Betti numbers of semi-Pfaffian sets

Journal of Pure and Applied Algebra 139 (1999) 323–338www.elsevier.com/locate/jpaa

Betti numbers of semi-Pfa�an sets

Thierry ZellUniversite de Rennes I, IRMAR, Campus de Beaulieu, 35042 Rennes Cedex, France

Received 15 February 1998; received in revised form 1 September 1998

Abstract

We consider semi-Pfa�an sets de�ned as closed sign conditions on a compact Pfa�an varietyV of dimension k ′ in Rk : If we consider a Pfa�an chain (f1; : : : ; f‘) of length ‘ and degree � onits domain U; we prove that all compact semi-Pfa�an sets de�ned on V by a sign condition ona family {p1; : : : ; ps} of Pfa�an functions that are polynomials of degree � in the Pfa�an chain(f1; : : : ; f‘); have the sum of their Betti numbers bounded by sk

′2‘(‘−1)=2 O(k�+min(‘; k)�)k+‘:

c© 1999 Elsevier Science B.V. All rights reserved.

MSC: 68Q25; 14P15

Semi-Pfa�an sets are a generalization of semi-algebraic sets, where polynomials arereplaced by analytic ‘Pfa�an’ functions, which are solutions of triangular systems ofpolynomial di�erential equations. These functions form an important class that containsmany usual functions.

Pfa�an functions were �rst introduced by Khovansk��i [19], who proved an ana-logue of the B�ezout theorem for them and deduced from it �niteness results for manytopological and geometrical characteristics of semi-Pfa�an sets, such as the number ofconnected components or the sum of their Betti numbers.

As in the algebraic case, the �niteness properties can be e�ectively estimated inthe Pfa�an setting, using a suitable notion of complexity: Gabrielov and Vorobjovintroduced this notion of Pfa�an complexity in [17].

Let (f1; : : : ; f‘) be a sequence of real analytic functions de�ned on subsets of Rk ;and call X the set of points in Rk where Taylor expansion of each fi has a non-zero

E-mail address: [email protected] (T. Zell)

0022-4049/99/$ – see front matter c© 1999 Elsevier Science B.V. All rights reserved.PII: S0022 -4049(99)00017 -1

324 T. Zell / Journal of Pure and Applied Algebra 139 (1999) 323–338

convergence radius. Let U be a connected component of X; the sequence (f1; : : : ; f‘)is a Pfa�an chain in U if there exists polynomials Pi;j such that the system:

@fi@xj

(x) = Pi;j(x; f1(x); : : : ; fi(x)); 1 ≤ i ≤ ‘; 1 ≤ j ≤ k;

holds for all x ∈ U: The set U is called the domain of the Pfa�an chain.We say that (f1; : : : ; f‘) has degree � if the degrees of the polynomials Pi;j are

bounded by �; and we call ‘ the length of the Pfa�an chain. The Pfa�an functionsin the chain (f1; : : : ; f‘) are by de�nition the functions that can be expressed in U aspolynomials in (x; f1; : : : ; f‘): The degree of a Pfa�an function in the chain (f1; : : : ; f‘)is the degree of its de�ning polynomial. We then call Pfa�an complexity of such afunction the triplet (‘; �; �):

This notion of complexity has been used in [17] to give estimates on weak strati�ca-tions of semi-Pfa�an sets, in [14] to bound the multiplicities of Pfa�an intersectionsand evaluate the Lojasiewicz exponents for semi-Pfa�an sets, and in [16], where adescription of the frontier of a semi-Pfa�an set is given in terms of the functionsde�ning the set.

Based on results from [19], we use Pfa�an complexity to bound the sum of theBetti numbers of compact semi-Pfa�an sets over compact Pfa�an varieties.

For a given Pfa�an chain (f1; : : : ; f‘); having U as a domain, we can de�ne thesemi-Pfa�an sets as the subsets of U given by sign conditions on the Pfa�an functionsthat are polynomials in that chain. If P is a family of s Pfa�an functions that havea degree in the chain (f1; : : : ; f‘) that is bounded by �; and S is a semi-Pfa�an setde�ned by sign conditions on P; we call (s; ‘; �; �) the format of S:

The �rst section elaborates on this introduction to give the more detailed de�nitionsand precise results we will use in this paper.

Using the method introduced by Thom [24] and Milnor [22] for real algebraic sets,Section 2 establishes a bound for compact Pfa�an varieties. If V is a compact Pfa�anvariety in U given by Pfa�an equations of complexity (‘; �; �); we show that the sumof its Betti numbers is bounded by 2‘(‘−1)=2 O(k� + min(‘; k)�)k+‘:

In Section 3, we show how to reduce the case of a general compact semi-Pfa�anset to the case of one in general position.

