Semi-algebraic triangulation over p-adically closed fields

Luck Darnière

February 24, 2017

Abstract

We prove a triangulation theorem for semi-algebraic sets over a p-adically closed field, quite similar to its real counterpart. We derive from it several applications like the existence of flexible retractions and splitting for semi-algebraic sets.

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . 1 2 Prerequisites and notation . . . . . . . . . . . . 4 3 Applications . . . . . . . . . . . . . . . . . . . . 14 4 Largely continuous cell decomposition . . . . . 19 5 Cellular complexes . . . . . . . . . . . . . . . . 24 6 Cellular monoplexes . . . . . . . . . . . . . . . 28 7 Cartesian morphisms . . . . . . . . . . . . . . . 39 8 Triangulation . . . . . . . . . . . . . . . . . . . 47 9 Open problems . . . . . . . . . . . . . . . . . . 49

1 Introduction

Our knowledge of geometric objects in affine spaces over p-adic fields, that is the field Q_{p} of p-adic numbers or a finite extension of it, has grown up intensively in the past decades. Several remarkable analogies have emerged with real geometry, in spite of the striking differences between the real and the p-adic metrics. The present paper raises a new such analogy: we prove a triangulation theorem over p-adically closed fields, quite similar to its real counterpart. Let us first recall the classical results in p-adic geometry which will be used here.

Semi-algebraic sets over a field K are the finite unions of set defined by finitely many conditions “f(x) = 0” or “f has a non-zero N-th root in K”, where f(x) is a polynomial function with m variables. Of course we can restrict to N = 2 if K is real closed (and to N = 1 if it is algebraically closed). Macintyre has proved in [Mac76] that semi-algebraic sets over Q_{p} are stable by boolean combinations and projection (from Q_{p}^{m} to Q_{p}^{m−1}, for every m). This is a p-adic version of Tarski’s theorem for real closed fields (and of Chevalley’s theorem for algebraically closed fields). Prestel and Roquette (see [PR84]) have generalized it to arbitrary p-adically closed fields (a p-adic version of real closed fields).

Denef has proved in [Den84] a cell decomposition theorem for p-adic semialgebraic sets very similar to its real counterpart, and derived from it the rationality of Poincarré series. Another important application of the cell decomposition is that it provides a good dimension theory for semi-algebraic sets (see [SvdD88]). All this material has been later widely generalized, first to arbitrary p-adically closed fields and then to richer structures on these fields (see [DvdD88] and [Clu04] for subanalytic sets, [HM97], [CKL16], [CKDL15] and [DH15] for definable sets in P -minimal and p-optimal structures).

By means of Denef’s cell decomposition, Cluckers has proved in [Clu01] that for every two semi-algebraic sets A, B over a p-adically closed field, there is a semi-algebraic bijection from A to B if and only if A, B have the same dimension and the same number of isolated points. This has been latter generalized to subanalytic sets [Clu04] and p-optimal sets [DH15]. Note that these semi-algebraic bijections are not continuous in general (for example Clucker’s theorem applies to the valuation ring Z_{p}, which is compact, and to Z_{p} \ {0}, which is not).

We present here a p-adic triangulation theorem quite similar to the real one. We prove it for semi-algebraic^{1} sets and maps over an arbitrary p-adically closed field K, which is fixed throughout all this paper. Since it requires numerous prerequisites, we state it here as a foretaste and refer the reader to Section 2 for precise definitions.

Theorem (Triangulation T_{m}). Given a finite family (θ_{i} : A_{i} ⊆ K^{m} → K)_{i∈I} of semi-algebraic functions and integers n, N ≥ 1, for some integers e, M which can be made arbitrarily large^{2}, there exists a simplicial complex T of index M and a semi-algebraic homeomorphism φ : ⨄ T → ⋃_{i∈I} A_{i} such that for every i in I:

1. {φ(T) : T ∈ T and φ(T) ⊆ A_{i}} is a partition of A_{i}.

2. ∀T ∈ T such that φ(T) ⊆ A_{i}, θ_{i} ∘ φ_{|T} is N-monomial mod U_{e,n}.

We call the pair (T , φ) given by T_{m} a triangulation of the θ_{i}’s with parameters (n, N, e, M). When a finite family (A_{i})_{i∈I} of semi-algebraic sets is given, the result of the application of T_{m} to the indicator functions of the A_{i}’s is called a triangulation of (A_{i})_{i∈I}.

Remark 1.1. The elements of the simplicial complex appearing in the conclusion of T_{m} are not contained in K^{m} but in finitely many copies of K^{q} for various q, usually much larger than m. This is the main, but harmless, structural difference with the triangulation in the real case.

In section 3 we derive from T_{m} the next four applications.

Theorem (Lifting). For every semi-algebraic function f : X ⊆ K^{m} → K such that |f| is continuous, there is a continuous semi-algebraic function h : X → K such that |f| = |h|.

The real counterpart of the above result is quite obvious. At the contrary, the two next results do not hold true in real geometry. In the same vein as Clucker’s result on classification of semi-algebraic sets up to semi-algebraic bijection [Clu01], they confirm the intuition that the lack of connectedness and of “holes” (in the sense of algebraic topology, see below) makes semi-algebraic sets over p-adically closed fields much more flexible than over real closed fields.

1For possible generalizations to subanalytic sets and maps, see Section 9.

2The exact meaning of “e, M can be made arbitrarily large” is a bit special here: it says that for any given integers e_{∗} ≥ 1 and M_{∗} ≥ 1, the integers e, M can be chosen so that e_{∗} divides e and M_{∗} ≤ M.

Recall that a retraction of a topological space X onto a subspace Y is a continuous map σ : X → Y such that σ(y) = y for every y ∈ Y. When such a retraction exists on a Hausdorff space X, then necessarily Y is closed in X.

Over the reals, the main obstruction for the existence of retractions is the existence of “holes” which are detected by homotopy. This does not work over p-adic fields. Indeed, given a non-empty semi-algebraic set X ⊆ K^{m} and a point a ∈ X, the semi-algebraic function H : X × R → X defined by H(x, s) = x if s ∈ R^{×} and H(x, s) = a if s ∈ πR is obviously continuous. In that sense, with the unit ball R replacing in K the unit interval [0, 1] in the reals, the identity function on X is homotopic to a constant function, that is X is “contractile”. But it is another story to prove that retractions actually exist.

Theorem (Retraction). For every non-empty semi-algebraic sets Y ⊆ X ⊆ K^{m}, there is a semi-algebraic retraction of X onto Y if and only if Y is closed in X.

It is worth mentioning that it is the next Splitting Theorem, already conjectured in [Dar06], which has motivated all the present paper. Here ∂X denotes the topological frontier of X, see Section 2.

Theorem (Splitting). Let X be a relatively open non-empty semi-algebraic subset of K^{m} without isolated points, and Y_{1}, ⋯, Y_{s} a collection of closed semialgebraic subsets of ∂X such that^{3} Y_{1} ∪ ⋯ ∪ Y_{s} = ∂X. Then there is a partition of X in non-empty^{4} semi-algebraic sets X_{1}, …, X_{s} such that ∂X_{i} = Y_{i} for 1 ≤ i ≤ s.

The trivial remark that every ball B ⊆ K^{m} is disconnected can be seen as a very special case of the above Splitting Property (applied to X = B with Y_{1} = Y_{2} = ∅). This property is actually (in a sense which can be made precise, see [Dar06]) the strongest possible form of disconnectedness that can be observed in a finite dimensional topological space whose points are closed. It is a versatile property which we encountered in different contexts with minor changes (see [Dar06], [DJ10]). In the present paper, it plays a key role in the induction step.

A limit value for a function f : X ⊆ K^{m} → K at a point x adherent to X, is a value l ∈ K such that (x, l) is adherent to the graph of f. Of course f tends to l at x if and only if l is the unique limit value of f at x. Let us say that f is largely continuous on a subset A of X if the restriction of f to A has a unique limit value at every point adherent to A, that is if f extends to a continuous function on the topological closure of A. If A is not mentioned it simply means that f is largely continuous on its domain X. Finally f is piecewise largely continuous if there exists a finite partition of X in semi-algebraic pieces on which f is largely continuous. Of course in that case f has finitely many limit values at every point adherent to X.

3Note that Y_{1}, …, Y_{s} are not assumed to be disjoint. All of them can be equal to ∂X, for example.

4A partition of a set X is for us a family of two-by-two disjoint subsets of X covering X. We do not assume that the pieces must be non-empty. So when it happens by exception, like here, that this property is required and does not follow from the context, we explicitly mention it.

Theorem (Largely Continuous Splitting). Let f : X ⊆ K^{m} → K be a semialgebraic bounded function with bounded^{5} domain. If f has finitely many limit values at every point adherent to X then f is piecewise largely continuous.

The real counterpart of this result is easily seen to be true, by means of a triangulation and the trivial remark that every real simplex is connected (see Section 3). This last argument is no longer valid in the p-adic case but, as we will see, the existence of retractions allows us to bypass this problem and recover the full result in the p-adic context.

This paper is a continuation of [Dar16], where p-adic simplexes were introduced and studied. However all the results of [Dar16] used here are recalled in Section 2, as well as all the classical prerequisites needed. The above applications are then derived from T_{m} in Section 3. In Section 4 we derive from Denef’s cell decomposition and T_{m} a “largely continuous cell decomposition up to a small deformation” for semi-algebraic functions in m + 1 variables (Theorem 4.6). Sections 5 to 7 are then devoted to our main constructions, which are summarized in Lemma 6.1 and Lemma 7.10. In Section 8, we finally derive T_{m+1} from T_{m} by means of these two technical lemmas. This finishes the proof of our p-adic triangulation theorem for every m. Section 9 lists some open problems on related topics.

Acknowledgement. The idea of the proof of the Good Direction Lemma 4.4 was given to me (at the beginning of this research, more than fifteen years ago), by my colleague Daniel Naie from the Université d’Angers.

2 Prerequisites and notation

We let N denote the set of positive integers and N^{∗} = N \ {0}. For every integers k, l we let [[k, l]] be the set of integers i such that k ≤ i ≤ l (hence an empty set if k > l).

A Z-group is a linearly ordered group Z such that, with additive notation, Z has a least strictly positive element 1 and Z/nZ has exactly n elements for every integer n ≥ 1. A p-adically closed field (see [PR84]) is a Henselian valued field of characteristic zero whose residue field is a finite field with characteristic p and whose value group is a Z-group. The most prominent examples of such fields, which are the p-adic counterpart of real closed fields, are obviously the field Q_{p} of p-adic numbers and all its finite extensions. But there are other natural examples such as the algebraic closure of Q inside Q_{p} (which is not complete), the t-adic completion of the field ⋃_{n≥1} Q((t^{1/n})) of Puiseux series over Q_{p} (whose value group is not Z, but Z × Q lexicographically ordered), and many others (every ultraproduct of p-adically closed fields is still p-adically closed).

