O-minimal structures-basics and some applications

Kobi Peterzil

Department of MathematicsUniversity of Haifa

11th Panhellenic Logic Symposium at Delphi

Kobi Peterzil 1

Map of the universe (thanks to Gabriel Conant)

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Map of the universe (thanks to Gabriel Conant)

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The genrel plan

Talk IDefinitions, examples and a variety of basic results about o-minimalstructures. Including some proofs.

Talk IIO-minimality and Diophantine geometry.

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O-minimal structures

DefinitionA linearly ordered structureM = 〈M;<, · · ·〉, is o-minimal if everydefinable subset of M (with parameters) is of the form

k⋃i=1

(ai ,bi) ∪ F︸︷︷︸finite

with ai ,bi ∈ M ∪ {±∞}.

Negative ExamplesI 〈Z;<,+〉 (why? 2Z)I 〈R;<,+, ·, sin x〉 (why? the set {x : sin x = 0} ).I 〈Q;<, ·〉 (why? The set {x : x2 < 2})I 〈γ;<〉, γ an ordinal > ω2 (Exercise).

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Main examples of o-minimal structures

For simplicity we restrict to dense linear orderings.

I 〈D;<〉 a dense linear ordering (by quantifier elimination).I 〈G;<,+〉 an ordered divisible abelian group (by quantifier

elimination).I 〈R;<,+, ·〉 a real closed field (for example, the field R).

why? By QE (Tarski) and the fact that every polynomial hasfinitely many roots.

I Ran = 〈R;<,+, ·, f � [−1,1]n〉, for f real analytic (v.d.Dries, basedon Gabrielov).

I Rexp = 〈R;<,+, ·,ex〉 (Wilkie).I Ran,exp = 〈Ran,ex〉 (vdD-Miller). Most of recent applications of

o-minimality take place in Ran,exp.

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Basic properties of o-minimal structures

We assume thatM = 〈M, <, · · ·〉 is o-minimal.

Observations• An ordered reduct of an o-minimal structure is still o-minimal.• Definable completeness: If X ⊆ M is definable and bounded belowthen X has a maximal lower bound.• Every definable infinite X ⊆ M contains an open interval.• Every definable unbounded X ⊆ M contains a ray (a,∞) or(−∞,a).Namely, assume that

M |= ∀x∃y(y > x ∧ φ(y , a)).

ThenM |= ∃x∀y(y > x → φ(y , a)).

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Definable Choice (Skolem functions)

Assume thatM = 〈M, <,+,1, · · ·〉 is an o-minimal expansion of anordered (abelian) group, with 1 a non-zero element.

TheoremLet {Xa : a ∈ T} be a definable family of nonempty subsets of Mn. I.e.for some φ(x , y) and definable T ⊆ Mk , Xa := {b ∈ Mn : φ(b, a)}.Then there is a definable f : T → Mn (i.e. a definable graph) such that(1) for all a ∈ T , f (a) ∈ Xa and (2) If Xa = Xb then f (a) = f (b).

Proof By induction on n. For n = 1, write Xa as an <-increasing unionof intervals Xa = J1,a ∪ J2,a · · · ......If J1,a is a bounded interval, let f (a) be its midpoint (use +). Otherwise,let f (a) = Inf (J1,a) + 1 or sup(Ja)− 1.

Exercise• Show that f is definable.• Finish by induction.

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Structural properties of definable sets I

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

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

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Lemma-an ordered Ramsey Theorem (no proof here)

Let I ⊆ M be an open interval, and assume I 2 ⊆ X1 ∪ · · · ∪ Xn, fordefinable X ′i s. Then there exists an open interval J ⊆ I and a fixed1 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 ♦ ∈ {>,<,=}such that for all x < y in J, f (x)♦f (y).

So, f is either constant, of strictly increasing, or strictly decreasing onJ. Piecewise montonicity follows.

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Topological properties of definable sets

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

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

definable open sets.I Mn is also definably connected.

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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, inwhich case C′ is a k -cell. OR(ii) C′ = {(x , y) ∈ C ×M : f (x) < y < g(x)}, for f ,g : C → M definablecontinuous, with f < g. In which case dim C′ is a k + 1-cell.

Exercise: Each cell is definably connected.