This then leads us onto our main result – in Section 4 – a combinatorial bound onthe Betti numbers of compact semi-Pfa�an sets, deducted from the bound on compactvarieties. We adapt the method Basu used in the semi-algebraic case [1] and the resultsof Section 2 to get the following:

Main result. Let S be a compact semi-Pfa�an set S in U⊂Rk ; given on a compactPfa�an variety V of dimension k ′ by sign conditions on a family P of s Pfa�anfunction. If all the functions de�ning S have a Pfa�an complexity that is at most(‘; �; �); the sum of the Betti numbers of S is:

B(S) = sk′

2‘(‘−1)=2 O(k� + min(‘; k)�) k+‘:

T. Zell / Journal of Pure and Applied Algebra 139 (1999) 323–338 325

As in [1], we obtain a bound that has a combinatorial part – sk′

– that depends onlyon the dimension of V and the cardinality of P: This combinatorial part is multipliedby a bound that is linked to the Betti numbers of V; and depends only on the Pfa�ancomplexity (‘; �; �) and the dimension k of the ambient space. This particular aspectfor the bound is due to the use of the combinatorial level of P over V (see also[3,4]).

The geometric setting necessary for the proof is exactly the one of o-minimality; (forreference, see any of [8–11]) Little emphasis will be made on this point for the rest ofthis paper, but all the geometric properties of semi-Pfa�an sets that will be needed canbe deduced from arguments of o-minimality for certain structures; the o-minimality ofthe expansion of R by Pfa�an functions was recently proved by Wilkie in [25] (seealso [18,23]).

1. Pfa�an functions and semi-Pfa�an sets

In this section, we go into more details about Pfa�an functions and semi-Pfa�ansets, giving the results we will need further on.

1.1. Pfa�an functions

De�nition 1. Let (f1; : : : ; f‘) be a sequence of real analytic functions in k variables.The sequence satis�es the Pfa�an chain condition if there is a non-empty subset O

of Rk such that for all x ∈ O:

@fi@xj

(x) = Pi;j(x; f1(x); : : : ; fi(x)); (1)

where 1 ≤ i ≤ ‘ and 1 ≤ j ≤ k and the Pi;j are polynomials in k + i variables ofdegree at most �:

If U is a maximal open set for which the Pfa�an condition holds for (f1; : : : ; f‘);the restriction of the sequence to U is called a Pfa�an chain on U; and U is calledthe domain of the Pfa�an chain.

Remark. If we call X the set where all the functions f1; : : : ; f‘ have converging Taylorseries, then U has to be a connected component of X (by the uniqueness of the analyticcontinuation, if (1) holds on a neighbourhood of some point x; it holds on the connectedcomponent of x in X). What we should stress at this point is that according to ourde�nition, if U1 and U2 are two di�erent connected components of X where (1) holds,we obtain two di�erent Pfa�an chains by restricting the functions to these domains.To avoid any ambiguity, we will write (f1; : : : ; f‘)U to say that the Pfa�an chain(f1; : : : ; f‘) is considered on the domain U:

A more general notion of Pfa�an chains arises naturally in [14–19]; sequences ofpolynomial 1-forms replace the equations (1) and Pfa�an chains are de�ned as nested


integral manifolds of this sequence. The two de�nitions give locally the same class offunctions.

De�nition 2. Let (f1; : : : ; f‘) be a Pfa�an chain of length ‘ and degree �; and let Ube its domain. A function q in U is called a Pfa�an function of degree � in the chain(f1; : : : ; f‘) if there exists a polynomial Q in k + ‘ variables of degree at most � suchthat

q(x) = Q(x; f1(x); : : : ; f‘(x)) ∀x ∈ U: (2)

1.2. Examples

1. The polynomials are the Pfa�an functions such that ‘ = 0:2. The exponential function f(x) = ex is Pfa�an, with ‘ = 1 and �= 1; because of

the equation f′ = f:3. The function f(x) = ln(|x|) is Pfa�an in R \ {0} with ‘ = 2 and � = 2: Indeed,g(x) = x−1 is solution of the equation y′ = −y2 on R \ {0}; and (f; g) forms aPfa�an chain with the equation f′ = g:

4. Let n ≥ 1 be an integer, and let t(x) = tan( x2n): This function is solution of theequation dt=dx=(1=2n)(1+t2) on the open set X={x ∈ R: x 6= n�+2kn�; k ∈ Z}:The function

u(x) =1

1 + t(x)2

has for derivativedudx

=dudt

· dtdx

=−1

(1 + t2)2 · 12n

(1 + t2) = − 12nt · u;

so the restriction of (t; u) to any connected component U of X is a Pfa�an chainof length ‘ = 2 and degree �= 2:On U; the function cn(x) = cos(x=n) veri�es:

cos( xn

)=

1 − t21 + t2

= (1 − t2) · u;

so cn(x) is Pfa�an with a degree � = 3 in the chain (t; u)U: If Tn is the nth

Chebyshev polynomial of the �rst kind, then we have cos(x) = Tn(cn(x)) on U;and this proves that cos (x) is Pfa�an with degree � = 3n in the chain (t; u)U:

5. Monomials are also Pfa�an functions with ‘ = 2 and � = 2 on (−∞; 0) and on(0;+∞); the sequence (x−1; xm) forming a Pfa�an chain on each of the domainsbecause of the equations d(x−1) = (x−1)2 dx and d(xm) = m · x−1 · xm dx:

Here is a more general result taken from [19]. Consider the following functions(in any �nite number of variables): polynomials, exponentials, trigonometric functionsand their composition inverses wherever applicable. Then, consider the real elemen-tary functions, the class obtained from these by taking the closure under arithmetical


operations and composition. If f is such a function where sin and cos appear onlythrough their restriction to bounded intervals, and if D is the set where f is properlyde�ned, then f is Pfa�an on any connected component of D: Note that because ofthe �niteness results that will be exposed later on, sin and cos are not Pfa�an on thewhole real line.

Proposition 1. The set of all Pfa�an functions on U is a subalgebra of A(U); thealgebra of analytic functions in U; that is closed under di�erentiation. Moreover; thedegree of the result of all the operations can be e�ectively estimated from the degreesof the functions operated on.

The proof of this proposition is very straightforward, and can be found in [17].Notice however that in order to prove this proposition, we can work in the setting of a�xed Pfa�an chain: if q1 and q2 are Pfa�an functions in two di�erent Pfa�an chains(f1; : : : f‘1 ) and (g1; : : : g‘2 ); then (f1; : : : f‘1 ; g1; : : : ; g‘2 ) is a Pfa�an chain in whichq1; q2; q1 + q2 and q1 · q2 can be expressed. In a similar fashion, the partial derivativesof a Pfa�an function q can be expressed in the same chain as q: This justi�es thefact that later on, we will consider semi-Pfa�an sets de�ned using functions that arepolynomials in a given chain.

Khovansk��i introduced Pfa�an functions while working on generalized versions ofthe Rolle theorem. This enabled him in [19] to prove the following: a system of k Pfaf-�an equations in k variables q1(x)=· · ·=qk(x)=0 has only �nitely many non-degenerateroots – that is, there is �nitely many x ∈ U such that q1(x)= · · ·=qk(x)=0 and the Ja-

cobian matrix(@qi@xj

(x))

1≤i;j≤kis nonsingular. This number of non-degenerate solutions

can be expressed in terms of the Pfa�an complexity.

Theorem 1 (Generalized B�ezout theorem). Let q1; : : : ; qk be Pfa�an functions of de-gree � in a common Pfa�an chain of length ‘ and degree �: The number ofnon-degenerate solutions in U of the system q1(x) = · · · = qk(x) = 0 is bounded by

�kO(k� + min(‘; k)�)‘2‘(‘−1)=2: (3)

This result appears in chapter 3 of [19], as an example of the application of more gen-eral estimates on integral manifolds of di�erential forms with polynomial co-e�cients.

Remark. For the statement of this theorem, it is very important to place oneself inthe domain U of the chain. As seen from example 4, there are sequences of analyticfunctions that satisfy the Pfa�an chain condition on an open set O that has in�nitelymany connected components. This means that we can construct from these sequencesin�nitely many Pfa�an chains, but the �niteness result cannot hold on the whole set O:

As explained in [19], this theorem lets us prove good topological properties for setsde�ned using Pfa�an functions: they have a �nite number of connected components,


�nite Betti numbers, : : : It’s also very important in proving that the expansion of thereal �eld by Pfa�an functions is o-minimal [25]. The property of o-minimality is itselfclosely linked with the ‘good’ topology for de�nable sets [8–11].

These considerations lead us to our next result, which says that a Pfa�an functionq : U → R has only �nitely many critical values. This statement has to be comparedwith Sard’s lemma, which says that the critical values of such a function q is ofmeasure zero; here, the topology of semi-Pfa�an sets lets us improve on the initialresult.

Lemma 1 (Sard’s lemma). Let q : U → R be a Pfa�an function. Then q has only�nitely many critical values.

Proof. Let C be the set of critical points of q: It is given by @q=@x1 = · · ·= @q=xk = 0;so it is a Pfa�an variety.

Let X be a connected component of C and let x and y be points in X: By thecurve selection lemma, there is a smooth arc contained in X joining x to y: As allpoints of this arc are in C; the mean value theorem shows that the restriction of q to is constant, so q is constant on X: The set C; has only a �nite number of connectedcomponents, so q has only a �nite number of critical values.

1.3. Semi-Pfa�an sets

We will now precise the notions of Pfa�an varieties and semi-Pfa�an sets. Let(f1; : : : ; f‘) be a Pfa�an chain of length ‘ and degree �; and let U be its domain ofde�nition.