5These boundedness assumptions could easily be removed if we add to K a point at infinity and require that f has finitely limit values in K̂ = K ∪ {∞} at every point of the closure of X in K̂^{m}, using the same construction as in the preparation of the proof of Lemma 3.3.

Throughout all this paper we consider a p-adically closed field K fixed once for all. We let v denote its (unique) p-adic valuation, R its valuation ring, R^{×} the multiplicative group of invertible elements of R, and π a fixed generator of the maximal ideal of R.

Remark 2.1. From now on the letter p will usually denote various integers unrelated to the characteristic of the residue field of v. The only exception is when it appears in expressions such as “the field Q_{p}”, “a p-adically closed field” or “a p-adic simplex”, which refer to p-adic analogues of notions coming from real algebraic geometry.

We let Z denote the value group of K, with additive notation. It is extended by a largest element +∞ for v(0) and we let Γ = v(K) = Z ∪ {+∞}. We let Q denote the divisible hull of Z and Ω = Q ∪ {+∞}. As an ordered group, Z identifies naturally to the smallest non-trivial convex subgroup of Z. We consider Z and Q as embedded into Q via this identification.

For every subset X of K we let X^{∗} = X\{0}. However, if X^{∗} is a subgroup of the multiplicative group of K, we denote it X^{×} in order to remind this property (so R^{∗} = R \ {0} ≠ R^{×} but K^{∗} = K \ {0} = K^{×}). For every subgroup G of K^{×} we let xG = {xg : g ∈ G} for every x ∈ K, and K/G = {xG : x ∈ K}. Abusing the notation, 0G = {0} will be denoted 0 whenever the context makes it unambiguous.

We let |a| = aR^{×} for every a ∈ K, and |K| = K/R^{×} = {|a| : a ∈ K}. The latter is naturally ordered by inclusion, and isomorphic to Γ with the reverse order : |a| ≤ |b| if and only if v(a) ≥ v(b). So |a| is just a multiplicative notation for v(a): we have |ab| = |a|.|b| and |a + b| ≤ max(|a|, |b|), and of course |a| = 0 if and only if a = 0.

In order to ease the notation, given a ∈ K^{m}, A ⊆ K^{m} and f : X → K^{m} we will often write va for v(a), vA for the direct image v(A), vf for the composite v ∘ f, and similarly for |A| and |f|.

At some rare places it will be convenient to add to K a new element ∞ (and to Γ and |K| new elements −∞ and +∞ respectively) with the natural convention that 0^{−1} = ∞, ∞^{−1} = 0, v(∞) = −∞, |∞| = +∞, and a.∞ = ∞ for every a ∈ K^{×}. We also let 0.∞ = 1 and 0^{0} = 1 when needed.

2.a Topology and coordinate projections

When an m-tuple a is given, it is understood that (a_{1}, …, a_{m}) are its coordinates, except if otherwise specified. For every a ∈ K^{m} we let

va = (va_{1}, …, va_{m}) and |a| = (|a_{1}|, …, |a_{m}|).

This has not to be confused with ∥a∥ = max(|a_{1}|, …, |a_{m}|). For r ∈ K^{×} the (clopen) ball of center a and radius r is defined as

B(a, r) = {x ∈ K^{m} : ∥x − a∥ ≤ |r|}.

The valuation induces a topology on K, which is inherited by |K| and Γ. The topology generated on Ω by the open intervals and the intervals ]a, +∞] for a ∈ Q, extends the topology of Γ. The direct products of these topological spaces endow the product topology. For every subset X of any of these spaces, \overline{X} denotes the topological closure of X. In particular \overline{Z} = Γ and \overline{Q} = Ω. Note that Γ is closed in Ω. The specialisation preorder on the subsets of X is defined by B ≤ A iff B ⊆ \overline{A}.

We let ∂X = \overline{\overline{X} \ X} denote the frontier of X. We say that X is relatively open if it is open in \overline{X}, that is if ∂X = \overline{X} \ X.

When a function f is largely continuous (see Section 1) we usually denote \overline{f} the continuous extension of f to the closure of its domain. At the contrary, the restriction of f to some subset A of its domain is denoted f_{|A}.

The support of a ∈ K^{m}, denoted Supp a, is the set of indexes k such that a_{k} ≠ 0. The support of an element of |K|^{m} or Γ^{m} is defined accordingly, so that

Supp |a| = Supp v(a) = Supp a.

For every subset S of K^{m} and every I ⊆ {1, …, m} we let

F_{I}(S) = \{a ∈ \overline{S} : Supp a = I\}.

When F_{I}(S) ≠ ∅ we call it the face of S with support I. Faces of subsets of |K|^{m} or Γ^{m} are defined accordingly. The coordinate projection of K^{m} (resp. |K|^{m}, Γ^{m}) onto its face with support I will be denoted π_{I}. So π_{I}(a) is the unique point b with support I such that b_{i} = a_{i} for every i ∈ I.

For every a ∈ K^{m+1} (resp. |K|^{m+1} or Γ^{m+1}) we let â denote the tuple of the first m coordinates of a, so that a = (â, a_{m+1}). If A is a set of (m+1)-tuples we let Â = {â : a ∈ A}, and if A is a family of such sets we let

Â = {Â: A ∈ A}.

We call Â (resp. Â) the socle of A (resp. A).

Given two families H, A of subsets of K^{m+1} we say that H is finer than A if every H ∈ H which meets a set A ∈ A is contained in A. If moreover H is a partition of ⋃ A we say that H refines A. We will often distinguish between “horizontal refinements” for which Ĥ = Â, and “vertical refinement” for which H is the family of A ∩ (X × K) where A ranges over A and X over a refinement of the socle of A.

2.b Semi-algebraic sets and formulas

For every integer N ≥ 1 let

P_{N} = {a ∈ K : ∃x ∈ K, a = x^{N}}.

P_{N}^{×} = P_{N} \ {0} is a clopen subgroup of K^{×} with finite index, and P_{1} = K^{×}. Hence a subset K^{m} is a semi-algebraic set if it is a boolean combination of finitely many sets S_{i} defined by conditions

f_{i}(x) ∈ P_{N_{i}}  (1)

where the f_{i}’s are m-ary polynomial functions. A semi-algebraic map is a function whose graph is semi-algebraic. Rational functions, root functions and monomial functions (see below) are semi-algebraic, among many others.

Abusing a little bit the terminology, we also say that a subset S of K^{m}×|K|^{n} is semi-algebraic if {(x, t) ∈ K^{m+n} : (x, |t|) ∈ S} is semi-algebraic. Similarly a function f : X ⊆ K^{m} → |K|^{n} is semi-algebraic if its graph is. When a map φ is defined on the disjoint union of finitely many semi-algebraic sets A_{i} living in different copies of K^{m}, we say that φ is semi-algebraic if its restriction to each A_{i} is semi-algebraic in the classical sense.

Remark 2.2. If N′ divides N then P_{N}^{×} is a clopen subgroup of P_{N′}^{×} with finite index. For this reason, all the integers N_{i,j} appearing in (1) can be replaced by any common multiple N. Note also that 0 ∈ P_{N} is an empty condition, equivalent to 1 ∈ P_{N}, hence all the f_{i,j}’s can be assumed to be non-zero polynomials.

The so-called first order formulas in the language of rings with parameters in K, formulas for short, are defined inductively as follows:

1. Every equation “f(x) = 0” with f an n-ary polynomial with coefficients in K is a formula with n free variables.

2. Every combination of formulas with n free variables using the logical connectives “and”, “or”, “not” (denoted ∧, ∨, ¬) is a formula with n free variables.

3. If φ(x) is a formula with an n-tuple x of free variables x_{1}, …, x_{n}, then ∃x_{n}φ(x) and ∀x_{n}φ(x) are formulas with n − 1 free variables.

In the last notation above it is understood that the quantified variable x_{n} only concerns elements of K. Thus a formula φ(x) with n free variables always states a property of the elements of K^{n}. When a tuple a ∈ K^{n} has this property we say that φ(x) is satisfied by a in K and denote it K ⊧ φ(a). A set S is definable over K if there exists a formula φ(x) with n free variables such that

S = {a ∈ K^{n} : K ⊧ φ(a)}.

A family (C_{a})_{a∈A} of semi-algebraic subsets of K^{n} is uniformly definable if A ⊆ K^{m} is definable and there is a formula φ(x, y) with m + n free variables such that C_{a} = {b ∈ K^{m} : K ⊧ φ(a, b)} for every a ∈ A.

Given m-ary definable functions f, g, the set of points in K^{m} satisfying the condition “|f(x)| ≤ |g(x)|” is known to be definable^{6}. Thus we will consider these expressions as formulas as well (more exactly as abbreviations of some first order formulas in the language of rings stating the same property). Similarly, if φ(x, y) is a formula with m + n variables and S ⊆ K^{n} is definable by a formula ψ(y) then we will consider ∃y ∈ S, φ(x, y) as a formula since it is an abbreviation for the genuine formula ∃y, ψ(y) ∧ φ(x, y). We refer the reader to [Hod97] for more information on this topic.

Theorem 2.3 (Macintyre). The definable subsets of K^{m} are exactly the semialgebraic ones.

Remark 2.4. This fundamental theorem says that, in this paper, “semialgebraic” and “definable” are synonymous. In particular a subset of K^{m} is semi-algebraic whenever it can be defined by a formula. This is an extremely convenient criterion for being semi-algebraic, which we will use everywhere without more comments.

6This follows from the non-trivial fact that R is definable by means of the Kochen operator (see [PR84]).

Another important consequence of Macintyre’s theorem is that every p-adically closed field is elementarily equivalent to a finite extension of Q_{p} (see [PR84]). With other words, there is a finite extension L of Q_{p} such that K and L satisfy exactly the same formulas. The following semi-algebraic version of the so-called “théorème des fermés emboîtés” transfers from L to K by means of this elementary equivalence.

Theorem 2.5. Let (C_{α})_{α∈R^{∗}} be a uniformly definable family of non-empty, closed and bounded subsets of K^{n}, such that |β| ≤ |α| implies that C_{β} ⊆ C_{α}. Then ⋂_{α∈R^{∗}} C_{α} is non-empty.

The next classical properties can easily be derived from this theorem (or transfered from L to K by elementary equivalence).

Theorem 2.6. For every continuous semi-algebraic function f : X ⊆ K^{m} → K^{n} whose domain X is closed and bounded, f(X) is closed and bounded. As a consequence:

1. ∥f∥ is bounded and attains its bounds.

