Effective cylindrical cell decompositions for restricted sub-Pfaffian sets
aa r X i v : . [ m a t h . L O ] A p r EFFECTIVE CYLINDRICAL CELL DECOMPOSITIONS FORRESTRICTED SUB-PFAFFIAN SETS
GAL BINYAMINI AND NICOLAI VOROBJOV
Abstract.
The o-minimal structure generated by the restricted Pfaffian func-tions, known as restricted sub-Pfaffian sets, admits a natural measure of com-plexity in terms of a format F , recording information like the number of vari-ables and quantifiers involved in the definition of the set, and a degree D record-ing the degrees of the equations involved. Khovanskii and later Gabrielov andVorobjov have established many effective estimates for the geometric complex-ity of sub-Pfaffian sets in terms of these parameters. It is often important inapplications that these estimates are polynomial in D .Despite much research done in this area, it is still not known whether celldecomposition, the foundational operation of o-minimal geometry, preservespolynomial dependence on D . We slightly modify the usual notions of formatand degree and prove that with these revised notions this does in fact hold.As one consequence we also obtain the first polynomial (in D ) upper boundsfor the sum of Betti numbers of sets defined using quantified formulas in therestricted sub-Pfaffian structure. Statement of the main results
Setup.
Let I := [0 , ⊂ R and for k, n ∈ N , k > n , denote by π kn : I k → I n the projection map. We sometimes omit k if its meaning is clear from the context.Pfaffian functions, introduced by Khovanskii in [14, 15], are analytic functionssatisfying triangular systems of Pfaffian (first order partial differential) equationswith polynomial coefficients. We refer the reader to [7] for precise definition ofPfaffian functions, based on Pfaffian chain , and examples of Pfaffian functions inan open domain G ⊂ R k , which we assume here for simplicity to be given by aproduct of intervals. Definition 1 (semi-Pfaffian set) . Let G be an open set in R k and I k ⊂ G . A set X ⊂ I k is called (restricted) semi-Pfaffian if it consists of points in I k satisfyinga Boolean combination of atomic equations and inequalities of the kind f = 0 or f > , where f is a Pfaffian functions defined in G . The format of X is the numberof variables k and the degree of X is the sum of degrees of all the Pfaffian functionsappearing in the atomic formulas (i.e. the degrees of the polynomials defining thesePfaffian functions). Mathematics Subject Classification.
Key words and phrases.
Pfaffian functions, cell decomposition.This research was supported by the ISRAEL SCIENCE FOUNDATION (grant No. 1167/17)and by funding received from the MINERVA Stiftung with the funds from the BMBF of theFederal Republic of Germany. This project has received funding from the European ResearchCouncil (ERC) under the European Union’s Horizon 2020 research and innovation programme(grant agreement No 802107).
Note that the degree of X bounds from above the number of all atomic equationsand inequalities. We assume that a Pfaffian chain has been fixed once and for all,and all semi-Pfaffian sets under consideration are defined from this single Pfaffianchain. Definition 2 (sub-Pfaffian set) . A set Y ⊂ I n is called (restricted) sub-Pfaffian if Y = π kn ( X ) for a semi-Pfaffian set X ⊂ I k . In the special case of a semi-algebraic set X , the Tarski-Seidenberg theoremstates that the set Y = π kn ( X ) is also semi-algebraic, i.e., is a set of points satisfyinga Boolean combination of polynomial equations and inequalities. By contrast, asub-Pfaffian set may not be semi-Pfaffian.It is customary in the literature to define the format and degree of a sub-Pfaffianset as in Definition 2 to be the format and degree of the set X . Below we introducea variant of these notions which turns out to behave better with respect to celldecompositions. To avoid confusion we refer to these modified notions as *-formatand *-degree. Definition 3 (*-format and *-degree of projections) . If { X α } , X α ⊂ I k α is a finitecollection of semi-Pfaffian sets and X ◦ α is a connected component of X α , we definethe *-format of the sub-Pfaffian set Y := S α π k α n ( X ◦ α ) ⊂ I n to be the maximumamong the formats of sets X α , and the *-degree of Y to be the sum of the degreesof sets X α . We remark that since the restricted sub-Pfaffian sets form an o-minimal struc-ture, the connected components of semi-Pfaffian (or even sub-Pfaffian) sets, as wellas their projections, are again sub-Pfaffian, so the sets Y for which *-format and*-degree are introduced in Definition 3 are indeed sub-Pfaffian. When we speakabout a sub-Pfaffian set Y of *-format and *-degree bounded by F and D , we im-plicitly mean that there exists a presentation in the form specified in Definition 3with the corresponding bounds on the format and degree. Different presentationsof the same set may of course give rise to different pairs of *-format and *-degree. Remark 4.