De�nition 3. The set V⊂U is a Pfa�an variety if there are Pfa�an functions q1; : : : ; qrin the chain (f1; : : : ; f‘) such that V = {x ∈ U : q1(x) = · · · = qr(x) = 0}:

Remark. It will be convenient for us to assume from now on that all Pfa�an varietiesare given by a single equation {q= 0} with q ≥ 0 on U: This can be done by takingq= q2

1 + · · · + q2r :

De�nition 4. A basic semi-Pfa�an set S on the variety V is a set given by the signconditions:

S = {x ∈ V : p1(x)�10 ∧ · · · ∧ ps(x)�s0};where p1; : : : ; ps are Pfa�an functions in (f1; : : : ; f‘) and �1; : : : ; �s are any sign con-dition from {〈; 〉;≤;≥;=}:

A semi-Pfa�an set on V is any �nite union of basic semi-Pfa�an sets on V:

De�nition 5. If S is a semi-Pfa�an set on the Pfa�an variety V= {q= 0} given bysign conditions on a family P= {p1; : : : ; ps} such that the functions q; p1; : : : ; ps havedegree at most � in the chain (f1; : : : ; f‘); we call (s; ‘; �; �) the format of S:


Notation. For any topological space X ⊂Rk ; we take Hi(X ) to be the i-th �Cech ho-mology group of X; and Bi(X ) to be the rank of Hi(X ): We write B(X ) for the sumB1(X ) + · · · + Bk(X ):

Remark. As semi-Pfa�an sets are triangulable, considering their Betti numbers interms of �Cech homology or in terms of singular homology does not make any di�erence(see [12]).

2. A bound for compact Pfa�an varieties

In this section, we shall show that the sum of the Betti numbers of a compact Pfa�anvariety is bounded by 2‘(‘−1)=2O(k� + min(‘; k)�)k+‘; if (‘; �; �) is the format of thisvariety. The method is inspired from the Thom–Milnor bounds for real algebraic sets,and the presentation owes a lot to [5]. (See also [6, 22, 24].)

Let (f1; : : : ; f‘) be a Pfa�an chain with a domain of de�nition U: Let V be acompact Pfa�an variety in U⊂Rk ; given by a single equation {q= 0}; with q(x) ≥ 0everywhere.

Let Kr = {x ∈ U: q(x) ≤ r}: According to Lemma 1, q has only a �nite number ofcritical values, and so the Kr are smooth manifolds with boundaries for all but �nitelymany values of r: Let K∗

r ⊂Kr be the union of the connected components of Kr thatintersect V: We want to show that B(V) is equal to B(K∗

r ) for small values of r:We shall start by proving that K∗

r is compact if r is small enough.

Lemma 2. Let dV(x) be the distance of x to V; and for all �¿ 0; let T (�) be theset of all x in U such that dV(x) ≤ �: There exists �1¿ 0 such that K∗

r = Kr ∩ T (�1)for all su�ciently small values of r:

Proof. As V is compact, it does not meet the boundary of U: If the distance of V

to @U is greater than �0; then T (�) is compact for all � ≤ �0:As dV is subanalytic, we can apply the Lojasiewicz inequality to T (�0) and V: So,

there is a real number a¿ 0 and a positive integer n such that:

q(x) ≥ a · dV(x)n ∀x ∈ T (�0):

Note that we can chose a so that this inequality is strict for x outside of V: Then, forx in @T (�0); the inequality becomes: q(x)¿a ·�n0: This means no connected componentof K = Ka�n0 meets @T (�0): As K has a �nite number of connected components, a�1 ≤ �0 can be found such that T (�1) contains only the connected components of K

that are in K∗: As K∗r is contained in K∗ for all r ≤ a�n0; it follows that K∗

r ⊂T (�1)for all r such that r ≤ a�n0; which yields the lemma.

Remark. It is important to consider K∗r ; since Kr itself is not necessarily compact. The

following example comes from [5]. Let P : R2 → R be the map:

P(x; y) = (x2 + y2)((y(x2 + 1) − 1)2 + y2):


P−1(0) = {0} is compact, but as P(x; (1 + x2)−1) goes to 0 as x goes to in�nity, thesets {P ≤ r} are not bounded for r ¿ 0:

Lemma 3. For small r ¿ 0; K∗r and V have same homology groups.