2. If f is a injective then it is a homeomorphism from X to f(X).

Corollary 2.7. For every bounded semi-algebraic subset X of K^{m} which is non-empty, there is an element x ∈ X such that ∥x∥ is maximal on X.

Another crucial property of p-adically closed fields is the existence of so called “built-in Skolem functions” (see [vdD84], or the appendix of [DvdD88] for a more constructive proof). Basically, it says that for every semi-algebraic subset A of K^{m+n}, the coordinate projection of A onto K^{m} has a semi-algebraic section.

Theorem 2.8 (Skolem functions). Let X ⊆ K^{m} be semi-algebraic set and φ(x, t) a formula with m + n free variables. If, for every a ∈ X there is b ∈ K^{n} such that K ⊧ φ(a, b), then there exists a semi-algebraic function ξ : X → K^{n} (called a Skolem function) such that K ⊧ φ(x, ξ(x)) for every x ∈ X.

For example, if a semi-algebraic function f : X → K takes values in P_{N}, then Theorem 2.8 applied the formula φ(x, t) saying that “f(x) = t^{N}” gives a semi-algebraic function ξ : X → K such that f = ξ^{N}.

2.c Root functions and monomial functions

Following Lemma 1.3 in [CL12] there is for each integer M > 0 a unique group homomorphism \overline{ac}_{M} from K^{×} to (R/π^{M}R)^{×} such that \overline{ac}_{M}(π) = 1 and \overline{ac}_{M}(u) = u + π^{M}R for every u ∈ R^{×}. The construction of \overline{ac}_{M} given in [CL12] shows that for each integer N > 0 the set

Q_{N,M} = \{0\} ∪ \{x ∈ P_{N}^{×} · (1 + π^{M}R) : \overline{ac}_{M}(x) = 1\}

is semi-algebraic. Q_{N,M}^{×} = Q_{N,M} \ {0} is a clopen subgroup of K^{×} with finite index. When v(K^{×}) = Z, it is worth mentioning that Q_{N,M} can be given a simpler description as follows:

Q_{N,M} = {0} ∪ ⋃_{k∈Z} π^{kN}(1 + π^{M}R).

If M > 2v(N), Hensel’s Lemma implies that 1 + π^{M}R ⊆ P_{N}, hence Q_{N,M} is contained in P_{N}. The importance of Q_{N,M} comes from the following property, which also follows from Hensel’s lemma (see for example lemma 1 and corollary 1 in [Clu01]).

Lemma 2.9. The function x ↦ x^{e} is a group endomorphism of Q_{N,M}^{×}. If M > v(e) this endomorphism is injective and its image is Q_{eN,v(e)+M}^{×}.

In particular x ↦ x^{e} defines a continuous bijection from Q_{1,v(e)+1} to Q_{e,2v(e)+1}. We let x ↦ x^{1/e} denote the reverse continuous bijection. In particular it is defined on Q_{N,M} for every N, M such that e divides N and M > 2v(e).

For every positive integers e, n we let

U_{e} = {x ∈ K : x^{e} = 1} and U_{e,n} = U_{e} · (1 + π^{n}R).

Analogously to Landau’s notation O(x^{n}) of calculus, we let U_{e,n}(x) denote any semi-algebraic function in the multi-variable x with values in U_{e,n}. Any such function is the product of two semi-algebraic functions, with values in U_{e} and 1 + π^{n}R respectively. So, given a family of functions f_{i}, g_{i} on the same domain X, we write that f_{i} = U_{e,n}g_{i} for every i, when there are semi-algebraic functions ω_{i} : X → R and χ_{i} : X → U_{e} such that for every x in X

f_{i}(x) = χ_{i}(x)(1 + π^{n}ω_{i}(x))g_{i}(x).

U_{1,n}(x) is simply denoted U_{n}(x).

Remark 2.10. If f(x) = U_{n}(x) for some n > 2v(e) then f^{1/e} is well defined and takes values in 1+π^{n−v(e)}R. Therefore we can write U_{n}(x) = (U_{n−v(e)}(x))^{e}.

A function g is N-monomial on S ⊆ K^{q} if either it is constantly equal to ∞ or there exists ξ ∈ K and β_{1}, …, β_{q} ∈ Z such that

∀x = (x_{1}, …, x_{q}) ∈ S, g(x) = ξ ∏_{i=1}^{q} x_{i}^{Nβ_{i}}.

In this definition we use when needed our convention that 0^{0} = 1. A function f is N-monomial mod U_{e,n} if f = U_{e,n}g with g an N-monomial function.

2.d Cell decomposition and dimension theory

Given a clopen semi-algebraic subgroup G of K^{×} with finite index, a presented cell A mod G in K^{m+1} is a tuple (c_{A}, ν_{A}, μ_{A}, G_{A}) with c_{A} a semi-algebraic function on a non-empty domain X ⊆ K^{m} with values in K (called the center of A), ν_{A} and μ_{A} either semi-algebraic functions on X with values in K^{×} or constant functions on X with values 0 or ∞ (called the bounds of A), and G_{A} an element of K/G (called the coset of A), having the property that for every x ∈ X there is t ∈ K such that:

|ν_{A}(x)| ≤ |t − c_{A}(x)| ≤ |μ_{A}(x)| and t − c_{A}(x) ∈ G_{A}  (2)

The set of tuples (x, t) ∈ X × K satisfying (2) is the cellular set underlying A. Abusing the notation we will also denote it A most often. The conditions enumerated above (2) ensure that the domain X of c_{A}, μ_{A}, ν_{A} is exactly the socle Â of A. When two presented cells A and B have the same underlying cellular set we write it A ≃ B.

A is of type 0 if G_{A} = {0}, of type 1 otherwise. We say that A is largely continuous if its center and bounds are so. We say that it is well presented if either vν_{A} − vμ_{A} is unbounded or ν_{A} = μ_{A}.

We call A a fitting cell if it has fitting bounds that is, for every x ∈ Â:

|μ_{A}(x)| = sup{|t − c_{A}(x)| : (x, t) ∈ A}
|ν_{A}(x)| = inf{|t − c_{A}(x)| : (x, t) ∈ A}

Cellular sets mod K^{×} and P_{N}^{×} have been first introduced in [Den84]. Cellular sets mod Q_{N,M}^{×} appear implicitly in [Clu01], and explicitly in further papers of Cluckers. In this paper we will use the word “cell” mostly for presented cells but also very often for the underlying cellular sets, the difference being clear from the context. For instance we will freely talk of disjoint (presented) cells, of bounded (presented) cells, of (presented) cells partitioning some set and so on, meaning that the corresponding cellular sets have these properties.

For any Z ⊆ Â we will write A ∩ (Z × K) both for this (cellular) set and for the presented cell (c_{A|Z}, ν_{A|Z}, μ_{A|Z}, G_{A}). The latter will also be denoted (c_{A}, ν_{A}, μ_{A}, G_{A})_{|Z}. Similarly Gr c_{A} both denotes the graph of c_{A} and the presented cell (c_{A}, 0, 0, {0}).

Sometimes it will be convenient to write G_{A} = λ_{A}G for some λ_{A} ∈ G_{A}. We will always do this uniformly, so that λ_{A} = λ_{B} whenever G_{A} = G_{B}. To that end a set Λ_{G} of representatives of K/G is fixed once for all, and when we consider a presented cell A mod G it is understood that λ_{A} is the unique element of G_{A} ∩ Λ_{G}. In addition, we require from this set of representatives that every λ ∈ Λ_{G} has the smallest possible positive valuation. In particular if G = P_{N} or Q_{N,M} and A is a cell mod G of type 1 then 0 ≤ vλ_{A} < N.

For every family A of presented cells in K^{m+1} we let CB(A) denote the family of all the functions c_{A}, μ_{A}, ν_{A} for A ∈ A. Given another family D of presented cells in K^{m+1} we say that:

1. D belongs to Vect A if for every D ∈ D, c_{D}, ν_{D} are linear combinations (with coefficients in K) of the restrictions to D̂ of the centers and bounds of the cells A ∈ A such that D̂ ⊆ Â, and either μ_{D} is so or μ_{D} = ∞.

2. D belongs to Alg_{n} A if D is finer than A and for every A ∈ A, every D ∈ D contained in A and every (x, t) ∈ D:

(a) either t − c_{A}(x) = U_{n}(x, t)(t − c_{D}(x));

(b) or t − c_{A}(x) = U_{n}(x, t)h_{D,A}(x) where h_{D,A} : D̂ → R is the product of (finitely many) linear combinations of functions c_{B}_{|D̂} such that

B ∈ A and D̂ ⊆ B̂.

Theorem 2.11 (Denef). Given a semi-algebraic subgroup G of K^{×} with finite index, let A be a finite family of presented cells mod G in K^{m+1}. Then for every positive integer n there exists a finite family D of fitting cells mod G refining A such that D̂ is a partition and D belongs to Vect A and to Alg_{n} A.

This is essentially theorem 7.3 of [Den84]. Indeed, for any given integer N, if n is large enough then 1+π^{n}R ⊆ P_{N} ∩R^{×}. Hence U_{n}(x, t) in conditions (2a), (2b) of the definition of Alg_{n} A can be written u(x, t)^{N} with u a semi-algebraic function from A to R^{×} (thanks to Theorem 2.8). This is how the above result is stated in [Den84] with G = K^{×}. Our slightly more precise form, as well as the additional properties involving Vect A and Alg_{n} A, appear only in the proof of theorem 7.3 in [Den84] (still with G = K^{×}). The generalization to fitting cells mod an arbitrary clopen semi-algebraic group G with finite index in K^{×} is straightforward.

The cell decomposition leads to a good dimension theory for semi-algebraic sets over p-adically closed fields, see [SvdD88] and [vdD89]. We will use repeatedly its following properties, for every semi-algebraic sets A, B and semialgebraic map f defined on A. By convention dim ∅ = −∞.

1. dim A = 0 if and only if A is finite non-empty.

2. dim A ∪ B = max(dim A, dim B).

3. If A ≠ ∅, dim ∂A < dim A.

4. dim f(A) ≤ dim A.

The local dimension of a semi-algebraic set A ⊆ K^{m} at a point a ∈ A is the minimum of dim U, for every semi-algebraic neighbourhood U of a in A (with respect to the relative topology, induced by K^{m} on A). A is pure dimensional if it has the same local dimension at every point. Note that if a semi-algebraic set B is open in A and A is pure dimensional then B is so, and that a cell is pure dimensional if and only if its socle is. This last point, combined with Denef’s Cell Decomposition Theorem 2.11 and a straightforward induction, shows that every semi-algebraic set A is the union of finitely many pure dimensional ones.

