O-minimal structures - unirioja.es...O-minimal structures O-minimal strutures: axiomatic...

O-minimal structures

Michel Coste, IRMAR (Rennes)

MAP 2010, Logrono

Michel Coste, IRMAR (Rennes) O-minimal structures

IntroductionDefinition

Cell decomposition

Tutorial 1

1 Introduction

2 Definition

3 Cell decomposition



Cell decomposition

Motivating example: semialgebraic and subanalytic sets

The semialgebraic subsets of Rn form the smallest class SAn ofsubsets of Rn such that:

1 If P ∈ R[X1, . . . ,Xn], then {x ∈ Rn ; P(x) = 0} ∈ SAn and{x ∈ Rn ; P(x) > 0} ∈ SAn.

2 If A ∈ SAn and B ∈ SAn, then A ∪ B, A ∩ B and Rn \ A arein SAn.

Semianalytic sets : locally described by analytic functions (assemialgebraic sets by polynomials). Subanalytic sets : locallyprojection of relatively compact semianalytic sets.



Cell decomposition

Features of semialgebraic geometry

Semialgebraic sets are stable under many constructions :projection (Tarski-Seidenberg), closure, taking connectedcomponents,. . .

Tame topology : no pathological behavior, semialgebraic setsare triangulable,. . .

Finiteness, uniform bounds: finitely many topological types ina semialgebraic family,. . .

The same holds for globally subanalytic sets (= subanalytic in acompactification of Rn)



Cell decomposition

O-minimal structures

O-minimal strutures: axiomatic generalization of the precedingexamples (semialgebraic and globally subanalytic), retaining theirnice features. They include other interesting examples:

with the exponential function,

more generally, pfaffian functions,

with some functions associated with non convergent powerseries.

The entire sine function cannot be in a o-minimal structure(infinitely many zeroes); only its restriction to a compact segment.



Cell decomposition

Real closed fields

Semialgebraic geometry works over real closed fields, whichretain all algebraic properties of R.

Real closed field R : ordered field such that every P ∈ R[X ]satisfies the Intermediate Value Theorem :

a < b and P(a)P(b) < 0⇒ ∃c ∈ (a, b) P(c) = 0 .

Examples: the field of real algebraic numbers, the field of realPuiseux series

⋃p∈N∗ R((x1/p)).

We shall introduce o-minimal structures in the framework ofreal closed fields. Real closed fields will appear related to“ideal points”.

Interval in a real closed field R: (a, b) where a < b inR ∪ {−∞,+∞}. Products of intervals generate the topologyof Rn.



Cell decomposition

Structure

Definition

A structure expanding the real closed field R is a collectionS = (Sn)n∈N, where each Sn is a set of subsets of the affine spaceRn, satisfying the following axioms:

1 All semialgebraic subsets of Rn are in Sn.

2 For every n, Sn is a Boolean subalgebra of the powerset of Rn.

3 If A ∈ Sm and B ∈ Sn, then A× B ∈ Sm+n.

4 If p : Rn+1 → Rn is the projection on the first n coordinatesand A ∈ Sn+1, then p(A) ∈ Sn.



Cell decomposition

O-minimal structure

Definition

The structure S is said to be o-minimal if, moreover, it satisfies:

5 The elements of S1 are precisely the finite unions of pointsand intervals.

The elements of Sn are called the definable subsets of Rn. A mapf : A→ B between definable sets is called definable if its graph isdefinable.

Exercise : an ordered field carrying an o-minimal structure is realclosed.



Cell decomposition

Elementary facts

The image of a definable set by a definable map is definable.The composite of definable maps is definable. Definablefunctions A→ R form an R-algebra.

The closure and the interior of a definable subset A ⊂ Rn aredefinable.

clos(A) = Rn \(pn+1,n

(Rn+1 \ p2n+1,n+1(B)

)),

where B is

(Rn×R×A)∩{(x , ε, y) ∈ Rn×R×Rn |n∑

i=1

(xi − yi )2 < ε2} .



Cell decomposition

Language of an o-minimal structure

First-order formula (x = (x1, . . . , xn)):1 P(x) = 0, P(x) > 0 for P ∈ R[X1, . . . ,Xn],

x ∈ A for A ⊂ Rn definable,2 Φ and Ψ, Φ or Ψ, notΦ, Φ⇒ Ψ .3 ∃x ∈ A Φ(y , x), ∀x ∈ A Φ(y , x) where A ⊂ Rn is definable

subset of Rn.