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

Many important corollariesI 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).

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A model theoretic corollary

TheoremIfM 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-minimalityLet {Xa : a ∈ T} be a definable family of subsets of M. I.e. for someformula φ(x , t) and a definable T ⊆ Mn, for every a ∈ T ,

Xa := {b ∈ M : φ(b,a)}.

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

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O-minimality and algebra

TheoremLet 〈G;<, ·〉 be an o-minimal ordered group. Then G is abelian anddivisible.

Proof1. There are no proper nontrivial definable subgroups of G:Indeed, G must be torsion-free (ordered), so every nontrivial subgroupis infinite. So, if H < G is definable then Int(H) 6= ∅. Hence, H is openin G. Hence, H is closed (its complement is open). G definablyconnected, so H = G.

2. G ia abelian: For every g ∈ G, its centralizer {h ∈ G : gh = hg} isdefinable, nontrivial subgroup so equals G.

3. G is divisible: For every n ∈ N, nG is a definable nontrivialsubgroup, so nG = G.

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O-minimality and algebra-definable groups

DefinitionA group G is called definable inM if the set G and the groupoperation are definable inM.

Examples of definable groups in o-minimal structures1. 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 u

0 et v0 0 1

: t ,u, v ∈ R

Solvable, not isomorphic to a semi-algebraic group.

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Definable groups, rich theory, many authors

Some general principlesDefinable 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 algebraicgroups).

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 fieldoperations on C ∼ R2 are polynomial in the real and imaginarycoordinates.

QuestionWhich holomorphic functions are definable in o-minimal structures?

FactIf z0 is an isolated singularity of a definable holomorphic function fthen 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 thendim(Cl(F ) \ F ) = dim F = 2. Contradicting topological fact.

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O-minimality and complex analysis II

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

Positive (and negative) results1. (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.

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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 ModelTheory, 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/kobi5. A survey on groups definable in o-minimal structures, M. Otero,

2006.

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O-minimal structures II

DefinitionA linearly ordered structureM = 〈M;<, · · ·〉, is o-minimal if everydefinable subset of M (with parameters) is of the form

k⋃i=1

(ai ,bi) ∪ F︸︷︷︸finite

with ai ,bi ∈ M ∪ {±∞}.

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Talk II-Diophantine applications of o-minimality

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

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

A general principleA definable subset of Rn should not have “many” integer or rationalpoints 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

DefinitionThe height of a reduced fraction m/k ∈ Q is max{|m|, |k |}.For q = (q1, . . . ,qn) ∈ Qn, ht(q) = 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).

DefinitionFor X ⊆ Rn and H ∈ N, we let #X (Q,H) = |{q ∈ X ∩Qn : ht(q) 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 inRexp-nonalgebraic).• For X = {(x , sin(πx)) : x ∈ R}, #X (Q,H) > H. (non-definable,non-algebraic).

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The Pila-Wilkie Theorem

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

lim supH

#X (Q,H)

Hε= +∞.

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

lim supH

#X trans(Q,H)

Hε<∞.

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Many Diophantine applications

DefinitionA subset X ⊆ Cn is called algebraic variety if it is the solution sets tofinitely many polynomial equations in n variables. It is irreducible ifcannot be written as union of two proper algebraic subvarieties.

(For later use)A closed X ⊆ Cn is called an analytic subvariety of Cn if it is givenlocally by finitely many complex analytic equations.E.g. the graph of exp(z) is an analytic subvariety of C2 (but notalgebraic).

General “principle”Given an abelian algebraic group G, an algebraic subvariety V ⊆ Gshould not contain “many” torsion points, unless for a “good reason”.

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Concrete conjectures and theorems

The Manin-Mumford Conjecture-Raynaud’s theoremLet A ⊆ Pn(C) be an abelian variety defined over Q (i.e. a projectivealgebraic group) and let V ⊆ A be an irreducible algebraic subvariety.If the torsion points of A are Zariski dense in V then V is a coset of analgebraic subgroup of A.

An analogous simpler result (Laurent 1984)Let Gn

m = ((C∗)n, ·). Its torsion subgroup Un = (u1, . . . ,un) (all tuples ofroots of unity).If V ⊆ Gn

m is an irreducible algebraic variety and V ∩ Un is Zariskidense in V then there exists an algebraic subgroup A 6 Gn

m andp ∈ Un such that V = A · p.