2.e Discrete and p-adic simplexes

We say that f : S ⊆ F_{I}(Γ^{m}) → Ω is affine if either it is constantly equal to +∞, or there are elements α_{0} ∈ Q and α_{i} ∈ Q for i ∈ I such that

∀x ∈ S, f(x) = α_{0} + ∑_{i∈I} α_{i}x_{i}.

In [Dar16] we introduced discrete polytopes in Γ^{q} as “largely continuous precells mod N”, for an arbitrary q-tuple N of positive integers. In this paper N = (1, …, 1) will not play any role so we can remove it in the next definition.

The only polytope in Γ^{0} is Γ^{0} itself (which is a one-point set). For every I ⊆ [[1, q + 1]], a subset A of F_{I}(Γ^{q+1}) is a discrete polytope of Γ^{q+1} if Â is a discrete polytope of Γ^{q} and if there is a pair (μ, ν) of largely continuous affine maps from Â to Ω, called a presentation of A, such that 0 ≤ μ ≤ ν and

A = {a ∈ F_{I}(Γ^{q}) : â ∈ Â and μ(â) ≤ a_{q+1} ≤ ν(â)}.

Example 2.12.

• A = {(x, y) ∈ Z^{2} : 0 ≤ y ≤ x/2} is a discrete simplex, with proper faces F_{{2}}(A) = {(+∞, y) ∈ {+∞} × Z : 0 ≤ y} and F_{∅}(A) = {(+∞, +∞)}.

• B = {(x, y, z) ∈ Z^{3} : (x, y) ∈ A and z = 2y − x} is a subset of Z^{3} defined by linear inequalities, whose proper faces F_{{3}}(B) = {+∞}^{2} × N and F_{∅}(B) are linearly ordered by specialization. However it is not a polytope, because the linear map φ(x, y) = 2y − x is not largely continuous on A: it has no limit when (x, y) tends to (+∞, +∞) in A.

All the references in the next proposition are taken from [Dar16].

Proposition 2.13. Let q ≥ 1 and A ⊆ F_{I}(Γ^{q}) be a discrete polytope. Let (μ, ν) be a largely continuous presentation of A, let J be a subset of I, and Ĵ = J \ {m}. Finally let Y = FĴ(Â). Then F_{J}(A) ≠ ∅ if and only if either m ∈ J and μ̄ < +∞ on Y, or m ∉ J and ν̄ = +∞ on Y (Proposition 3.11). When this happens:

1. F_{J}(A) = π_{J}(A) (Proposition 3.3).

2. The socle of F_{J}(A) is a face of Â: { \atop F_{J}(A)} = FĴ(Â) = Y (Proposition 3.7).

3. F_{J}(A) is a discrete polytope and (μ̄_{|Y} , ν̄_{|Y}) is a presentation of it (Proposition 3.11):

F_{J}(A) = {b ∈ F_{J}(Γ^{m}) : b̂ ∈ Y, and μ̄(b̂) ≤ b_{m} ≤ ν̄(b̂)}.

We will also use the next result (Proposition 3.5 in [Dar16]).

Proposition 2.14. Let A ⊆ Γ^{q} be a discrete polytope, f : A → Ω be an affine map and B = F_{J}(A) = π_{J}(A) a face of A. Assume that f extends to a continuous map f^{∗} : A ∪ B → Ω. Then f^{∗} is affine and if f^{∗} ≠ +∞ then f = f_{|B}^{∗} ∘ π_{J}_{|A}. In particular if f^{∗} ≠ +∞ then f(A) = f^{∗}(B).

A discrete simplex is a discrete polytope whose faces are linearly ordered by specialization. This is a “monohedral largely continuous precell mod (1, …, 1)” in [Dar16]. Of course every face of a simplex is a simplex (see Remark 3.12 of [Dar16]).

For every M ≥ 1 we let D^{M}R^{q} = (R ∩ Q_{1,M})^{q} and define p-adic simplexes of index M as the inverse images of discrete simplexes by the restriction of the valuation to D^{M}R^{q}. The faces of a simplex S of index M are obviously the pre-images in D^{M}R^{q} of the faces of vS. In particular they are linearly ordered by specialization. S is closed if and only if vS is a singleton in Γ^{q}. If S is not closed, its largest proper face T is called its facet and ∂S = \overline{T}.

Remark 2.15. With the notation of Proposition 2.14, if S = v^{−1}(A) ∩ D^{M}R^{q} and T = v^{−1}(B) ∩ D^{M}R^{q} then T = F_{J}(S), and so by Proposition 2.14 T = π_{J}(S). We will sometimes refer to the restriction of π_{J} to A (resp. S) as to “the coordinate projection of A onto B (resp. of S onto T)”.

2.f Simplicial complexes

We will have to consider complexes of sets, of cells and of simplexes. All of them are finite families of subsets of a topological space X, organised in a such a way that one controls how the closures of these sets intersect.

Recall first that an ordered set A is a tree if for every A in A, the set of elements in A smaller than A is linearly ordered. It is a rooted tree if it has one smallest element. A lower subset of A is a subset B of A such that whenever an element of A is smaller than an element of B, it belongs to B.

Now, given a finite family A of pairwise disjoint subsets of X, we call A a closed complex if every A ∈ A is relatively open and if it frontier ∂A is a union of elements of A. The specialization preorder is then an order on A. If A, ordered by specialization, is a tree (resp. a rooted tree) we call it a (resp. rooted) closed monoplex. A complex (resp. monoplex) is then an arbitrary subfamily of a closed complex (resp. closed monoplex). Of course a complex A is a closed complex if and only if ⋃ A is closed.

Remark 2.16. Using that every semi-algebraic set is the disjoint union of finitely many pure dimensional ones, and that dim ∂A < A for every semialgebraic set A, a straightforward induction shows that every finite family of semi-algebraic subsets of K^{m} refines in a complex of pure dimensional semialgebraic sets.

A simplicial complex in D^{M}R^{q} (resp. in Γ^{q}) is a complex of simplexes in D^{M}R^{q} (resp. in Γ^{q}). We say that S is well dispatched if for every S, S′ ∈ \overline{S}, S′ ≤ S if and only if Supp S′ ⊆ Supp S.

Let S be a finite family of simplexes in D^{M}R^{q} (or Γ^{q}). Then S is a simplicial complex if and only if for every S, T ∈ S, \overline{S} ∩ \overline{T} is the union of the common faces of S and T. When this happens:

1. S is a monoplex;

2. every subset S_{0} of S in D^{M}R^{q} is again a simplicial complex;

3. ⨄ S_{0} is closed in ⨄ S if and only if S_{0} is a lower subset of S.

Let \overline{S} denote the family of all the faces of the elements of S. We call it the closure of S, and say that S is closed if S = \overline{S}. Note that S is a complex (resp. a closed complex) if and only if S ⊆ \overline{S} (resp. S = \overline{S}) and the elements of \overline{S} are pairwise disjoint.

If S is a simplicial complex, we say T is a simplicial subcomplex of S of if T is a simplicial complex such that \overline{T} refines a decreasing subset of \overline{S}, and ⋃ T is a closed subset of ⋃ S.

The following results are respectively Theorem 6.3 and Proposition 6.4 of [Dar16].

Theorem 2.17 (Monotopic Division). Let S be a simplex in D^{M}R^{m} and T a simplicial complex in D^{M}R^{m} which is a partition of ∂S. Let ε : ∂S → K^{×} be a definable function such that the restriction of |ε| to every proper face of S is continuous. Then there exists a finite partition U of S such that U ∪ T is a simplicial complex in D^{M}R^{m}, U contains for every T ∈ T a unique cell U with facet T, and moreover ∥u − π_{J}(u)∥ ≤ |ε(π_{J}(u))| for every u ∈ U, where J = Supp(T).

Proposition 2.18. Let A ⊆ D^{M}R^{q} be a relatively open set. Assume that A is the union of a simplicial complex A in D^{m}R^{q}. Then for every integer n ≥ 1 there exists a finite partition of A in semi-algebraic sets A_{1}, …, A_{n} such that ∂A_{k} = ∂A for every k.

Finally, a simplicial complex of index M is a collection S of finitely many (possibly zero) rooted simplicial complexes S_{i} in D^{M}R^{q_{i}}, for various integers q_{i} (indexed by some finite set I). Its closure is the collection of the closures of the S_{i}’s. It is well dispatched if each S_{i} is. We say that T = (T_{i})_{i∈I} is a simplicial subcomplex of S if each T_{i} is a simplicial subcomplex of S_{i}.

Given a semi-algebraic homeomorphism φ from ⨄ S to a subset X of K^{m}, we will let

φ(S) = {φ(S) : S ∈ S}.

If S is closed, φ(S) is obviously a closed monoplex of pure dimensional semialgebraic sets partitioning X.

Remark 2.19. With φ as above, S is closed if and only if X is closed and bounded. Indeed, each ⋃ S_{i} is clopen in ⨄ S, hence its homeomorphic image X_{i} by φ is clopen in X. In particular X is closed and bounded in K^{m} if and only if each X_{i} is so. Let φ_{i} be the semi-algebraic homeomorphism from ⋃ S_{i} to X_{i} induced by restriction of φ. Note that ⋃ S_{i} is bounded (it is contained in R^{q_{i}}). By Theorem 2.6 applied to φ_{i} and φ_{i}^{−1} it follows that ⋃ S_{i} is closed in K^{q_{i}}, that is S_{i} is closed, if and only if X_{i} is closed and bounded in K^{m}.

3 Applications

The proof of our main result T_{m} goes by induction on m. Some of the applications given in this section are actually needed in the induction step, hence it is important to emphasize that all of them follow from T_{m} for a fixed (throughout all this section) integer m.

Theorem 3.1. If f : X ⊆ K^{m} → K is semi-algebraic and |f| is continuous, then there exists a function h : X → K semi-algebraic and continuous such that |f| = |h| on X.