It may be useful below for the reader to consider the notions of *-formatand *-degree as sub-Pfaffian analogs of the notions of dimension and degree in thetheory of algebraic or semi-algebraic geometry. Our main objective is to obtain, asin the semialgebraic case, bounds that depend polynomially on the degree for a fixedformat.
In what follows, for a, b, c ∈ N , we will write: a is const( b ) (resp., a is poly c ( b ))if there is a function γ : N → N such that a γ ( b ) (resp., a ( b + 1) γ ( c ) ).This notation extends naturally to several arguments in const( · ) and poly c ( · ). Allimplied constants in this paper can be effectively and explicitly computed from thedata defining the Pfaffian chain (degrees of the differential equations involved), butwe omit these computations for brevity.1.2. Cell decompositions.
We recall the standard definitions of a cylindrical celland a cylindrical cell decomposition. Later in the paper we consider only cylindricaldecompositions and omit the prefix “cylindrical” for brevity.
Definition 5 (Cylindrical cell) . A cylindrical cell is defined by induction as follows. Below we will consider only restricted sub-Pfaffian sets, and refer to them simply as sub-Pfaffian.
ELL DECOMPOSITIONS FOR RESTRICTED SUB-PFAFFIAN SETS 3 (1) Cylindrical 0-cell in R n is an isolated point.(2) Cylindrical 1-cell in R is an open interval ( a, b ) ⊂ R .(3) For n > and ℓ < n a cylindrical ( ℓ + 1) -cell in R n is either a graphof a continuous bounded function f : C → R , where C is a cylindrical acylindrical ( ℓ + 1) -cell in R n − , or else a set of the form { ( x , . . . , x n ) ∈ R n : ( x , . . . , x n − ) ∈ C and f ( x , . . . , x n − ) < x n < g ( x , . . . , x n − ) } , (1) where C is a cylindrical ℓ -cell in R n − , and f, g : C → R are continuousbounded functions such that f ( x , . . . , x n − ) < g ( x , . . . , x n − ) for all points ( x , . . . , x n − ) ∈ C . This definition implies that any ℓ -cell is homeomorphic to an open ℓ -dimensionalball. Definition 6 (Cylindrical cell decomposition) . A cylindrical cell decomposition D of a subset A ⊂ R n is defined by induction as follows.(1) If n = 1 , then D is a finite family of pair-wise disjoint cylindrical cells (i.e.,isolated points and intervals) whose union is A .(2) If n > , then D is a finite family of pair-wise disjoint cylindrical cellsin R n whose union is A and there is a cylindrical cell decomposition of π nn − ( A ) such that π nn − ( C ) is its cell for each C ∈ D .Let B ⊂ A . Then D is compatible with B if for any C ∈ D we have either C ⊂ B or C ∩ B = ∅ (equivalently, some subset D ′ ⊂ D is a cylindrical cell decompositionof B ). Our main result is as follows.
Theorem 1.
Let { Y α } , Y α ⊂ I n be a collection of N sub-Pfaffian sets of *-format F and *-degree D . Then there exists a sub-Pfaffian cell decomposition of I n com-patible with each Y α such that the number of cells is poly F ( N, D ) , their *-format is const( F ) , and their *-degree is poly F ( D ) . The proof of Theorem 1 is given in § Sub-pfaffian sets defined by quantified formulas.