For the rest of the talk I will present the Pila-Zannier proof of this testcase.

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Translating to the analytic context

The exponential covering mapConsider exp : (Cn,+)→ ((Gm)n, ·), given by exp(z) = (ez1 , . . . ,ezn ).So exp is a holomorphic group homomorphism.Γ := Ker(exp) = (2πiZ)n, is an infinite discrete subgroup.

From torsion to rationalBecause exp is a homomorphism, exp(z) is a torsion point of order kin Gn

m if and only if kz ∈ (2πiZ)n.So, exp(z) is a torsion of Gn

m if and only if there exists k such thatz ∈ 1/k(2πiZ)n iff z ∈ (2πiQ)n.

Subgroups to linear subspacesIf L ⊆ Cn is a C-subspace with a basis in Qn then exp(L) is aconnected algebraic subgroup of Gn

m.

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A rough description of the strategy

1. We start with an irreducible algebraic subvariety V ⊆ Gnm such that

V ∩ Un is Zariski dense in V . Using Galois Theory we shall seethat V contains “many” torsion points in the sense of Pila-Wilkie.

2. We let X = exp−1(V ). As we shall see, a subset of X (but not allof X !) is definable in an o-minimal structure. Thus the set Xcontains “many” points in 2πiQn.

3. By Pila-Wilkie, X contains a semialgebraic set of dimension one.Taking Zariski closure, X contains also a complex algebraic curveS ⊆ X .

4. By o-minimality, the Zariski closure of exp(S) is a coset of analgebraic subgroup of V . Thus V contains a coset of a subgroup.

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Definability in an o-minimal structure

• Because exp : Cn → Cn has an infinite discrete kernel Γ, it cannot bedefinable in any o-minimal structure.• Instead, we consider a restriction of exp to a fundamental setF ⊆ Cn, namely F such that F + Γ = Cn.

F = {(z1, . . . , zn) ∈ Cn : 0 6 Im(zi) 6 π}.

0

2πi

F = {(z1, . . . , zn) : 0 ≤ |zi| ≤ 2π}.F = {(z1, . . . , zn) : 0 ≤ |zi| ≤ 2π}.

The restricted function exp � F is definable in Ran,exp:ex+iy = ex (cosy + isiny). The function ex : R→ R is definable in Rexpand sin x , cos x � [0, π] is definable in Ran.

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From “infinity” to “polynomially many” torsion pts

I Assume that V ⊆ Gnm is algebraic, say defined over Q, and V ∩ Un

is infinite (weaker than Zariski dense).I Hence, there are m1 < m2 < · · · < · · · in N and g′i s in V such that

the order of gi in Gnm equals mi .

I By elementary considerations, [Q(gi) : Q] = φ(mi), where φ is theEuler totient function.

I Since V is Q, all conjugates of gi are in V , hence V containsφ(mi)-many conjugates of gi , all elements of order mi .

Move to the analytic side: Recall X = exp−1(V ). It follows thatX0 := X ∩ F contains at least φ(mi)-many “rational” points of height mi(actually points in 2πiQn).

It is known that φ(n)/n1/2 →∞.

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Applying Pila-Wilkie

I The set X = exp−1(V ) is a analytic subset of Cn (in general notdefinable in an o-minimal structure). We have Γ + X = X .

I The set X0 = X ∩ F is definable in Ran,exp.

I We saw: #X0(2πiQ,mi )

m1/2i

→∞.

I By Pila-Wilkie, there exists a semialgebraic set S0 ⊆ X0 ofdimension 1.

I Let S be the Zariski closure of S0 (the smallest algebraicsubvariety of Cn containing S0).

I It is a complex algebraic curve that is contained in X .

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Summary thus far

We thus have

• A Γ-invariant analytic set X ⊆ Cn (not definable!).• The image exp(X ) = V is an algebraic variety in Gn

m.• The set X0 = X ∩ F is definable.• In X we have a complex algebraic curve (so definable!) S, such thatdim(S ∩ F) = 1.