Proof: T_{m} gives a triangulation (T , φ) of f with parameters (1, 1, e, M). On every T ∈ T , f ∘ φ_{|T} = U_{e,1}ψ with ψ : T → K a 1-monomial function. Thus for some q_{T} such that T is contained in D^{M}R^{q_{T}}, there are λ_{T} in K and α_{1,T} , …, α_{q_{T} ,T} in Z such that:

∀x ∈ T, |f ∘ φ(x)| = |λ_{T} ∏_{i=1}^{q_{T}} x_{i}^{α_{i,T}} |  (3)

Let α_{0,T} = vλ_{T} and ξ_{T} : vT → Γ be defined by:

∀a ∈ vT, ξ_{T} (a) = α_{0,T} + ∑_{i=1}^{q_{T}} α_{i,T} a_{i}

By construction ξ_{T} (vx) = vf(φ(x)) for every x ∈ T (in particular ξ_{T} only depends on f ∘φ, even if the coefficients α_{i,T} in (3) are not uniquely determined by f ∘φ on T). By assumption vf is continuous on X hence so is vf ∘φ on ⨄ T . In particular ξ_{T} extends continuously to vS for every face S of T in T , and the restriction to S of such an extension \overline{ξ}_{T} is precisely ξ_{S}. By proposition 2.14 it follows that if ξ_{S} ≠ +∞ (that is if f ∘ φ ≠ 0 on S) then ξ_{T} = ξ_{S} ∘ π_{S} where π_{vS} denotes the coordinate projection of vT to vS (see Remark 2.15).

Now, for every T in T let g_{T} : T → K be defined (by induction on T ordered by specialization) as follows:

1. If f ∘ φ = 0 on T, g_{T} = 0.

2. If T is minimal (with respect to the specialisation preorder) among the simplexes in T on which f ∘ φ ≠ 0 then for every x ∈ T:

g_{T} (x) = π^{α_{0,T}} ∏_{i=1}^{q_{T}} x_{i}^{α_{i,T}}

3. Otherwise g_{T} = g_{S} ∘ π_{S} where π_{S} is the coordinate projection (see Remark 2.15) of T onto its larger proper face S in T on which f ∘ φ ≠ 0.

By construction vg_{T} (x) = ξ_{T} (vx) for every x ∈ T hence |g_{T} | = |f| on T. Moreover for every S ≤ T and y ∈ S, g_{T} (x) tends to g_{S}(y) as x tends to y in T (because g_{T} (x) = g_{S}(π_{S}(x)) if g_{S} ≠ 0, and otherwise because |g_{T} | = |f ∘ φ| on T, |g_{S}| = |f ∘ φ| = 0 on S and |f ∘ φ| = |f| ∘ φ is continuous by assumption).

The function h : X → K defined by h = g_{T} ∘ φ^{−1} on every φ(T) with T in T , is clearly semi-algebraic. By construction |f| = |h| on X, and by the above argument h is continuous on X.

Theorem 3.2. For every non-empty semi-algebraic sets Y ⊆ X ⊆ K^{m}, there is a semi-algebraic retraction of X onto Y if and only if Y is closed in X.

Proof: One direction is general. For the converse we assume that Y is closed in X. Let (S, φ) be a triangulation of X, Y given by T_{m}, and T be the family of simplexes T in S such that φ(T) ⊆ Y. It suffices to construct a continuous retraction of ⨄ S onto ⨄ T .

Let S_{0} = T and σ_{0} be the identity map on ⨄ T . Because Y is closed in X, T is a lower subset of S. Let k be a positive integer and assume that there is a lower subset S_{k−1} of S containing T , and a retraction σ_{k−1} of ⨄ S_{k−1} to T . If S_{k−1} = S we are done. Otherwise let S be a minimal element (with respect to the specialisation order) in S \ S_{k−1}, and let^{7} S_{k} = S_{k−1} ∪ {S}. It only remains to build a retraction τ of ⨄ S_{k} onto ⨄ S_{k−1}. Indeed σ_{k−1} ∘ τ will then be a continuous retraction of S_{k} onto T , and the result will follow by induction.

If S has no proper face in S then it is clopen in ⨄ S_{k}. So the map τ which is the identity map on ⨄ S_{k} and which sends every point of S to an arbitrary given point of ⨄ S_{k−1} is continuous on S_{k}, and a retraction of ⨄ S_{k} onto S_{k−1}.

Otherwise let T be the largest proper face of S in S. By minimality of S, T belongs to S_{k−1}. Let π_{T} be the coordinate projection of S onto T. The frontier of S inside ⨄ S_{k} is the closure of T in ⨄ S_{k}, hence the function τ which coincides with the identity map on ⨄ S_{k−1} and with π_{T} on S is continuous. It is then a retraction ⨄ S_{k} onto S_{k−1}, which finishes the proof.

7We are abusing the notation here: S is a finite collection of simplicial simplexes S_{i} in D^{M} R^{q_{i}} for various q_{i}, S_{k} is a collection of lower subsets S_{k,i} of S_{i}, there is an index i_{0} such that S belongs to S_{i_{0}}, and what we have denoted abusively S_{k−1} ∪ {S} is actually the collection of all the S_{k,i}’s for i ≠ i_{0} and of S_{k,i_{0}} ∪ {S}.

The Splitting Theorem 3.4 is a strengthening of the next lemma using retractions.

Lemma 3.3. Let X ⊆ K^{m} be a relatively open semi-algebraic set without isolated points and n ≥ 1 an integer. Then there exists a partition of X in semialgebraic sets X_{k} for 1 ≤ k ≤ n such that ∂X_{k} = ∂X for every k.

We are going to prove Lemma 3.3 by using a triangulation (U, φ) of (X, ∂X) and applying Proposition 2.18 to φ^{−1}(X). In order to ensure that this set is still relatively open, we reduce first to the case where X is bounded by means of the following construction.

Let K̂ = K ∪ {∞} and for every I ⊆ {1, …, m} let K_{I}^{m} = K̂_{I}^{m} ∩ K^{m} where

K̂_{I}^{m} = {x ∈ K̂^{m} : x_{k} ∈ R ⟺ k ∈ I}.

Let R_{I}^{m} = {x ∈ R^{m} : ∀k ∉ I, x_{k} ≠ 0}, and for every x ∈ R_{I}^{m} let ψ_{I}(x) = (y_{k})_{1≤k≤m} be defined by y_{k} = x_{k} if k ∈ I, and y_{k} = 1/(πx_{k}) otherwise. Clearly ψ_{I} is semi-algebraic homeomorphism from R_{I}^{m} to K_{I}^{m} which extends uniquely to a homeomorphism ψ̂_{I} from R^{m} to K̂_{I}^{m}.

Proof: Note first that if it is given a partition of X in finitely many semialgebraic pieces U_{1}, …, U_{r} which are clopen in X, then it suffices to prove the result separately for each U_{j}. Indeed, each U_{j} will then be relatively open with ∂U_{j} ⊆ ∂X (because U_{j} is clopen in X), and ⋃_{j≤r} ∂U_{j} = ∂X (because ∂U_{j} = \overline{U}_{j} \ X and ⋃_{j≤r} \overline{U}_{j} = \overline{X}). So, if a partition of each U_{j} in semi-algebraic pieces (U_{j,k})_{1≤k≤n} is found such that ∂U_{j,k} = ∂U_{j} for every k, then the union X_{k} of U_{j,k} for 1 ≤ j ≤ r defines a partition of X in semi-algebraic pieces and we have ∂X_{k} = ⋃_{j≤r} ∂U_{j,k} (same argument as above) hence ∂X_{k} = ⋃_{j≤r} ∂U_{j} = ∂X.

Now, as I ranges over the subsets of {1, …, m}, the sets X ∩ K_{I}^{m} form a partition of X in semi-algebraic sets clopen in X. By the argument above we can deal separately with each of these sets, which reduces to the case where X ⊆ K_{I}^{m} for some I.

Let Y = ψ_{I}^{−1}(X) and X̂ be the closure of X in K̂_{I}^{m}. Note that ψ̂_{I}(\overline{Y}) = X̂. The fact that \overline{X} \ X is closed K^{m}, hence in K_{I}^{m}, implies that X̂ \ X is closed in K̂_{I}^{m}. It follows that its image by ψ̂_{I}^{−1}, which is precisely \overline{Y} \ Y, is closed in R^{m}, hence in K^{m}. Thus Y is relatively open. It then suffices to prove the result for Y, that is we can assume that X = Y is bounded. Of course we can assume as well that ∂X is non-empty (otherwise X_{1} = X and X_{k} = ∅ for 2 ≤ k ≤ n is obviously a solution).

T_{m} gives a triangulation (U, φ) of (X, ∂X). U is the disjoint union of finitely many simplicial complexes U_{j} in D^{M}R^{q_{j}} for 1 ≤ j ≤ r. Let U_{j} = φ(⋃ U_{j}) ∩ X for every j, this defines a partition of X in semi-algebraic sets clopen in X. By using again the initial remark in this proof, it suffices to check the result for each U_{j} separately. So we can assume that U itself is a simplicial complex in D^{M}R^{q} for some q.

By construction \overline{X} is semi-algebraic, closed and bounded, and φ^{−1} is semialgebraic and continuous, so φ^{−1}(\overline{Y}) = \overline{φ^{−1}(Y)} for every semi-algebraic^{8} Y ⊆ \overline{X}. Let A = φ^{−1}(X), we have \overline{A} = φ(\overline{X} hence \overline{A} \ A = φ(\overline{X} \ X) is closed, that is A is relatively open. Proposition 2.18 then applies to A and gives a partition of A in semi-algebraic sets A_{1}, …, A_{n} such that ∂A_{k} = ∂A for every k.

For 1 ≤ k ≤ n let X_{k} = φ(A_{k}). This semi-algebraic sets form a partition of X, because A_{1}, …, A_{n} form a partition of A. Moreover, since ⋃ U is semialgebraic, closed and bounded, we have φ(\overline{B}) = \overline{φ(B)} for every semi-algebraic^{9} set B contained in ⋃ U. It follows that for 1 ≤ k ≤ n we have ∂X_{k} = φ(∂A_{k}) = φ(∂A) = ∂X, which proves the result.

Theorem 3.4. Let X be a relatively open non-empty semi-algebraic subset of K^{m} without isolated points, and Y_{1}, ⋯, Y_{s} a collection of closed semi-algebraic subsets of ∂X such that Y_{1} ∪ ⋯ ∪ Y_{s} = ∂X. Then there is a partition of X in non-empty semi-algebraic sets X_{1}, …, X_{s} such that ∂X_{i} = Y_{i} for 1 ≤ i ≤ s.

Proof: X is non-empty and has no isolated point, hence is infinite. The result is obvious for s = 0 (there is nothing to prove) and s = 1 (take X_{1} = X). By induction it suffices to prove it for s = 2. Indeed, if s ≥ 3 and the result is proved for s−1, then the result for s = 2 applied to X with Z_{1} = Y_{1} ∪⋯∪Y_{s−1} and Z_{2} = B_{s} gives a partition in two pieces X′_{1}, X′_{2} such that ∂X′_{l} = Z_{l} for l = 1, 2, and the induction hypothesis applied to X′_{1} with Y_{1}, …, Y_{s} gives a partition of X′_{1} in pieces X_{1}, …, X_{s−1} such that ∂X_{i} = Y_{i} for 1 ≤ i ≤ s. The conclusion follows, by taking X_{s} = X′_{2}. So from now on we assume that s = 2.