We introduce the no-tion of *-format and *-degree for quantified formulas in the language of restrictedsub-Pfaffian sets, as follows.
Definition 7 (*-format and *-degree of quantified formulas) . Let a quantified for-mula φ , not necessarily in a prenex form, have atomic predicates of the form ( x ∈ Y ) where Y is a restricted sub-Pfaffian set. We define the *-degree D ( φ ) to be the sumof D ( Y ) for all Y appearing in the atomic predicates of φ . We inductively definethe *-format F ( φ ) as follows. • For φ = ( x ∈ Y ) we define F ( φ ) to be the *-format of Y . • For φ = W kj =1 φ j we define F ( φ ) = max j F ( φ j ) . • For φ = V kj =1 φ j we define F ( φ ) = 1 + max j F ( φ j ) . • For φ = ∃ x φ ′ we define F ( φ ) = F ( φ ′ ) . • For φ = ¬ φ ′ , ∀ x φ ′ we define F ( φ ) = 1 + F ( φ ′ ) . GAL BINYAMINI AND NICOLAI VOROBJOV
In particular, the format F ( φ ) is bounded from above by the maximum over theformats of the atomic predicates of φ , plus the depth of the parse-tree for φ .Definition 7 is motivated by the following theorem, which shows that the *-format and *-degree of a set defined by a sub-Pfaffian formula φ can be boundedin terms of F ( φ ) and D ( φ ). Theorem 2.
Let φ be a sub-Pfaffian formula as above, with *-format F and *-degree D . Then the set defined by φ has *-format const( F ) and *-degree poly F ( D ) . The proof of Theorem 2 is given in § Upper bound on homologies.
Theorem 1 implies in particular an upperbound poly F ( D ) for the number of connected components of a sub-Pfaffian set of*-format F and *-degree D . Below we generalize this to an upper bound for thehomology of a sub-Pfaffian set in terms of the *-format and *-degree. Theorem 3.
The sum of the Betti numbers of a sub-Pfaffian set of *-format F and *-degree D is bounded by poly F ( D ) . The proof of Theorem 3 is given in § § Applications and motivation.
Our goal in this paper is to provide a frame-work that can be used to effectivize most of the results of o-minimal geometry inthe restricted sub-Pfaffian structure, with polynomial dependence on the degree.Indeed since definition using first-order formulas and cell decomposition form twoof the common technical tools of o-minimal geometry, many o-minimal proofs canbe carried out verbatim using Theorems 1 and 2 to obtain such polynomial bounds.As an example we have the following.
Corollary 8.
Let A ⊂ I n be a restricted sub-Pfaffian set of *-format F and *-degree D . Then the topological closure and the smooth part of A have *-format const( F ) and *-degree poly F ( D ) .Let f : A → B be a restricted sub-Pfaffian map of *-format F and ∗ -degree D . Then for every r ∈ N the C r -smooth locus of f has *-format const( F , r ) and*-degree poly F ,r ( D ) .Proof. All of the statements follow by defining the relevant sets using first-orderformulas in the restricted sub-Pfaffian language and applying Theorem 2.For instance, to define the smooth part of a set A having dimension m ∈ N ateach point, we can define it as a locus of points x ∈ A such that a neighbourhoodof x in A is the graph of a C -smooth map. More precisely, the smooth part of A is the set of all points x ∈ A each having a neighbourhood U in A such that thereexists a linear map T : R n → R m such that the restriction T | U is one-to-one, T ( U )contains a neighbourhood of T ( x ) in R m , and the inverse to T | U is a C -smooth ELL DECOMPOSITIONS FOR RESTRICTED SUB-PFAFFIAN SETS 5 map with the Jacobian non-vanishing at every point in T ( U ). It is straightforwardto reduce all of this to first-order formulas (an “ ǫ - δ definition”), and we leave thedetails to the reader. (cid:3) One of our main motivations for developing the theory in this paper is in rela-tion to the Pila-Wilkie counting theorem [17] and its applications in diophantinegeometry (see the survey [19]).Several applications of the counting theorem involve the geometry of ellipticcurves and abelian varieties — the most famous example perhaps being the proofof the Manin-Mumford conjecture by Pila-Zannier [18]. In these applications oneconsiders sets defined using elliptic and abelian functions. Since these functions arerestricted sub-Pfaffian by an observation of Macintyre [16], the definable sets arerestricted sub-Pfaffian. One can therefore hope to effectivize the counting theoremin this context, and subsequently obtain effective results for diophantine problems.Toward this end Jones and Thomas [13] have established a version of the count-ing theorem for certain surfaces definable in the restricted sub-Pfaffian structure,and Jones and Schmidt have explored several applications of this in diophantinegeometry [11, 12].In an upcoming paper by the first author with Jones, Schmidt and Thomas,we use the framework developed in the present paper to extend the result of [13]to arbitrary restricted sub-Pfaffian sets of arbitrary dimension, and improve thedependence of the effective constants to make them polynomial in the degrees. Thiscan be viewed as a case of effectivizing an o-minimal proof in the sense describedabove, albeit for a much more technically involved statement. We expect thatthis result will greatly extend the scope of potential applications in diophantinegeometry, as well as give rise to more reasonable (indeed, polynomial in degrees)estimates in these applications.1.6.