If Φ(x) is a first-order formula, then {x ∈ Rn | Φ(x)} isdefinable.

Exercise: x 7→ dist(x ,A) = inf{‖y − x‖ | y ∈ A}, for A ⊂ Rn

non empty definable, is a continuous definable function on Rn.

{(x , y) ∈ R2 | ∃n ∈ N y = nx} is not definable.



Cell decomposition

Monotonicity Theorem

Theorem

Let f : (a, b)→ R be a definable function. There exists a finitesubdivision a = a0 < a1 < . . . < ak = b such that, on each interval(ai , ai+1), f is continuous and either constant or strictly monotone.

If there is no subinterval on which f is constant, there is asubinterval on which f is injective, hence there is a subinterval onwhich f is strictly monotone, hence there is a subinterval on whichf is continuous strictly monotone.

Remark : one can ask f to be C k on each (ai , ai+1).Exercise: every continuous definable f : (a, b)→ R has left andright derivatives in R ∪ {−∞,+∞} at every point.



Cell decomposition

CDCD (Cylindrical Definable Cell Decomposition)

Definition

A cdcd of Rn is a finite partition of Rn into definable cells, given:

n = 1: by a finite subdivision a1 < . . . < a` of R.Cells : the singletons {ai}, 0 < i ≤ `, and theintervals (ai , ai+1), 0 ≤ i ≤ ` (a0 = −∞,a`+1 = +∞).

n > 1: by a cdcd of Rn−1 and continuous definable functionsζD,1 < . . . < ζD,`(D) : D → R for each cellD ⊂ Rn−1.Cells : the graphs of the ζD,i , and the bands(ζD,i , ζD,i+1), 0 ≤ i ≤ `(D) cut in D × R by thesegraphs (ζD,0 = −∞, ζD,`(D)+1 = +∞).



Cell decomposition

Dimension of cells

Definition by induction:

dim(point) = 0, dim(interval)=1,dim(graph in D × R) = dim(D),dim(band in D × R) = dim(D)+1.

A cell C of a cdcd is definably homeomorphic to Rdim(C).

A cell C of a cdcd of Rn is open in Rn iff dim(C ) = n.



Cell decomposition

Adapted cdcd

Figure: cdcd adapted to thesphere

A cdcd is said to be adaptedto a definable set A if A is aunion of cells.

Theorem (Cell DecompositionCDCDn)

Let A1, . . . ,Ak be definablesubsets of Rn. There is a cdcdof Rn adapted to A1, . . . ,Ak .



Cell decomposition

Uniform Finiteness and Piecewise Continuity

The Cell Decomposition Theorem is proved by induction on thedimension n, together with the following results:

Theorem (Uniform Finiteness UFn)

Let A be a definable subset of Rn such that, for every x ∈ Rn−1,the set Ax = {y ∈ R | (x , y) ∈ A} is finite. Then there existsk ∈ N such that ]Ax ≤ k for every x ∈ Rn−1.

Theorem (Piecewise Continuity PCn)

Let A be a definable subset of Rn and f : A→ R a definablefunction. There is a cdcd of Rn adapted to A such that, for everycell C contained in A, f |C is continuous.

First UFn, then CDCDn and finally PCn.


Compactness, connectedness, dimensionGood coordinates

Triangulation of definable sets and functions

Tutorial 2

In Tutorial 1 : o-minimal structures, cylindrical definable celldecomposition.Today: tameness properties following from this decomposition(connected components, dimension, . . . ) and description of thetopology of definable sets in finite terms

4 Compactness, connectedness, dimension

5 Good coordinates

6 Triangulation of definable sets and functions




Definable choice, curve selection

Theorem (Definable Choice)

A ⊂ Rm × Rn definable, p : Rm × Rn → Rm the projection. Thereis a definable function f : p(A)→ Rn such that, for everyx ∈ p(A), (x , f (x)) belongs to A.