It suffices to prove the weaker result that a partition (X′_{1}, X′_{2}) exists with all the required properties for (X_{1}, X_{2}) except possibly the condition that they are non empty. Indeed, if such a partition is found and for example X′_{2} = ∅ then necessarily Y_{2} = ∂X′_{2} = ∅. In that case pick any x ∈ X, and choose a clopen neighbourhood V of x such that V ∩ ∂X is empty (this is possible because X is relatively open). Then X_{1} = X \ V and X_{2} = X ∩ V give the conclusion.

Let ρ : \overline{X} → ∂X be a continuous retraction of \overline{X} onto ∂X given by Theorem 3.2. Let V ⊆ ∂X be any semi-algebraic set open in ∂X, Z its closure and A = ρ^{−1}(Z) ∩ X. We are claiming that ∂A = Z. Note that A is closed in X by continuity of ρ, because A is the inverse image of the closed set Z by ρ_{|X}. So it suffices to prove that \overline{A} ∩ ∂X = Z, or equivalently that \overline{A} ∩ ∂X contains V and is contained in Z. For the first inclusion let y be any element of V, and W any neighbourhood of y. We have to prove that W ∩ A ≠ ∅. By continuity of ρ at y = ρ(y) there is a neighbourhood U of y such that U ∩ \overline{X} is contained in ρ^{−1}(W ∩ V). In particular

U ∩ W ∩ X ⊆ U ∩ \overline{X} ⊆ ρ^{−1}(W ∩ V) ⊆ ρ^{−1}(V) = A

so U ∩ W ∩ A = U ∩ W ∩ X. On the other hand, U ∩ W ∩ X ≠ ∅ because U ∩ W is a neighbourhood of y and y ∈ V ⊆ \overline{X}. A fortiori W ∩ A is non-empty. This proves that y ∈ \overline{A}, hence that V ⊆ \overline{A} ∩ ∂X. Conversely, if y′ is any element of ∂X \ Z, there is a neighbourhood W′ of y′ such that W′ ∩ ∂X is disjoint from Z. By continuity of ρ, ρ^{−1}(W) is then a neighbourhood of y′ in \overline{X}. It is disjoint from ρ^{−1}(Z) = A hence y′ ∉ \overline{A}. So \overline{A} is disjoint from ∂X \ Z. That is \overline{A} ∩ ∂X ⊆ Z, which proves our claim.

8For every continuous map f : X ⊆ K^{q} → K^{r} and every Y ⊆ X, if X is closed then f(\overline{Y}) ⊆ \overline{f(Y)}. The reverse inclusion holds true if X is compact, or if f, Y, X are semialgebraic and X is closed and bounded (see Theorem 2.6).

9See footnote 8

For k = 1, 2 let Z_{k} be the closure of Y_{k}\Y_{2−k} = ∂X\Y_{2−k} and A_{k} = ρ^{−1}(Z_{k}). Let Z_{0} be the closure of ∂X \ (Z_{1} ∪ Z_{2}) and A_{0} = ρ^{−1}(Z_{0}). The above claim gives that ∂A_{k} = Z_{k} for 0 ≤ k ≤ 2. Let B_{0} be the set of non-isolated points of A_{0}. Clearly ∂B_{0} = ∂A_{0} = Z_{0} since A_{0}\B_{0} is finite. In particular B_{0} is relatively open, and Lemma 3.3 gives two semi-algebraic sets B_{1}, B_{2} partitioning B_{0} such that ∂B_{1} = ∂B_{2} = Z_{0}. So if we set X_{1} = A_{1} ∪ B_{1} and X_{2} = A_{2} ∪ B_{2} ∪ (A_{0} \ B_{0}) we get the conclusion.

Theorem 3.5. Let f : X ⊆ K^{m} → K be a semi-algebraic function with bounded graph (that is f is a bounded function on a bounded domain). If it has finitely many limit values at every point of \overline{X} then f is piecewise largely continuous.

Note that the counterpart of Theorem 3.5 for real-closed fields holds true. Indeed, by triangulation we are reduced to the case of a continuous function f on a simplex S ⊆ K^{m}. The assumption that f has finitely many limit values at every point of \overline{S} then implies directly that f is largely continuous. Indeed, this follows easily from the fact that over real-closed fields the direct image by a continuous semi-algebraic map of any semi-algebraically connected set (such as S ∩ B with B a ball centered at any point of \overline{S}) is again semi-algebraically connected.

At the contrary, p-adic simplexes are not at all semi-algebraically connected and it can happen that a function satisfying all these assumptions on a p-adic simplex is not largely continuous. For example on the simplex S = D^{1}R^{∗} the semi-algebraic function f defined by f(x) = 0 if v(x) ∈ 2Z and f(x) = 1 otherwise is a continuous, bounded function having two distinct limit values at 0. Thus f is not largely continuous. It is obviously piecewise largely continuous, though.

Proof: Every semi-algebraic function is piecewise continuous (see for example [Mou09]). So, replacing f by its restriction to the pieces of an appropriate partition of X is necessary, we can assume that f is continuous. Removing X ∩ ∂X if necessary (using a straightforward induction on dim X and the fact that dim ∂X < dim X) we can even assume that X is relatively open. The proof then goes by induction on the lexicographically ordered tuples (e, e′) where e = dim X and e′ = dim ∂X. If ∂X is empty, that is X is closed, then f is largely continuous and the result is obvious. So let us assume that e′ ≥ 0 (hence e ≥ 1) and the result is proved for smaller tuples (e, e′).

Let D = (∂X × K) ∩ \overline{Gr f}. The projection of D onto ∂X has finite fibers hence D is a union of cells of type 0. The number of these cells, say N, then bounds the cardinality of these fibers, that is the number of limit values of f at every point of ∂X. For every a ∈ ∂X let D_{a} = {t ∈ K : (a, t) ∈ D}. We first show that D̂ = ∂X, that is D_{a} ≠ ∅ for every a ∈ ∂X. For every ε ∈ R^{∗} let C_{ε} = (B(a, ε) × K) ∩ \overline{Gr f}. This is a uniformly semi-algebraic family of closed and bounded semi-algebraic subsets of K^{n}. Each of them is non-empty because C_{ε} contains (x, f(x)) for any x in B(a, ε)∩X (which is non-empty since a ∈ ∂X). Obviously C_{ε_{1}} ⊆ C_{ε_{2}} whenever |ε_{1}| ≤ |ε_{2}|, so ⋂_{ε∈R^{∗}} C_{ε} is non-empty by Theorem 2.5. The latter is equal to D_{a}, which proves our claim.

For 1 ≤ i ≤ N let W_{i} be the set of a ∈ ∂X such that D_{a} has exactly i elements. These sets W_{i} form a partition of D in semi-algebraic pieces. By Theorem 2.8 (and a straightforward induction) there are semi-algebraic functions f_{i,j} : W_{i} → K such that D_{a} = {f_{i,j}(a)}_{1≤j≤i} for every a ∈ D_{a}. Since dim ∂X < dim X, by induction hypothesis these functions f_{i,j} are piecewise largely continuous. This gives a partition of ∂X in semi-algebraic pieces V_{k} for 1 ≤ k ≤ r, and a family of largely continuous semi-algebraic functions g_{k,l} : V_{k} → K for 1 ≤ l ≤ s_{k} such that V_{k} ⊆ W_{s_{k}} and D is the union of the graphs of all these functions g_{k,l}.

Theorem 3.4 applied to X and the sets \overline{V_{k}} for 1 ≤ k ≤ r gives a partition of X in semi-algebraic pieces X_{k} such that ∂X_{k} = \overline{V_{k}}. It suffices to prove that the restrictions of f to each X_{k} is piecewise largely continuous. So we can assume that r = 1 and X = X_{1}. That is, we have a semi-algebraic set V = V_{1} dense in ∂X and largely continuous functions g_{l} = g_{1,l} : V → K for 1 ≤ l ≤ s = s_{1} such that D_{a} = {g_{l}(a)}_{1≤l≤s} has s elements for every a ∈ D_{a}. Replacing V by V \ ∂V if necessary we can assume that V is relatively open.

Let ρ : \overline{X} → \overline{V} be a continuous retraction given by Theorem 3.2. For 1 ≤ l ≤ s let

U_{l} = {x ∈ X : ∀k ≠ l, |f(x) − ḡ_{l}(ρ(x))| < |f(x) − ḡ_{k}(ρ(x))|}.

Each U_{l} is open in X by continuity of f, ρ and the ḡ_{k}’s. Their complement X′ = X \ ⋃_{k=1}^{l} is closed in X, hence ∂X′ ⊆ ∂X. Moreover, for every a ∈ V, the limit values g_{k}(a) of f at a being distinct by construction, there exists ε ∈ R^{∗} such that every point of B(a, ε) ∩ X belongs to one of the U_{k}’s. With other words B(a, ε) ∩ X′ = ∅ hence a does not belong to the closure of X′. So ∂X ⊆ ∂X \ V = ∂V, in particular dim ∂X′ < dim V = dim ∂X hence the induction hypothesis applies to the restriction of f to X′.

It only remains to check that the restrictions of f to each U_{l} is piecewise largely continuous. We are claiming that f has only one limit value at every point a of \overline{U}_{l} \∂V. Note that \overline{U}_{l} is the disjoint union of \overline{U}_{l} ∩X and \overline{U}_{l} ∩∂X, and that ∂X = V ∪∂V. Obviously, if a ∈ \overline{U}_{l} ∩X then by continuity of f, f(x) tends to f(a) as x tends to a in U_{l}. Now if a ∈ (\overline{U}_{l} \ ∂V) \ X then a ∈ V, ρ(a) = a and ḡ_{k}(ρ(a)) = g_{k}(a) for every k. Hence by definition of U_{l}, f(x) is closer to g_{l}(a) than to every other g_{k}(a), so g_{l}(a) is the only possible limit value of f(x) as x tends to a in U_{l}, which proves our claim. So the semi-algebraic function g which coincides with f on \overline{U}_{l} ∩ X and with g_{l} on V is continuous. The frontier of its domain is contained in \overline{U_{l}} ∩ ∂X ⊆ ∂X = V ∪ ∂V and is disjoint from V, hence is contained in ∂V. By induction hypothesis, g is then piecewise largely continuous, hence so is f_{|U_{l}} since f and g coincide on U_{l}.