Proof. Note that K∗r and V are both triangulable, so that the singular and the �Cech

homology coincide on them.Fix r0¿ 0 such that r0 is inferior to all positive critical values of q; and such that

K∗r0 is compact. Then, by a standard argument of Morse theory, K∗

r is a retract of K∗r0

for all 0¡r¡r0: As V and the K∗r are compact and V =

⋂0¡r¡r0 K

∗r ; the �Cech

homology groups of V are the inverse limit of the �Cech homology groups of K∗r

(see [12]). This proves that H (V) ∼= H (K∗r0 ) as the groups H (K∗

r ) are isomorphic forr ≤ r0:

Lemma 4. Let K = K∗r : Then B(@K) = 2B(K):

Proof. Let Kc = Rk \ K: The Mayer-Vietoris sequence in reduced homology of(K; Kc) is:

· · · → H̃ i+1(Rk) → H̃ i(@K) → H̃ i(K) ⊕ H̃ i(Kc) → H̃ i(Rk) → · · ·As H̃∗(Rk) = 0; this yields H̃ i(@K) ∼= H̃ i(K) ⊕ H̃ i(Kc); and as @K has a collar in Kc;we have H̃ i(Kc) ∼= H̃ i(Kc):

Alexander duality gives H̃ i(Kc) ∼= �Hk−i−1

(K) ∼= Hom(Hk−i−1(K);R): This yieldsthe relations:

Bi(@K) = Bi(K) + Bk−i−1(K); 1 ≤ i ≤ k − 1; (4)

B0(@K) − 1 = B0(K) − 1 + Bk−1(K): (5)

As dim(V)¡k; we have Bk(V)=Bk(K)=0: For the same reason, we have Bk(@K)=0;so summing all the equalities in (4), we get

B(@K) = 2B(K);

which proves the lemma.

Theorem 2. Let V be a compact Pfa�an variety in U⊂Rk in the format (‘; �; �)given by a single equation {q= 0} where q is nonnegative on U: Then:

B(V) = 2‘(‘−1)=2 O(k� + min(‘; k)�)k+‘: (6)

Proof. According to the results already proved in this section, we can choose r ¿ 0such that K∗

r is a compact smooth manifold with boundary that has the Betti numbersof V: According to Lemma 4, we only need to give an estimate on B(@K∗

r ):Let W = @K∗

r : Up to a rotation of the coordinate system,1 we can assume thatthe projection map � of W on the x1 axis has only non-degenerate critical points

1 Note that this does not alter the format of V:


with distinct critical values. The sum of the Betti numbers of W is bounded by thenumber of these critical points (see [21]), which in turn is bounded by the number of(nondegenerate) solutions of:

q(x) − r =@q@x2

(x) = · · · =@q@xk

(x) = 0;

which is a system of Pfa�an equations with degree bounded by �+ � − 1: Using theKhovansk��i bound given in Theorem 1, B(V) follows.

Remark. In the algebraic case, the bound can easily be extended to the case whereV is not compact considering the intersection of V with a ball of large radius. Inour case, it is still true that any Pfa�an variety V has the same homology as itsintersection with a ball of su�ciently large radius, but the presence of the domain ofde�nition U poses a problem: if that ball is not entirely inside U; that intersection maynot be compact, and we cannot make the reduction to the compact case that way.

3. Going to general position

In this section, we show how to reduce the case of any compact semi-Pfa�an setto a problem where the functions are in general position.

Let V = {q= 0} be a compact Pfa�an variety, P = {p1; : : : ; ps} be Pfa�an func-tions and S be a compact semi-Pfa�an set in V de�ned by sign conditions �(P) indisjunctive normal form, meaning � =

∨Li=1 �i; where �i(P) is of the form:

�i(P) = p1�10 ∧ · · · ∧ ps�s0; �j ∈ {≤;=;≥}:We will de�ne here the notion of combinatorial level of a semi-Pfa�an set. This

notion, that is actually a purely combinatorial concept, was introduced by Basu in hisPh. D. thesis [2] for the semi-algebraic case.

De�nition 6. The combinatorial level of the system (q;P) is the largest integer msuch that there exists x in V and m functions in P vanishing at x: The set P is saidto be in general position with V if the combinatorial level of (q;P) is bounded bydim(V):

We will prove now that for any compact semi-Pfa�an set, we can �nd a set that isde�ned with a system in general position and has the same homology groups:

Proposition 2. Let S be a compact semi-Pfa�an set on V given by sign conditionson P={p1; : : : ; ps}: Then; there exists a compact semi-Pfa�an set S∗ on V de�nedwith sign conditions on a family P∗ of 2s Pfa�an functions; such that S∗ has samehomology as S and P∗ is in general position with V:

Proof. We know from [15] that V is the union of a �nite number of smooth manifoldsthat can be given by equations with functions from the algebra generated by q and its


partial derivatives of all orders. Let V� be the union of all strata of dimension �: Wewill prove a slightly stronger result, that P∗ can be chosen in general position withV� for �= 0; : : : ; k; i.e. the combinatorial level for P∗ will be at most � on V�:

Let p� be the restriction of the mapping (p1; : : : ; ps) to V�: Take t = (t1; : : : ; ts) in(0;+∞) s such that if

C = {(|y1|; : : : ; |ys|): ∃�; y is a critical value of p�};then [0; t] ∩C has measure zero (for the Lebesgue measure on [0; t]). It is possible to�nd such a t because – according to Sard’s lemma – the set C is a �nite union of setsof measure zero.