Comparison with previous results.
Gabrielov and Vorobjov have estab-lished many results on effectivity of operations in the restricted sub-Pfaffian cate-gory. For a survey we refer the reader to [7]. In these works, the notion of formatand degree for sub-Pfaffian sets is similar to ours but more straightforward: oneconsiders simply projections of semi-Pfaffian sets, rather than projections of theirconnected components. In particular, in [6] Gabrielov and Vorobjov prove a celldecomposition result quite similar to our Theorem 1, as follows (where we omit theexplicit, doubly-exponential dependence on the format).
Theorem 4.
Let { Y α } , Y α ⊂ I n be a collection of N sub-Pfaffian sets of format F and degree D . Then there exists a linear transformation L : R n → R n and acylindrical cell decomposition of R n , compatible with each L ( Y α ) , such that each cellin the decomposition is sub-Pfaffian and has format poly F ( N, D ) , and the numberof cells and their degree is poly F ( N, D ) . There are two main differences compared to Theorem 1: first, the applicationof the linear transformation L ; and second, more crucially, the dependence of theformat of the cells on the degree D . These are crucial limitation, which make itimpossible to apply this result to obtain analogs of Theorems 2 and 3: first, becausein the recursive proofs it is essential that the cells preserve the order of coordinates;and second, more crucially, because after the first recursive application the formatbecomes dependent on D , and the complexity of any further operations performedon such cells is no longer polynomial in D . GAL BINYAMINI AND NICOLAI VOROBJOV
We are also unable to sharpen Theorem 4 to eliminate the dependence of the for-mat on D , and this appears to be a fundamental difficulty . Our main observationin the present paper is that with the revised notions of *-format and *-degree itis possible, at a crucial point in the cell decomposition algorithm, to produce cellswith format independent of D . However other aspects of the algorithm becomemore delicate with these notions. The main reason is that the known approaches toeffective cell decomposition involve reductions using topological closure and fron-tier, and in our setting one must take care to avoid different components becomingglued along their common boundary when performing these operations. For thisreason the strategy of cell decomposition employed in the present paper differssignificantly from that of [6].We also note that analogs of Theorem 3 for quantified formulas have been pur-sued, using an entirely different topological approach, in the work of Gabrielov,Vorobjov and Zell [10]. This paper establishes similar bounds (and also with an ex-plicit dependence on the format) for sets defined by quantified formulas in a prenexform: { x ∈ [0 , n : Q y Q y · · · Q ν y ν (( x, y ) ∈ X ) } (2)where X is a semi-Pfaffian set that is either open or closed, and Q , . . . , Q ν ∈ {∃ , ∀} .For formulas involving only existential quantifiers, Gabrielov and Vorobjov [9] havemanaged to remove the topological condition on X by an approximation method. Remark 9.