4 Largely continuous cell decomposition

This section recalls the main theorem of [Den84] in order to emphasize some details which appear only in its proof. These details are important for us because they ensure that the center and bounds of the cells involved in the conclusions inherit certain properties, defined below, from the center and bounds of the cells in the assumptions. Using them we finally derive from T_{m} a new preparation theorem for semi-algebraic functions “up to a small deformation” (Theorem 4.6). The point is that after such a deformation, we get a preparation theorem involving only largely continuous cells.

Given a basic function f, we say that a function h : X ⊆ K^{m} → K belongs to coalg(f) if there exists a finite partition of X into definable pieces H, on each of which the degree in t of f(x, t) is constant, say e_{H}, and such that the following holds. If e_{H} ≤ 0 then h(x) is identically equal to 0 on H. Otherwise there is a family (ξ_{1}, …, ξ_{r_{H}}) of K-linearly independent elements in an algebraic closure of K and a family of definable functions b_{i,j} : H → K for 1 ≤ i ≤ e_{H} and 1 ≤ j ≤ r_{H}, and a_{e_{H}} : H → K^{∗} such that for every x in H

f(x, T) = a_{e_{H}} (x) ∏_{1≤i≤e_{H}} (T − ∑_{1≤j≤r_{H}} b_{i,j}(x)ξ_{j})

and

h(x) = ∑_{1≤i≤e_{H}} ∑_{1≤j≤r_{H}} α_{i,j}b_{i,j}(x)

with the α_{i,j}’s in K. If F is any family of basic functions we let coalg(F) denote the set of linear combinations of functions in coalg(f) for f in F.

Theorem 4.1 (Denef). Let F ⊆ K[X, T] be a finite family of polynomials, with X an m-tuple of indeterminates and T one more indeterminate. Let N ≥ 1 be an integer and A a family of boolean combinations of subsets of the form f^{−1}(P_{N}) with f ∈ F. For every integer n ≥ 1 there is a finite family of fitting cells mod P_{N}^{×} refining A, with center and bounds in coalg(F), and for every such cell H a positive integer α_{f,H} and a semi-algebraic function h_{f,H} : Ĥ → K such that for every (x, t) ∈ H:

f(x, t) = U_{n}(x, t)h_{f,H}(x)(t − c_{H}(x))^{α_{f,H}} .

Proof: W.l.o.g. we can assume that every f in F is non constant and that n is large enough so that 1 + π^{n}R ⊆ P_{N}^{×}. Theorem 7.3 in [Den84] gives a finite family of cells B mod K^{×} partitioning K^{m}, and for each of them a positive integer α_{f,B} and semi-algebraic functions u_{f,B} : B → R^{×} and h_{f,B} : B̂ → K such that:

∀(x, t) ∈ B, f(x, t) = u_{f,B}(x, t)^{N}h_{f,B}(x)(t − c_{B}(x))^{α_{f,B}}  (4)

Moreover it appears along the line of the proofs of lemma 7.2 and theorem 7.3 in [Den84] that these functions u_{f,B}^{N} are precisely of the form 1 + π^{n}ω_{f,B} for some semi-algebraic function ω_{f,B} on B, and that the center and bounds of B belong to coalg(F). Refining the socle of B if necessary we can ensure that h_{f,B}(x)P_{N}^{×} is constant as (x, t) ranges over B. On the other hand B splits into finitely many cells mod P_{N}^{×}, with the same center and bounds as B, because P_{N}^{×} as finite index in K^{×}. On each of these cells H, f(x, t)P_{N}^{×} is constant by (4). Hence H is either contained or disjoint from A, for every A ∈ A. So the family of all these cells H which are contained in ⋃ A gives the conclusion.

Using that every semi-algebraic function is piecewise continuous, the cells mod P_{N}^{×} given by Theorem 4.1 can easily be chosen with continuous center and bounds. However it is not possible to ensure that they are largely continuous (think to the case where A consists in a single semi-algebraic set which is itself the graph of a semi-algebraic function which is not largely continuous). Our aim, in the remaining of this section, is to find a work-around. We are going to prove that it can be done, not exactly for θ but for a function θ ∘ u_{η} where η ∈ K^{m} can be chosen arbitrarily small and u_{η} is the linear automorphism of K^{m+1} defined by:

∀(x, t) ∈ K^{m} × K, u_{η}(x, t) = (x + tη, t).  (5)

Remark 4.2. The smaller is η, the closer is u_{η} from the identity map since ∥η∥ is also the norm (in the usual sense for linear maps) of u_{η} − Id. So the functions θ ∘ u_{η} can be considered as “arbitrarily small deformations” of θ.

In [vdD98] a good direction for a subset S of K^{m+1} is defined as a non-zero vector x = (x_{1}, …, x_{m+1}) ∈ K^{m+1} such that every line directed by x has finite intersection with S. It is more convenient to identify collinear such vectors hence we redefine good directions for S as the points x = [x_{1}, …, x_{m+1}] in the projective space P^{m}(K) such that every affine line in K^{m+1} directed by x has finite intersection with S.

Analogously we call x ∈ P^{m}(K) a geometrically good direction for a family F of polynomials in K[X, T] if for every algebraic extension F of K and every f ∈ F, x is a good direction for the zero set of f in F^{m+1}.

Remark 4.3. With the above notation, [η, 1] is a good direction for S if and only if the projection of u_{η}^{−1}(S) onto K^{m} has finite fibers. Indeed for every a ∈ K^{m} and every t ∈ K we have:

(a, 0) + t(η, 1) ∈ S ⟺ (a + tη, t) ∈ S ⟺ (a, t) ∈ u_{η}^{−1}(S)

Therefore [η, 1] is a geometrically good direction for F if and only if for every algebraic extension F of K and every f ∈ F, the projection onto F^{m} of the zero set of f ∘ u_{η} in F^{m+1} has finite fibers.

Lemma 4.4 (Good Direction). For every finite family F of non-zero polynomial in K[X, T], the set of geometrically good directions for F contains a non empty Zariski open subset of P^{m}(K). In particular, for every non-zero ε ∈ R there is η ∈ R^{m} such that ∥η∥ ≤ |ε| and [η, 1] is a good direction for F.

Proof: Let p_{F} be the product of the polynomials in F, and d its total degree. Then p_{F} writes p_{F} = p_{F}^{∘} − q_{F} with p_{F}^{∘} a non zero homogeneous polynomial of degree d and q_{F} a polynomial of total degree < d.

Let b ∈ K^{m+1} be non-zero and x be the corresponding point in P^{m}(K). It is not a geometrically good direction for F if and only if for some algebraic extension F of K and some a ∈ F^{m} the line a + F.b is contained in the zero set of p_{F} in F^{m+1}, that is p_{F}(a + tb) = 0 for every t ∈ F or equivalently p_{F}^{∘}(a + T b) = q_{F}(a + T b). This implies that the degree in T of p_{F}^{∘}(a + T b) is < d. In particular the coefficient of T^{d} in p_{F}^{∘}(a + T b) is zero. A straightforward computation shows that this coefficient is just p_{F}^{∘}(b).

So every element in P^{m}(K) which is outside the zero set of p_{F}^{∘} is a geometrically good direction for F. This proves the main point. Now if K^{m} is identified to its image in P^{m}(K) by the mapping a ↦ [a, 1] then every ball in K^{m} is Zariski dense in P^{m}(K), so the last point.

Lemma 4.5. Assume T_{m}. Let η ∈ K^{m} be such that [η, 1] is a geometrically good direction for F. Let u_{η} be as in (5) and F_{η} = {f ∘u_{η} : f ∈ F}. Then every function in coalg(F_{η}) whose graph is bounded is piecewise largely continuous.

Proof: The functions in coalg(F_{η}) are linear combinations of functions in coalg(f_{η}) for f ∈ F, hence it suffices to fix any f in F and prove the result for coalg(f_{η}). Let d be the degree in T of f, and F a Galois extension of K in which every polynomial in K[T] of degree ≤ d splits in linear factor. Given a basis B = (ξ_{1}, …, ξ_{r}) of F over K, for each integer e ≤ d let a_{e} ∈ K[X] be the coefficient of T^{e} in f_{η}, let A_{e} ⊆ K^{m} be the set of elements x ∈ K^{m} such that d_{T}^{∘} f_{η}(x, T) = e, and choose a family of semi-algebraic functions b_{i,j} : A_{e} → K such that for every x ∈ A_{e}

f_{η}(x, T) = a_{e}(x) ∏_{i≤e} (T − ∑_{j≤r} b_{i,j}(x)ξ_{j}).  (6)

Let Z_{F} (f_{η}) denote the zero set of f_{η} in F, and σ_{1}, …, σ_{r} be the list of K-automorphisms of F. Fix an integer i ≤ e, and for every x ∈ A_{e} let

λ_{i}(x) = ∑_{j≤r} b_{i,j}(x)ξ_{j}.

For every k ≤ r we have

σ_{k}(λ_{i}(x)) = ∑_{j≤r} b_{i,j}(x)σ_{k}(ξ_{j}).

Inverting the matrix (σ_{k}(ξ_{j}))_{j≤r,k≤r} gives for every j ≤ r the function b_{i,j} as a linear combination of σ_{k} ∘ λ_{i} for k ≤ r. By construction Gr σ_{k} ∘ λ_{i} is contained in Z_{F} (f_{η}). The later is closed, hence \overline{Gr σ_{k} ∘ λ_{i}} is contained in Z_{F} (f_{η}) too.

The projection of Z_{F} (f_{η}) onto F^{m} has finite fibers since η is a good direction for F (see Remark 4.3). So the same holds true for the closure of the graph of σ_{k} ∘ λ_{i}. This means that each σ_{k} ∘ λ_{i} has finitely many different limit values at every point of \overline{A_{e}}. Obviously each b_{i,j} inherit this property, hence so does every h ∈ coalg f_{η}. If moreover the graph of h is bounded, it then follows from Theorem 3.5 (using T_{m}) that h is piecewise largely continuous.

Now we can turn to the “largely continuous cell decomposition up to small deformation” which was the aim of this section. We obtain it by combining the above construction based on good directions and the classical cell preparation theorem for semi-algebraic functions from Denef (Corollary 6.5 in [Den84]) revisited by Cluckers (Lemma 4 in [Clu01]).

Theorem 4.6. Assume T_{m}. Let (θ_{i} : A_{i} ⊆ K^{m+1} ↦ K)_{i∈I} be a finite family of semi-algebraic functions whose domains A_{i} are bounded. Then for some integer e ≥ 1 and every integers n, N ≥ 1 there exists a tuple η ∈ K^{m}, an integer M_{0} > 2v(e), an integer N_{0} divisible by eN, and a finite family D of largely continuous fitting cells mod Q_{N_{0},M_{0}}^{×}, such that D refines {u_{η}^{−1}(A_{i}) : i ∈ I} and such that for every i ∈ I, every D ∈ D contained in u_{η}^{−1}(A_{i}) and every (x, t) ∈ D

θ_{i} ∘ u_{η}(x, t) = U_{e,n}(x, t)h_{i,D}(x)[λ_{D}^{−1}(t − c_{D}(x))]^{\frac{α_{i,D}}{e}}

where u_{η} is as in (5), h_{i,D} : D̂ → K is a semi-algebraic function and α_{i,D} ∈ Z.