Theorem (Curve Selection Lemma)

A ⊂ Rn definable, b ∈ clos(A). There is a continuous definablemap γ : [0, 1)→ Rn such that γ(0) = b and γ((0, 1)) ⊂ A.

Apply definable choice toX = {(t, x) ∈ R × Rn ; x ∈ A and ‖x − b‖ < t}, and thenmonotonicity.




Definable compactness

Theorem

Let A ⊂ Rn definable. TFAE:

1 A is closed and bounded.

2 Every definable continuous map (0, 1)→ A extends bycontinuity to a map [0, 1)→ A.

3 For every definable continuous function f : A→ R, f (A) isclosed and bounded.

Such a A is called definably compact. This is an intrinsic notion(invariant by definable homeomorphism). Being locally closed ( =locally definably compact) is also intrinsic.




Definable connectednes

Definition

A definable set A is said to be definably connected if, for alldisjoint definable open subsets U and V of A such thatA = U ∪ V , one has A = U or A = V .It is said to be definably arcwise connected if, for all points a and bin A, there is a definable continuous map γ : [0, 1]→ A such thatγ(0) = a and γ(1) = b.

Cells of a cdcd are definably arcwise connected.

Exercise: definable + definably connected ⇒ definably arcwiseconnected.




Connected components

Theorem

Let A be a definable subset of Rn. There is a partition of A intofinitely many definable subsets A1, . . . ,Ak such that each Ai isnonempty, open and closed in A, and definably arcwise connected.Such a partition is unique. The A1, . . . ,Ak are called the definableconnected components of A.

Theorem (Uniform Finiteness revisited)

Let A be a definable subset of Rm × Rn. There is β ∈ N such that,for every x ∈ Rm the number of definable connected componentsof Ax = {y ∈ Rn | (x , y) ∈ A} is not greater than β.




Dimension

The dimension of cells coincide with the following intrinsic notion:

Definition

The dimension of a definable set A is the sup of d such that thereexists a injective definable map from Rd to A.

If C is a cdcd adapted to A, then dim(A) = maxC∈C dim(C ).

Theorem

Let A be a definable subset of Rm × Rn. For d ∈ N ∪ {−∞}, setXd = {x ∈ Rm | dim(Ax) = d}. Then Xd is a definable subset ofRm, and dim(A ∩ (Xd × Rn)) = dim(Xd) + d.




Dimension (2)

Theorem

Let A be a nonempty definable subset of Rn. Thendim(clos(A) \ A) < dim(A).

Dimension is invariant by definable bijection (not necessarilycontinuous). Another such invariant is Euler characteristic: forA ⊂ Rn definable, set χ(A) =

∑C⊂A(−1)dim(C), where the sum is

taken over the cells of an adapted cdcd contained in A. This doesnot depend on the cdcd.

Exercise: there is a definable bijection A→ B iff dim(A) = dim(B)and χ(A) = χ(B).




Extension by continuity

Adjacency between cells in different cylinders of a cdcd is not clear.Problem : extend by continuity ζ : D → R to clos(D).

Figure: Γ ⊂ R3 near the origin

Ex: ζ defined byζ(x , y) = 2xy/(x2 + y2) onx > 0 does not extendcontinuously to (0, 0). Theclosure Γ of the graph of ζ hasdimension 2, but the projectionΓ→ R2 is not finite-to-one.

What if we tilt it a little?




Good change of coordinates

Given: F ⊂ Rn definable, closed and bounded, such that therestriction to F of the projection p : Rn → Rn−1 is finite-to-one.X ⊂ p(F ) definable, such that every x ′ ∈ clos(X ) has a basis ofneighborhoods U such that U ∩ X is definably connected.Then every continuous definable function ζ : X → R with graphcontained in F extends continuously to clos(X ).

Lemma

G ⊂ Rq × Rn definable such that, for every t in Rq, the dimensionof Gt ⊂ Rn is < n. Then there is a polynomial automorphism u ofRn such that the restriction to G of the projectionRq × Rn → Rq × Rn−1 is finite-to-one.

If q = 0, the change of coordinates u can be taken linear.




Triangulation of definable sets

Using induction on nand the“good coordinates”lemma for q = 0,one obtains:

Theorem

Let A be a closed and bounded definable subset of Rn and Bi ,i = 1, . . . , k, definable subsets of A. Then there exist a finitesimplicial complex K with vertices in Qn and a definablehomeomorphism Φ : |K |R → A such that each Bi is a union ofimages by Φ of open simplices of K .