Let P�={p1 +�t1; p1−�t1; : : : ; ps+�ts; ps−�ts}; for all �¿ 0; and let, for 1 ≤ i ≤ L;��i (P�) be the sign condition built from �i(P) using the following replacement rules:

pi ≥ 0 → pi ≥ −�ti;pi ≤ 0 → pi ≤ �ti;pi = 0 → −�ti ≤ pi ≤ �ti:

Then, let S� be the set:

S� = {x ∈ V : ��1(P�) ∨ · · · ∨ ��L(P�)}:Note that all the sets S� are compact and that S =

⋂�¿0 S�: Using the o-minimality

of the expansion by Pfa�an functions [25,18,23], the generic triviality theorem [10]applied to the map S� 7→ � yields a decomposition of R into de�nable cells over whichthe topological type of S� is constant. In particular, there exists �0¿ 0 such that thetopological type of S� is constant over (0; �0]:

As the homology groups of S are the projective limits of the homology groups ofS�; (see [12]) S has the same homology as the S�s for � ∈ (0; �0]: As the segment[0; �0t] cannot be contained in C; there exists � ≤ �0 such that P� is in general positionwith V; and we can take S∗ = S�:

Remark. As the proof is, we only require the generic triviality theorem for Ran ; theexpansion of the real �eld by restricted analytic functions. This is the structure obtainedby adding on to (R;¡ ;+; ·) all functions f :Rn → R that are restrictions of analyticfunctions on [−1; 1]n and identically zero outside that cube. The set V being compact,it is de�nable in that structure. The o-minimality of Ran is a consequence of its modelcompleteness [13,15] and results from Lojasiewicz [20] (see also [7,8]).

4. Betti numbers of compact semi-Pfa�an sets

4.1. Preliminary results

We will start with some easy statements from algebraic topology.


Proposition 3. Let S1 and S2 be two compact semi-Pfa�an sets. We have the fol-lowing inequalities:

B(S1) + B(S2) ≤ B(S1 ∪S2) + B(S1 ∩S2); (7)

B(S1 ∪S2) ≤ B(S1) + B(S2) + B(S1 ∩S2): (8)

Proof. These inequalities follow from the Mayer-Vietoris sequence of the pair(S1;S2):

Proposition 4. Let S be a semi-Pfa�an set given by a set of sign conditions∨Li=1

�i(P) on a Pfa�an variety V; with the �i(P) being distinct conjunctions of strictinequalities on the elements of P; and for all 1 ≤ i ≤ L; let Si = {x ∈ V: �i(P)(x)}:Then; we have

H∗(S) =L⊕i=1H∗(Si):

Proof. This can be easily seen as – the �is being mutually exclusive – the singularchain complex of S is the direct sum of the chain complexes of the Sis: Note thatthis yields B(S) = B(S1) + · · · + B(SL):

We will now prove a bound for the case of a semi-Pfa�an set given by a conjunctionof strict sign conditions on P: This bound is slightly better than the one in the generalcase, and we will need it to prove the main theorem.

Lemma 5. Let S be a semi-Pfa�an set onV given by the sign conditions p1¿ 0; : : : ;ps ¿ 0; and let the combinatorial level of the system (q;P) be bounded by m ≤ k:Then;

B(S) =(sm

)2‘(‘−1)=2 O(k� + min(‘; k)�)k+‘:

Proof. Let S� be the set:

S� = {x ∈ U : q(x) = 0 ∧ p1(x) ≥ � ∧ · · · ∧ ps(x) ≥ �}:For small enough �¿ 0; the set S� has same homology as S: Indeed, as S=

⋃t¿0 St�;

the homology of S is the inductive limit of the homology of St�: And if � is smallerthan the least positive critical value of the pis; it is clear that St� is a deformationretract of S� for all t ∈ (0; 1); showing that the inductive limit is ultimately constant.

Consider now the sets:

T = {q= 0; p2 ≥ �; : : : ; ps ≥ �}⊃S�;

S−� = T ∩ {p1 ≤ −�};

U = S� ∪S−� ;


V = T ∩ {−� ≤ p1 ≤ �};W+ = T ∩ {p1 = �};W− = T ∩ {p1 = −�};W =W+ ∪W−:

As T = U ∪ V and W = U ∩ V and U and V are compact, the inequality (7) gives:

B(U ) + B(V ) ≤ B(T ) + B(W ):

As the union U =S�∪S−� ; is a disjoint union, we have B(U ) =B(S�) +B(S−

� ): Thisyields:

B(S) = B(S�) ≤ B(U ) ≤ B(T ) + B(W ): (9)

From [19], we know that B(S) can e�ectively be bounded with a function of the formatof S: Let B(s; m) be the maximum of B(S�) for S� running over all semi-Pfa�an setsof format(s; ‘; �; �) de�ned as in the statement of the lemma, and with a combinatoriallevel of m:

By de�nition, T is such a set with combinatorial complexity m and s−1 inequalities,and W is the disjoint union of two such sets, with combinatorial level m− 1 and s− 1inequalities. Using (9), we obtain for B:

B(s; m) ≤ B(s− 1; m) + 2B(s− 1; m− 1): (10)

When s = 0; we have no inequalities, so S = V; and when m = 0; the pis haveconstant sign on V; so that S=V if they are all positive and S=∅ otherwise. Usingthe results from Section 2, this gives for (10) the initial conditions:

B(0; m) = B(s; 0) = 2‘(‘−1)=2 O(k� + min(‘; k)�)k+‘: (11)

To end the proof of this lemma, we shall prove by induction on s that B(s; m) =

2m(sm

)2‘(‘−1)=2 O(k� + min(‘; k)�)k+‘:

The case s = 0 is just the initial condition (11). Assume the result is true for allintegers up to s− 1: We have

B(s; m) ≤B(s− 1; m) + 2B(s− 1; m− 1)

≤ 2m(s− 1m

)2‘(‘−1)=2 O(k� + min(‘; k)�)k+‘

+ 2 · 2m−1(s− 1m− 1

)2‘(‘−1)=2 O(k� + min(‘; k)�)k+‘

≤ 2m( sm

)2‘(‘−1)=2 O(k� + min(‘; k)�)k+‘ (from the Newton identity):

As we assumed m ≤ k; the lemma follows.


4.2. The main result

The following theorem will imply the result stated in the introduction:

Theorem 3. Let S⊂U be a compact semi-Pfa�an set given by sign conditions �(P)on a compact Pfa�an variety V; where V is given by the equation q=0; with q non-negative on U: Assume that the format of S is (s; �; �; ‘); and that the combinatoriallevel of (q;P) is m ≤ k: Then;

B(S) = sm 2‘(‘−1)=2 O(k� + min(‘; k)�)k+‘:

This statement clearly implies the main result as it was announced in the introduction,since according to Proposition 2, the set S is homotopic to a semi-Pfa�an set on V offormat (2s; �; �; ‘); de�ned by Pfa�an functions in general position with q: Combiningthe two, we obtain:

B(S) = sk′

2‘(‘−1)=2 O(k� + min(‘; k)�)k+‘:

Proof of Theorem 3. Let P = {p1; : : : ; ps}: We will start by bounding B(S) withBetti numbers of sets where the sign condition {p1 = 0} does not appear. Assume thatm¡s; and let I = {I = (i1; : : : ; im): 2 ≤ i1¡ · · ·¡is ≤ s}; and for all I ∈ I; letXI = {x ∈ U; q(x) = pi1 (x) = · · · = pim(x) = 0}: If X is the union of the XI s for allI ∈ I; then it is a compact set such that the restriction of p1 to X is never zero. 2

Let �1 = minX |p1|: The set X being compact, we have �1¿ 0: If m= s; we can takeany positive real for �1:

Let 0¡�1¡�1=2: and consider the sets:

U1 = S ∩ ({p1 ≤ −�1} ∪ {p1 ≥ �1});W+

1 = S ∩ {p1 = �1};W−

1 = S ∩ {p1 = −�1};V1 = S ∩ {−�1 ≤ p1 ≤ �1};V ′

1 = S ∩ {p1 = 0}:As S = U1 ∪ V1 and B(W+

1 ∪W−1 ) = B(W+

1 ) + B(W−1 ); we have:

B(S) ≤ B(U1) + B(V1) + B(U1 ∩ V1); from (8);

≤ B(U1) + B(V1) + B(W+1 ) + B(W−

1 ):

We can take �1 small enough so that V1 is homotopic to V ′1 ; and since W+

1 ∩W−1 = ∅;

the last relation can be written as

B(S) ≤ B(U1) + B(V ′1) + B(W+

1 ) + B(W−1 ): (12)

The sets V ′1 ; W

+1 ; and W−

1 are semi-Pfa�an sets with same format as S : V ′1 is a

subset of the variety Z(q; p1) given by sign conditions on P \ {p1}; and W+1 ; and

W−1 are given by sign conditions over the family P on the varieties Z(q; p1 − �1) and

2 Since the combinatorial level of (q;P) is bounded by m:


Z(q; p1 +�1) respectively. It is clear for V ′1 that – if it is not empty – the corresponding

system has a combinatorial level that is at most m − 1: if you could �nd x ∈ V ′1 and

pi1 ; : : : ; pim in {p2; : : : ; ps} such that pi1 (x) = · · · = pim(x) = 0; then x would be suchthat q(x) =p1(x) =pi1 (x) = · · ·=pim(x) = 0: This would contradict the hypothesis thatthe combinatorial level of (q;P) is bounded by m:

According to the choice made for �1; the varieties Z(q; p1 − �1) and Z(q; p1 +�1) do not meet X that is the union of all varieties obtained by setting exactly mof the functions in {p2; : : : ; ps} to zero. This means that W+

1 and W−1 have also a

combinatorial level that is at most m − 1: Note that in the case where m = s; thisdiscussion is not quite correct, but the combinatorial level for W+

1 and W−1 is still

m − 1 all the same: the sign condition p1 = · · · = ps = 0 is obviously empty sincep1 =�1 on W+

1 (and p1 =−�1 on W−1 ), so at most m−1 functions from P can vanish

at the same time.Now, the set U1 is de�ned by sign conditions where the atom p1 =0 does not appear

any more. We can use a similar treatment on this set to eliminate the atom p2 = 0 byde�ning the following sets:

U2 = U1 ∩ ({p2 ≤ −�2} ∪ {p2 ≥ �2});W+

2 = U1 ∩ {p2 = �2};W−

2 = U1 ∩ {p2 = −�2};V2 = U1 ∩ {−�2 ≤ p2 ≤ �2};V ′

2 = U1 ∩ {p2 = 0}:Here, �2 is any positive real smaller than �2=2; where �2 is the minimum of |p2| overall the varieties given by m equations on V chosen among p1 = �1; p1 = −�1; andpi=0 for i ≥ 3: Using the same arguments as in the beginning of the proof, it followsthat

B(U1) ≤ B(U2) + B(V ′2) + B(W+

2 ) + B(W−2 ) (13)

for small enough �2: Note that again, the sets V ′2 ; W

+2 and W−

2 are de�ned with systemsthat have a combinatorial level at most m − 1: Repeating this until all pis have beenprocessed, we end up with a bound of the form:

B(S) ≤ B(Us) +s∑i=1

B(V ′i ) + B(W+

i ) + B(W−i ): (14)

As all sign conditions for Us have atoms of the form pi ≤ −�i or pi ≥ �i; takingthe �is small enough will give a homotopy equivalence between Us and a set de�nedon V by sign conditions of the form pi ¿ 0 or pi ¡ 0: All such sets are subsets of theset �= {q= 0; p2

1¿ 0; : : : ; p2s ¿ 0}; and from Proposition 4, we have B(Us) ≤ B(�):

Lemma 5 yields B(�) =(sm

)2‘(‘−1)=2 O(k� + min(‘; k)�)k+‘:

Let B(s; m) be the maximum of B(S) for a compact semi-Pfa�an set of format(2s; ‘; �; �) and a combinatorial level bounded by m: As all the terms B(V ′

i )+B(W+i )+

B(W−i ) in (14) are sums of Betti numbers of semi-Pfa�an sets with format at most


(2s; �; �; ‘); (the sets V ′i ; W

+i ; and W−

i being subsets of Ui−1; they are given by signconditions that involve pj±�j for j¡ i). Their combinatorial level being at most m−1;we have

B(s; m) =( sm

)2‘(‘−1)=2 O(k� + min(‘; k)�)k+‘ + 3s B(s; m− 1);

B(s; 0) = 2‘(‘−1)=2 O(k� + min(‘; k)�)k+‘:(15)

We shall end this proof by an induction on m to show that B(s; m)=(4s)m 2‘(‘−1)=2O(k� + min(‘; k)�)k+‘:

First, we note that this is true for m= 0: Assume it is true for m− 1; using the �rstequation in (15), we get:

B(s; m) =( sm

)k‘ 2‘(‘−1)=2 O(k� + min(‘; k)�)k+‘

+ 3s(4s)m−1 2‘(‘−1)=2 O(k� + min(‘; k)�)k+‘

=[( sm

)+ 3 · 4m−1sm

]k‘ 2‘(‘−1)=2 O(k� + min(‘; k)�)k+‘:

We can bound the binomial coe�cient with( sm

)=

s!m!(s− m)!

=s(s− 1) · · · (s− m+ 1)

m!≤ sm

m!;

which gives us( sm

)+ 3 · 4m−1sm ≤

(1m!

+ 3 · 4m−1)sm ≤ (4s)m:

This concludes the induction, and to get the result as stated in the theorem, we onlyneed to notice that since we supposed that m ≤ k; we have 4m = O(1)k :

As a special case of this result, a corollary can be proved for semi-algebraic setsde�ned by sparse polynomials, using their representation as Pfa�an functions outsidethe coordinate hyperplanes. This bound was proved in [1].

Corollary 1. Let S be a closed semi-algebraic set on a compact algebraic varietyV⊂Rk ; de�ned by sign conditions on polynomials p1; : : : ; ps: Assume V={q= 0} isof dimension k ′; and that only m distinct monomials appear in {q; p1; : : : ; ps}: Then;

B(S) = s k′

2O(k2m4):

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Betti numbers of semi-Pfaffian sets