S

0

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Finding the coset inside V

Without loss of generality, exp(S) is Zariski dense in V .Let

Σ = {g ∈ Cn : (g + S) ∩ F 6= ∅& dim(g + S) ∩ X0 = 1}.

In fact, by the analyticity of X and the irreducibility of S,

Σ = {g ∈ Cn : (g + S) ∩ F 6= ∅& g + S ⊆ X}.

For g ∈ Σ, exp(S) ⊆ exp(X − g) = V · exp(g)−1.

Because V = Zarcl(expS), we have

V · exp(g)−1 = V .

It follows that exp(Σ) is contained in the group-stabilizer of V , in thegroup Gn

m (={h ∈ Gnm : h · V = V}).

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Ending the proof

So,we have

Σ = {g ∈ Cn : (g + S) ∩ F 6= ∅& dim(g + S) ∩ X0 = 1},

and we saw that exp(Σ) is contained in StabGnm

(V ).

Note: Since S and X0 are definable, then Σ is definable.

And because Σ contains all γ ∈ Γ such that S ∩ (−γ + F) 6= ∅, Σ isalso infinite.

By o-minimality, |Σ| = 2ℵ0 , so exp(Σ) is still infinite.

Thus, the algebraic group A = StabGnm

(V ) has positive dimension. Itfollows that A · p ⊆ V for every p ∈ V .

Using a supped-up version of the above argument one can showV = A · p.

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Conclusions and references

Similar strategy was used by Pila (2008) to prove unconditionallyunknown cases of the Andre-Oort Conjecture.Since then, there were many different applications of this strategy inDiophantine Geometry.

SurveysT. Scanlon A proof of the Andre-Oort Conjecture via mathematicallogic [after Pila, Wilkie and Zannier], Seminaire Bourbaki avril 201163em anee. 2010-2011, no 1037.T. Scanlon Counting special points: Logic, Diophantine geometry andtranscendence theory, BAMS (N.S) 49 (2012), no 1 51-71.K. Peterzil Video recording of MSRI talks,https://www.msri.org/people/8959, 2014.

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Recent work on “Diophantine approximation”

Let Λ be a lattice in Rn, namely Λ = ⊕ni=1Zωi , for ω1, . . . , ωn a basis for

Rn.Let T = Rn/Λ be a compact real torus.Let π : Rn → T be the covering map (a transcendental map), and letX ⊆ Rn be some definable set in an o-minimal structure (e.g. a realalgebraic variety).

A question of Ullmo and YafaevWhat is the topological closure of π(X ) in T?I.e. Which points in Rn can be approximated by elements in Λ and X?

A simple observationThe topological closure of π(X ) is precisely π(Cl(Λ + X )).So we need to answer: What is the topological closure of Λ + X in Rn?Namely, which points in Rn can be approximated by elements of Λ + X

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Some examples

Assume that Λ = Z2 ⊆ R2 and X ⊆ R2 closed and definable.

I If X ⊆ Rn is bounded the Λ + X is closed.I If X ⊆ Rn is an R-subspace then Cl(Λ + X ) = Λ + X Λ,where X Λ is

the smallest R-subspace with a basis in Λ containing X .I If X = {(x , y) ∈ R2 : x · y = 1} then

Cl(Λ + X ) = (Λ + X ) ∪ {0} × R ∪ R× {0}.

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The main theorem

Theorem (Pe-Starchenko)Let X ⊆ Rn be a closed definable set in an o-minimal structure. Thenthere are R-subspaces V1, . . . ,Vr ⊆ Rn of positive dimension, anddefinable closed sets C1, . . . ,Cr ⊆ Rn such that for every latticeΛ ⊆ Rn,

1. Cl(Λ + X ) = (Λ + X ) ∪⋃ri=1 Λ + V Λ

i + Ci . I.e.Cl(π(X )) = π(X ) ∪⋃r

i=1 Ti + π(Ci), for Ti real subtori of T .2. For each i = 1, . . . , r , dim Ci < dim X .3. If Vi is maximal among V1, . . . ,Vr then Ci is bounded.

In particular, if dim X = 1 (e.g. X is a real algebraic curve) then eachCi is finite, so

Cl(π(X )) = π(X ) ∪r⋃

i=1

Ti + pi

, for pi ∈ T .39