Triangulation of definable functions

Still using induction on n and the “good coordinates” lemma (nowwith q = 1) for the graph of a function, we obtain:

Theorem

Let X ⊂ Rn be definable, closed and bounded, f : X → Rcontinuous definable. Then there exist a finite simplicial complexK in Rn+1 and a definable homeomorphism ρ : |K |R → X suchthat f ◦ ρ is an affine function on each simplex of K.Given B1, . . . ,Bk ⊂ X definable, we may choose the triangulationso that each Bi is a union of images of open simplices of K .

Beware! No triangulation of definable continuous maps X → Y ifdim Y > 1. Ex.: show that f : [0, 1]2 → R2 defined byf (x , y) = (x , xy) cannot be triangulated.




Local conic structure

Theorem

Let A ⊂ Rn be a closed definable set, a ∈ A. There is r > 0 suchthat there exists a definable homeomorphism h from the cone withvertex a and base S(a, r) ∩ A onto B(a, r) ∩ A, satisfyingh|S(a,r)∩A = Id and ‖h(x)− a‖ = ‖x − a‖ for all x in the cone.

Easily obtained by triangulating the function x 7→ ‖x − a‖.


Ideal pointsResidual o-minimal structure at an ideal point

Ideal points as generic pointsTriviality theorems

Tutorial 3

First two lectures: o-minimal structures, Cylindrical DefinableCell Decomposition, tameness properties, triangulation.

Today: uniformity in definable families X ⊂ Rm × Rn, using“generic fibers” Xα at “ideal points” α of the parameterspace.

7 Ideal points

8 Residual o-minimal structure at an ideal point

9 Ideal points as generic points

10 Triviality theorems




Ultrafilters

Ideal point α of Rm = ultrafilter of the Boolean algebra Sm.

1 Rm ∈ α2 A ∩ B ∈ α if and only if A ∈ α and B ∈ α3 ∅ 6∈ α4 A ∪ B ∈ α if and only if A ∈ α or B ∈ α

Notation: Rm for the set of ideal points. Note Rm ⊂ Rm.Examples:

in R: {A ∈ S1 | ∃a ∈ R (a,+∞) ⊂ A}; this ideal point maybe called +∞.

in R2 : ultrafilter generated by all curvilinear triangles{(x , y) ∈ R2 | 0 < x < a, 0 < y < f (x)} where a > 0 andf : (0, a)→ (0,+∞) definable.




Topologies on Rm

Notation: for A ∈ Sm, set A = {α ∈ Rm | A ∈ α}.Stone space of the Boolean algebra. Compact, Hausdorff, totally

disconnected. Clopens: all A for A ∈ Sm.

Stone space of the lattice of definable opens. Compact, notHausdorff (spectral space). Compact opens: all U forU ⊂ Rm open definable.

In the semialgebraic case, ideal points = prime cones ofR[X1, . . . ,Xm] = couples (prime p, ordering of k(p)). The real

spectrum of R[X1, . . . ,Xm] is Rm with the second topology.




Residual field at an ideal point

For α ∈ Rm, the elements of κ(α) are definable functionsf : A→ R where A ∈ α, modulo identification of f and gwhen they coincide on some B ∈ α (germs of definablefunctions along α). Denote by f (α) ∈ κ(α) the class of f .

κ(α) is a real closed field.

If α = t ∈ Rm, then κ(α) = R.

In the semialgebraic case, κ(+∞) is the field of algebraicPuiseux series in 1/x . More generally, if α = (p,≤), thenκ(α) is the real closure of k(p) for ≤.




Fiber of a definable family at an ideal point

X ⊂ Rm × Rn definable = definable family of subsets of Rn

parametrized by Rm. If A ⊂ Rm, set X |A = X ∩ (A× Rn).

If t ∈ Rm, Xt = {x ∈ Rm | (t, x) ∈ X}.For α ∈ Rm, define the fiber Xα to be the set of f (α) ∈ κ(α)m

such that there exists A ∈ α with (t, f (t)) ∈ X for all t ∈ A.