In an unpublished manuscript Clutha [1] extended the method of [9] toan arbitrary number of quantifiers, combining with the ideas of [10] , and claimed inparticular a result implying Theorem 3 for arbitrary formulas. However the proofof this result contains a substantial gap and we are presently not able to repair it. Preliminaries on semi-Pfaffian sets
We will require the notion of a (weak) stratification of a semi-Pfaffian set.
Definition 10 ([5, Definition 5]) . A (weak) stratification of a semi-Pfaffian set X isa subdivision of X into a disjoint union of smooth, not necessarily connected, semi-Pfaffian subsets X α called strata . The system of equalities and inequalities for eachstratum X α of codimension k includes a set of k equalities h α, = · · · = h α,k = 0 whose differentials define the tangent space of X α at every point of X α . We will use the following formulation of the main result of [5].
Theorem 5 ([5, Theorem 1]) . Let X ⊂ I n be semi-Pfaffian of format F and degree D . Then there is a semi-Pfaffian stratification X = S α X α of X where the numberof strata and their degrees are poly F ( D ) and their format is const( F ) . Remark 11.
We will also need a parametric version of Theorem 5: if X ⊂ I n × I m is semi-Pfaffian of format F and degree D , then there exists a collection { S α } , S α ⊂ I n × I m of semi-Pfaffian sets with their number and degree poly F ( D ) and theirformat const( F ) such that the following holds. For any x ∈ I n , the collection { S α ∩ π − n ( x ) } forms a stratification of the fiber X ∩ π − n ( x ) , with dim( S α ∩ π − n ( x )) independent of x ∈ π n ( S α ) for each α . The proof of this is the same as the proofof Theorem 5, treating the coordinates in I n as parameters and performing theconstruction in I m . The linear transformation problem seems less fundamental, and could probably be avoidedusing a strategy similar to the one used in the present paper.
ELL DECOMPOSITIONS FOR RESTRICTED SUB-PFAFFIAN SETS 7
We also require the following result on the complexities of the closure X and frontier ∂X := X \ X of a semi-Pfaffian set X . Theorem 6 ([4]) . Let X ⊂ I n be a semi-Pfaffian set of format F and degree D .Then the closure X and the frontier ∂X are semi-Pfaffian of format const( F ) anddegree poly F ( D ) . Remark 12.
We will also need a parametric version of Theorem 6: if X ⊂ I m × I n is semi-Pfaffian of format F and degree D then the union of the fiberwise-closures { ( x, y ) ∈ I n + m : y ∈ X x } , where X x := { y : ( x, y ) ∈ X } (3) is semi-Pfaffian of format const( F ) and degree poly F ( D ) . This follows from theproof of Theorem 6, see [7, Remark 5.4] . We will also need a bound on the sum of Betti numbers of semi-Pfaffian sets (infact we will only require the zeroth Betti number, i.e. the number of connectedcomponents). This type of bound was proved by Khovanskii [15] for Pfaffian sets,and extended by Zell to the semi-Pfaffian class. We state only the part we need,omitting the more precise dependence on the parameters which is achieved in [21].
