans = X r all positive dimensional connnected semialgebraic susets of X , then for every � > 0,

lim sup H

#X trans(Q,H) H�

• Many Diophantine applications

Definition A subset X ⊆ Cn is called algebraic variety if it is the solution sets to finitely many polynomial equations in n variables. It is irreducible if cannot be written as union of two proper algebraic subvarieties.

(For later use) A closed X ⊆ Cn is called an

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