Set Sn(α) to be the set of fibers Xα for all definable familiesX ⊂ Rm × Rn.




Residual o-minimal structure

Theorem

The collection (Sn(α))n∈N is an o-minimal structure expanding thereal closed field κ(α).

Main points of the proof:

stability by projection: definable choice implies that theprojection of the fiber at α is the fiber at α of the projection.

elements of S1(α) are unions of points and intervals in κ(α):CDCD in Rm × R.

The extension of a definable subset A ⊂ Rn to κ(α) isAκ(α) = (Rm × A)α.From a model-theoretic point of view, κ(α) is an elementaryextension of the o-minimal structure on R.




What means validity at an ideal point?

In algebraic geometry, the properties of the generic fiber holdfor almost all fibers.Here : a property (expressible by a first-order formula) holdsfor Xα iff there exists A ∈ α such that it holds for all Xt witht ∈ A.For instance (property = being the graph of a function):definable functions Yα → Zα are precisely the fibers at α ofdefinable families of functions, i.e. definable functionsf : Y → Z commuting with projection to Rm :

Y -fZ

@@@@R

��

��Rm




Fiberwise and global properties

If Xα ⊂ κ(α)n, we know that there exists A ∈ α such that Xt ⊂ Rn

is closed for every t ∈ A. Actually, someting better holds:

Theorem

Let X ⊂ Y be definable families. The fiber Xα is closed in Yα ifand only if there exists A ∈ α such that X |A is closed in Y |A.Let f : X → Y be a definable family of maps. Then fα iscontinuous if and only if there exists A ∈ α such thatf |A : X |A → Y |A is continuous.

Exercise: Suppose f : X → Y is a definable family such that ft iscontinuous for each t ∈ Rm. Then there exists a finite definablepartition Rm =

⋃Ci such that f |Ci

is continuous for each i . (Hint:

use compactness of Rm).




Trivialisation

Let A ⊂ Rm definable. The definable family X ⊂ Rm × Rn is saidto be definably trivial over A if there exist a definable set F and adefinable homeomorphism h : A× F → X |A such that thefolllowing diagram commutes:

A× F -hXA

@@@@R

projection��

��

projection

A ⊂ Rm

The trivialization h is said to be compatible with a definable subsetY ⊂ X if there is a definable subset G of F such thath(A× G ) = Y |A.




Hardt’s theorem

Theorem (Hardt’s Theorem for Definable Families)

Let X ⊂ Rm × Rn be a definable family, Y1, . . . ,Y` definablesubsets of X . There exists a finite partition of Rm into definablesets C1, . . . , Ck such that X is definably trivial over each Ci and,moreover, the trivializations over each Ci are compatible withY1, . . . ,Y`.

Proof:

Can assume Xt closed and bounded for every t.

For every α ∈ Rm, triangulate Xα: hα : |K |κ(α) → Xα.

This gives a trivialisation h : A× |K |R → X |A for some A ∈ α.

Hardt’s theorem follows using the compactness of Rm.




Finiteness of topological types in definable families

Example: given n and d , there are finitely many topologicaltypes of V ⊂ Rn where V is an algebraic subset described byequations of degrees ≤ d .

There is also a “Hardt’s theorem” for families of definablefunctions (with values in R), using the triangulation offunctions.

Example of consequence: finiteness of topological types offewnomials. For n and p fixed, there is a finite number oftopological types of fewnomials Rn → R with ≤ p monomials(whatever are the degrees). These fewnomials can be put in adefinable family for the o-minimal structure Rexp.

There are semialgebraic families of maps R2 → R2 withinfinitely many topological types.


Complexity and familiesEffectiveness/non effectiveness of uniform bounds

Effectiveness in proofs of o-minimality

Tutorial 4

Yesterday: how to obtain triviality results for definable families(Hardt’s theorem) almost for free from triangulation theorems,using ideal points of the parameter space.Today: relations between complexity and families, some effectivebounds including a constant depending on the family. Problem ofeffectiveness of uniform bounds. Finally, a few words concerningeffectiveness issues in proving that a given structure is o-minimal.