Theorem 7 ([21, Main result]) . The sum of the Betti numbers of a semi-Pfaffianset of format F and degree D does not exceed poly F ( D ) . Cell decomposition of sub-Pfaffian sets
In this section we prove a result on cell decomposition of sub-Pfaffian sets, that isthe main ingredient of the proof of Theorem 1. As a shorthand, if ∆ is a collectionof sets we write π k (∆) for the collection of the projections of the elements of ∆.Furthermore, S ∆ will stand for the union of sets in ∆. Thus, S π k (∆) is the unionof projections of sets in ∆. We will freely use the results on effective bounds forstratifications and frontiers from § Lemma 13 (Effective fiber cutting) . Let X ⊂ I n + m be a semi-Pfaffian set offormat F and degree D . Then there exists a semi-Pfaffian set b X of format const( F ) and degree poly F ( D ) such that π n ( X ) = π n ( b X ) and π n | b X has zero-dimensionalfibers.Proof. We proceed by induction on the maximal fiber dimension k of π n | X . If k = 0we are done, and otherwise we will construct a set X ′ with π n ( X ) = π n ( X ′ ) andmaximal fiber dimension smaller then k .Let { S α } , S α ⊂ I n + m denote a fiberwise stratification of X , as in Remark 11.For any S α with fiber dimension smaller then k , we put S α into X ′ . For any S α with fiber dimension k we do the following.(1) We put the fiberwise frontier of S α into X ′ .(2) For any j = 1 , . . . , m let C α,j be the set of fiberwise critical points of thecoordinate function x n + j on the fibers of π n | S α . Then we stratify the fibersof π n | C α,j again by Remark 11 and put all strata of fiberwise dimensionsmaller than k into X ′ .Clearly π n ( X ′ ) ⊂ π n ( X ). To prove the converse, let x ∈ π n ( X ). If the fiberof X over x has strata of dimension less then k , or strata of dimension k withnon-empty frontier, we are done. Otherwise this fiber consists of smooth compact GAL BINYAMINI AND NICOLAI VOROBJOV k -dimensional manifolds. Then one of the coordinate functions x n + j must be non-constant on this fiber, and the corresponding set C α,j is non-empty and locallyclosed in the fiber. Then the fiberwise stratification of C α,j must contain strata ofdimension less than k that we put into X ′ . This proves the claim. (cid:3) The following proposition is quite similar to Theorem 1: the difference is thathere we essentially assume that we are given sub-Pfaffian sets with bounded formatand degree (rather than their *-analogs), but produce cell decompositions withbounded *-format and *-degree. In § Proposition 14.
Let { X α } , X α ⊂ I ℓ , where ℓ > n , be a collection of N semi-Pfaffian sets of format F and degree D . Then there exists a sub-Pfaffian cell de-composition of I n compatible with each π n ( X α ) such that the number of cells is poly F ( N, D ) , their *-format is const( F ) and their *-degree is poly F ( D ) .Proof. We will work by lexicographic induction on ( n, k ) where k := max α dim π n − ( X α ) . By Lemma 13 we may assume that π n | X α has zero-dimensional fibers. Refining each X α into its stratification, we may further assume without loss of generality that X α is smooth, that π n | X α has rank dim π n ( X α ), and that π n − | X α has constant rank.Let Π := { X α } . We may also assume that Π is closed under taking frontiers.Let Π k (resp. Π k +1 and Π 1, andthe frontier of the strata of dimension k , to Σ.Now apply induction to obtain a cell decomposition of I n − compatible with π n − (Π) and in π n − (Σ). If a cell C is disjoint from S π n − (Π) then C × I is a cellcompatible with every π n ( X α ).Let C be a cell contained in G = S π n − (Π) \ S π n − (Σ). We will show howto construct cells over C that are compatible with every π n ( X α ). First consider X α ∈ Π k +1 . Recall that X α ∩ ( G × I ) is open in G × I . It follows that its boundaryin G × I agrees with its frontier. If a cell in C × I is compatible with the boundaryof π n ( X α ) (or, equivalently in this case, with the frontier) then it is also compatiblewith π n ( X α ) by elementary topology. Since we assume that Π is closed under takingfrontiers, it will be enough to construct cells over C compatible with π n (Π k ).We claim that for any X α ∈ Π k , the set π n ( X α ) is a union of graph cells over C .Consider an arbitrary point c ∈ C . Then the fiber of π n ( X α ) over c consists offinitely many points. By construction, π n ( X α ) maps submersively to C , so as wemove c these points locally move continuously. Moreover the points must remainin π n ( X α ) as we continue globally to C , for otherwise one of these points wouldmeet ∂π n ( X α ) contradicting (5) since C ⊂ G is disjoint from π n − (Σ). Note thatfor the same reasons X α itself is also a union of graphs of continuous maps over C .If s : C → I is such a section of π n ( X α ) we denote by ˆ s the corresponding extensionto a section of X α .If s α , s β : C → I is a pair of such sections obtained from X α , X β then one of s α < s β s α = s β s α > s β (8)holds uniformly over C . Indeed, otherwise there should be a point c ∈ C where s α ( c ) = s β ( c ) and a neighborhood where this does not hold identically. Thisimplies that ( c , c α ( c )) belongs to a strata of dimension at most ( k − k -dimensional strata of Z α,β , and hence c G contradicting C ⊂ G .By the above, the set of sections obtained from any of the X α ∈ Π k is { s < · · · < s q } , where we may suppose for simplicity that s = 0 and s q = 1. Also note since π n ( X α ) is a union of disjoint graphs, their number is bounded by the number ofconnected components of X α . By Theorem 7 we therefore get at most poly F ( D )sections from each X α , and in total q = poly F ( N, D ). A cell decomposition of C × I , compatible with each X α , is given by the cells { x n = s ( x , . . . , x n − ) } , { s ( x , . . . , x n − ) < x n < s ( x , . . . , x n − ) } ,. . . , { x n = s q ( x , . . . , x n − ) } . (9)We show that each of these cells is a sub-Pfaffian set with appropriately bounded*-format and *-degree. Specifically we show that the graph of each section s j over C is a sub-Pfaffian set with the appropriate *-bounds, and it is then a simple exerciseto construct each of the cells above (where for the interval cells one repeats theconstruction for s j , s j +1 and works in the direct product).Recall that C = π ℓn ( Z ◦ ) is the projection of the connected component Z ◦ ofsome semi-Pfaffian set Z ⊂ I ℓ (for an appropriate ℓ ). Suppose that s j is a sectioncorresponding to X α . Then the graph of s j is a connected component of the set π n ( X α ) ∩ ( C × I ). In fact, since the section s j extends to a section of X α , the graphof s j the projection π n ( W ◦ j ) of a connected component W ◦ j of the set W j := { ( x, y ) ∈ Z × X α : x = y , . . . , x n − = y n − } (10)given by Z ◦ in the x -coordinates and by the graph of y = ˆ s j ( x ). Here we ordered thecoordinates in such a way that π n ( x, y ) = ( y , . . . , y n ). This proves that the graphof s j is indeed sub-Pfaffian with appropriately bounded *-format and *-degree.Since each cell is constructed using one or two of these graphs, their *-degree isindeed bounded by poly F ( D ) (noting in particular that there is no dependence on N ).Finally, we must construct cells covering S π n − (Σ) × I and compatible withevery π n ( X α ). Let Y α := (cid:16)[ π n − (Σ) × I (cid:17) ∩ π n ( X α ) = π n ( b Y α ) , (11)where it is easy to choose b Y α to be a semi-Pfaffian set of bounded complexity. Itwill suffice to construct a cell decomposition of I n compatible with S π n (Σ) × I and { Y α } and take from it only the cells covering S π n (Σ) × I . This can now be achievedby induction on k , noting that for any Z in this collection π n − ( Z ) ⊂ π n − (Σ) hasdimension strictly less than k . (cid:3) Proofs of the main results Proof of Theorem 1. We will require a couple of elementary lemmas. Lemma 15. Suppose that a cell C is compatible with a set X . Then it is compatiblewith each connected component of X .Proof. Immediate because cells are connected. (cid:3) Lemma 16. Suppose that a cell decomposition of I k is compatible with a set A .Then the induced decomposition on I n is compatible with π kn ( A ) .Proof. Assume not, and let C be a cell meeting both π kn ( A ) and I n \ π kn ( A ). Picksome point y ∈ π kn ( A ) ∩ C . Then since we have a cell decomposition of I k , theremust be a cell with base C that contains the point x ∈ A with y = π kn ( x ). Butthen such a cell must lie strictly in A uniformly over C , which is impossible overthe points where C meets I n \ π kn ( C ). (cid:3) ELL DECOMPOSITIONS FOR RESTRICTED SUB-PFAFFIAN SETS 11 We are now ready to finish the proof. Let X α,β ⊂ I k denote semi-Pfaffian setsand X ◦ α,β connected components such that Y α = S β π kn ( X ◦ α,β ). Use Proposition 14to find a sub-Pfaffian cell decomposition of I k compatible with { X α,β } with suitablybounded *-format and *-degree. Then by the preceding lemmas this cell decompo-sition is compatible with X ◦ α,β , and the induced decomposition on I n is thereforecompatible with Y α . Remark 17. Note that in the proof of Theorem 1, even though we are interested ina cell decomposition of I n , we apply Proposition 14 to obtain a cell decompositionof the full space I k . This allows us to ensure the compatibility with the projec-tions of each connected component X ◦ α,β separately. Applying Proposition 14 tothe projections π n ( X α,β ) would not suffice. It is therefore crucial at this point thatProposition 14 produces a cell decomposition in the original coordinates, withoutapplying a linear transformation (or at least preserving the projection π kn ). For thisreason, the cell decomposition algorithm of Gabrielov-Vorobjov [6] , which appliessuch a linear transformation to the coordinates, would not suffice for our purposes. Proof of Theorem 2. The proof is a routine recursive argument on thestructure of φ . The statements for existential quantifiers and disjunctions hold bydefinition. Up to re-writing ∀ x φ ≡ ¬∃ x ( ¬ φ ) , (12) ∧ ki =1 φ i ≡ ¬ ∨ ki =1 ( ¬ φ i ) , (13)everything else follows from the statement for negations φ = ¬ φ ′ . This is provedby constructing a cell decomposition of I n compatible with the set X defined by φ ′ , and taking the union of all cells disjoint from X in this decomposition.4.3. Proof of Theorem 3. We first establish an effective result on the existenceof triangulations in the restricted sub-Pfaffian class. Recall that a finite simplicialcomplex in R n is a finite collection K = { ¯ σ , . . . , ¯ σ p } of (closed) simplices ¯ σ i ⊂ R n such that the intersection of any pairs ¯ σ i ∩ ¯ σ j , if not empty, is a common face of ¯ σ i and ¯ σ j ; and such that any face of any ¯ σ i also belongs to K . We write | K | for theunion of all simplices in K . Theorem 8. Let Y ⊂ [0 , n be a closed sub-Pfaffian set, and X , . . . , X k ⊂ Y besub-Pfaffian subsets, and suppose all these sets have *-format bounded by F and*-degree bounded by D .Then there exists a finite simplicial complex K with vertices in Q n and a definablehomeomorphism Φ : | K | → Y such that each X i is a union of images by Φ of opensimplices of K . Moreover K contains poly F ( D ) simplices, and Φ has *-format const( F ) and *-degree poly F ( D ) . A proof of the triangulation theorem in the o-minimal setting can be found in [20],where it follows a similar proof for the semialgebraic class in [3]. For conveniencewe refer the reader to the alternative presentation given in [2, Theorem 4.4], whichalso establishes Theorem 8 without the effective estimates for arbitrary o-minimalstructures. Deriving the effective estimates from this proof in the restricted sub-Pfaffian context is a routine exercise in the application of Theorems 1 and 2: oneonly nees to verify that in the proof of [2, Theorem 4.4], the triangulating mapΦ is indeed defined by a first-order formula φ with F ( φ ) = const( F ) and D ( φ ) = poly F ( D ). As this verification is entirely straightforward from the presentation ofloc. cit., we leave the details as an exercise for the reader.To deduce Theorem 3 we first apply Theorem 8 with Y = [0 , n and X the givensub-Pfaffian set. We obtain a homeomorphism Φ : | K | → [0 , n . In particular, thesum of the Betti numbers of X is equal to that of Φ − ( X ), which is a union of atmost poly F ( D ) simplices. This set being semialgebraic, the bound on the sum ofBetti numbers now follows, e.g., from [8, Theorem 1]. References [1] M. Clutha. Bounding betti numbers of sets definable in o-minimal structures over the reals,2012.[2] M. Coste. 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ELL DECOMPOSITIONS FOR RESTRICTED SUB-PFAFFIAN SETS 13 Weizmann Institute of Science, Rehovot, Israel E-mail address : [email protected] Department of Computer Science, University of Bath, Bath, BA2 7AY, UK E-mail address ::