11 Complexity and families

12 Effectiveness/non effectiveness of uniform bounds

13 Effectiveness in proofs of o-minimality




Complexity and families

Say a semialgebraic set in Rn has complexity ≤ (c , d) if it can bedescribed by a boolean combination of sign conditions on ≤ cpolynomials. of degrees ≤ d . Then:

Given a semialgebraic family, there is (c , d) such that everysemaialgebraic set in the family has complexity ≤ (c , d).

Semialgebraic subsets of Rn of complexity ≤ (c, d) form asemialgebraic family.

So uniform bounds in term of complexity = uniform bounds infamilies.




A metric uniform bound

Theorem

Let X ⊂ Rm × R2 be a family of curves, definable for someo-minimal structure. Then there is a constant c(X ) dependingonly on the family X such that, for every disk D(a, r) ⊂ R2 andevery t ∈ Rm, one has length(Xt ∩ D(a, r)) ≤ c(X ) r .

Proof: Uniform finiteness for the definable family ofXt ∩ D(a, r) ∩ L (L a line) parameterized by t, a, r and L +Cauchy-Crofton formula which computes the length of a curve byintegrating over the Grassmannian of lines its number ofintersections with a line.




Combinatorial complexity (S. Basu)

If X1, . . . ,Xs are definable subsets of Rm, denote by C(X1, . . . ,Xs)the collection of all non empty intersections of the formY1 ∩ . . . ∩ Ys where Yi = Xi or Yi = Rm \ Xi .

Theorem

Let X ⊂ Rm × Rn be a definable family for some o-minimalstructure. Then there is a constant c(X ) depending only on thefamily X such that, for every positive integer s and every t1, . . . , tsin Rm, we have ∑

Y∈C(Xt1 ,...,Xts )

βi (Y ) ≤ c(X ) sm−i

S. Basu, Proc. London Math. Soc. (2010) 100: 405-428.




h-cobordism

Figure: Cobordismbetween M and N

A cobordism between M and Nis a compact manifold W withboundary equal to the disjointunion of M and N.

It is called h-cobordism if bothM and N are deformationretract of W .

Theorem (Smale, 1961)

Let (W ,M,N) be a simplyconnected h-cobordism, dim W ≥ 6.Then W ' M × [0, 1].




Semialgebraic h-cobordism with complexity

Theorem

There exists Ψ(m, n, p, q) ∈ N2 such that for every simplyconnected semialgebraic h-cobordism (W ,M,N) in Rn withdim(W ) = m ≥ 6, of complexity ≤ (p, q), there exists asemialgebraic homeomorphism h : W → M × [0, 1] of complexity≤ Ψ(m, n, p, q).

Sketch of proof:

Put all semialgebraic triples (W ,M,N) of subsets of Rn ofcomplexity ≤ (p, q) in a semialgebraic family.

Apply Hardt’s trivialisation theorem and retain the pieces ofthe parameter space over which the assumptions of the PL-hcobordism theorem are fulfilled.

Over each piece, apply PL-h cobordism theorem to one fiber.




Non-effectiveness of the bound

Theorem

Ψ(m, n, p, q) cannot be recursive.

Proof. Reduction of the problem ofrecognizing PL balls to h-cobordism:K a simplicial complex such that |K | isa PL m-ball. There is a subdivision K ′

of K simplicially isomorphic to asubdivision of the m-simplex; for m ≥ 6,]K ′ cannot be bounded by a recursivefunction of ]K .

K. Demdah, to appear in Ann. Inst. Fourier




Proving o-minimality

Usual tools for proving that a structure (presented by a familyof functions) is o-minimal are quantifier elimination and modelcompleteness.

Quantifier elimination: every definable set may be describedby a quantifier free formula (combination of f > 0, f = 0).Example: semialgebraic sets, Tarski-Seidenberg theorem.Then one has to check that such a set has finitely manyconnected components.

Model-completeness: every definable set may be described byan existential formula (projection of a set defined by acombination of f > 0, f = 0). Example: subanalytic sets,theorem of the complement (A. Gabrielov). It is then sufficientto check that sets defined by a combination of f > 0, f = 0have finitely many connected components (Khovanskii, . . . ).




Algorithmic issues

The case of quantifier elimination for real closed fields is wellknown.

Effective (algorithmic) aspects of the theorem of thecomplement for a specified structure? Gabrielov, Wilkie,Macintyre, Speisseger, Rolin. . .


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