Moreover the set of η ∈ K^{m} having this property is Zariski dense (in particular η can be chosen arbitrarily small), and the integers e, M can be chosen arbitrarily large (in the sense of footnote 2).

Remark 4.7. The above expression of θ_{i} ∘ u_{η} is well defined because e divides N_{0}, M_{0} > 2v(e) and λ_{D}^{−1}(t − c_{D}(x)) belongs to Q_{N_{0},M_{0}} for every (x, t) ∈ D (see the definition of x ↦ x^{\frac{1}{e}} on Q_{N_{0},M_{0}} after Lemma 2.9). Of course if D is of type 0, then λ_{D} = t − c_{D}(x) = 0 and we use our conventions that 0^{−1} = ∞ and ∞.0 = 1.

If we were only interested in the existence of such a preparation theorem with largely continuous cells for θ_{i} ∘ u_{η}, the integer N would be of no use and could be taken equal to 1. However it will be convenient to allow different values of N when we will use Theorem 4.6 in the proof the Triangulation Theorem.

Proof: Let e_{∗}, M_{∗} ≥ 1 be arbitrary integers. Corollary 6.5 in [Den84] applied to each θ_{i} gives an integer e_{i} ≥ 1 and a family A_{i} of semi-algebraic sets partitioning A_{i} such that for every every A in A_{i} and every (x, t) in A:

θ_{i}^{e_{i}}(x, t) = u_{i,A}(x, t)\frac{f_{i,A}(x, t)}{g_{i,A}(x, t)}  (7)

where u_{i,A} is a semi-algebraic function from A to R^{×} and f_{i,A}, g_{i,A} are polynomial functions such that g_{i,A}(x, t) ≠ 0 on A. Replacing if necessary each e_{i} by a common multiple e of them and of e_{∗}, we can assume that e_{i} = e for every i and e is divisible by e_{∗}. Let A be a refinement of ⋃_{i∈I} A_{i}’s.

Fix any two integers n, N ≥ 1 and any integer n_{0} such that n_{0} ≥ n + v(e) and n_{0} > 2v(e). Since D^{n_{0}}R^{×} is a subgroup of R^{×} with finite index, every A ∈ A splits into finitely many semi-algebraic pieces on each of which u_{i,A} is constant modulo D^{n_{0}}R^{×} (for every i ∈ I such that A ⊆ A_{i}). Thus, refining A if necessary, (7) can be replaced, for every A in A contained in A_{i} and every (x, t) in A, by

θ_{i}^{e}(x, t) = U_{n_{0}}(x, t)ũ_{i,A} \frac{f_{i,A}(x, t)}{g_{i,A}(x, t)}  (8)

with ũ_{i,A} ∈ R^{×}.

Macintyre’s Theorem 2.3, applied to each A in A gives a finite family B of semi-algebraic sets refining A, an integer N_{0} ≥ 1 and a finite list F of nonzero polynomials in m + 1 variables such that every element of B is a boolean combinations of sets f^{−1}(P_{N_{0}}) with f ∈ F. By Remark 2.2, N_{0} can be chosen divisible by eN. Expanding F if necessary, we can assume that all the polynomials f_{i,A} and g_{i,A} in (8) also belong to F (except of course those which are null).

Lemma 4.4 gives η ∈ K^{m+1} such that [η, 1] is a geometrically good direction for F_{η}, where F_{η} = {f ∘ u_{η} : f ∈ F}. Note that every set in A_{η} = {u_{η}^{−1}(A) : A ∈ A} is a boolean combination of sets f_{η}^{−1}(P_{N_{0}}) with f_{η} ∈ F_{η}. Denef’s Theorem 4.1 applied to F_{η} gives a finite family C of fitting cells mod P_{N_{0}}^{×} which refines A_{η} and whose center and bounds belong to coalg F_{η}, such that for every f ∈ F, every C ∈ C and every (x, t) ∈ C

f_{η}(x, t) = U_{n_{0}}(x, t)h_{f,C}(x)(t − c_{C}(x))^{α_{f,C}}  (9)

where h_{f,C} : Ĉ → K is a semi-algebraic function and α_{f,C} is a positive integer. We removed the zero polynomial from F, but obviously (9) holds true for f = 0 as well, by taking h_{f,C} = 0 in that case. Each A_{i} is bounded hence so is their union ⋃ A as well as ⋃ A_{η}. So the center and bounds of every cell in C must be bounded functions with bounded domain. By Lemma 4.5 (assuming T_{m}) these functions are piecewise largely continuous. Refining the socle of C if necessary, and C accordingly, we are then reduced to the case where every cell in C is largely continuous. Note that U_{n_{0}} ∘ u_{η} = U_{n_{0}}, so by combining (8) and (9) we get that for every i ∈ I, every C ∈ C contained in u_{η}^{−1}(A_{i}) and every (x, t) ∈ C

θ_{i,η}(x, t)^{e} = U_{n_{0}}(x, t)h_{i,C}(x)(t − c_{C}(x))^{α_{i,C}}  (10)

where θ_{i,η} = θ_{i} ∘ u_{η}, h_{i,C} : Ĉ → K is a semi-algebraic function and α_{i,C} ∈ Z. For any integer M_{0} > 2v(e), Q_{N_{0},M_{0}}^{×} is a subgroup with finite index in P_{N_{0}}^{×} hence every such cell C mod P_{N_{0}}^{×} splits into finitely many cells D mod Q_{N_{0},M_{0}}^{×} with the same center, bounds and type as C. The integer M_{0} can be chosen arbitrarily large, in particular greater than M_{∗}. Let D be the family of all these cells D. From (10) and Lemma 2.9 we derive that for every i ∈ I, every D ∈ D contained in u_{η}^{−1}(A_{i}) and every (x, t) ∈ D

θ_{i,η}(x, t)^{e} = U_{n_{0}}(x, t)h̃_{i,D}(x)([λ_{D}^{−1}(t − c_{D}(x))]^{\frac{α_{i,D}}{e}} )^{e}  (11)

where h̃_{i,D} = h_{i,C} and α_{i,D} = α_{i,C} with C the unique cell in C containing D. The factor U_{n_{0}} in (11) can be written U_{n_{0}−v(e)}^{e} by Remark 2.10. Thus (11) implies that h̃_{i,D} takes values in P_{e}. So by Theorem 2.8 there is a semi-algebraic function h_{i,D} such that h̃_{i,D} = h_{i,D}^{e}. As a consequence, from (11) it follows that there is a semi-algebraic function χ_{i,D} with values in U_{e} such that for every (x, t) ∈ D

θ_{i,η}(x, t) = χ_{i,D}(x, t)U_{n_{0}−v(e)}(x, t)h_{i,D}(x)[λ_{D}^{−1}(t − c_{D}(x))]^{\frac{α_{i,D}}{e}}  (12)

By construction n_{0} − v(e) ≥ n hence the factor U_{n_{0}−v(e)} can a fortiori be replaced by U_{n}. Then χ_{i,D}U_{n} (which is just U_{e,n}) replaces χ_{i,D}U_{n_{0}−v(e)} in (12), which proves the result.

5 Cellular complexes

For this and the next section, let G be a fixed semi-algebraic clopen subgroup of K^{×} with finite index. Then vG is a subgroup of Z with finite index, hence vG = N_{0}Z for some integer N_{0} ≥ 1. Our aim in these two sections is to prove that every finite family of bounded largely continuous fitting cells mod G, such as the one given by Theorem 4.6, can be refined in a complex of cells mod G satisfying certain restrictive assumptions defined below.

Notation. For every largely continuous fitting cell A mod G in K^{m+1} with socle X, and every semi-algebraic set Y contained in \overline{X} let:

• ∂_{Y}^{0} A = (c̄_{A}, 0, 0, {0})_{|Y} if ν̄_{A} = 0 on Y, ∂_{Y}^{0} A = ∅ otherwise;

• ∂_{Y}^{1} A = (c̄_{A}, ν̄_{A}, μ̄_{A}, G_{A})_{|Y} if μ̄_{A} ≠ 0 on Y, ∂_{Y}^{1} A = ∅ otherwise.

Provided that on Y, ν̄_{A} and μ̄_{A} either take values in K^{×} or are constant, ∂_{Y}^{0} A and ∂_{Y}^{1} A are (if non-empty) largely continuous fitting cells mod G contained in \overline{A}∩(Y ×K). Intuitively, we can represent them (when ν_{A} ≠ 0 and ν̄_{A} = 0 < μ̄_{A} on Y) like this:

\begin{matrix}Y \\ X \end{matrix} •c_{A} A •∂_{Y}^{0} A ∂_{Y}^{1} A

Remark 5.1. If X is a partition of \overline{X}, the family of non-empty ∂_{Y}^{i} A for i ∈ \{0, 1\} and Y ∈ X form a partition of \overline{A}.

Given two cells A, B in K^{m+1} and an integer n ≥ 1, we write B ◁^{n} A if B ⊆ A and if there exists α ∈ {0, 1} and a semi-algebraic function h_{B,A} : B̂ → K such that for every (x, t) in B:

t − c_{A}(x) = U_{n}(x, t)h_{B,A}(x)^{α}(t − c_{B}(x))^{1−α}

We call h_{B,A} a ◁^{n}-transition for (B, A). If A, B are families of cells in K^{m+1} we write B ◁^{n} A if B ◁^{n} A for every B ∈ B and A ∈ A such that B meets A. A ◁^{n}-system for (B, A) is then the data of one ◁^{n}-transition for each possible (B, A) in B × A.

Remark 5.2. For any two finite families A, B of cells mod G, if B refines A and belongs to Alg_{n} A then B ◁^{n} A.

A closed ◁^{n}-complex of cells mod G is a finite family A of largely continuous fitting cells mod G such that ⋃ A is closed, the socle of A is a complex of sets and for every A, B ∈ A if B meets \overline{A} then for some i ∈ \{0, 1\}, ∂_{Y}^{i} A is a cell^{10} and B ◁^{n} ∂_{Y}^{i} A, with Y = B̂. If moreover B = ∂_{Y}^{i} A we call A a closed cellular complex mod G. As the terminology suggests, we are going to prove that closed ◁^{n}- and cellular complexes are complexes of sets in the general sense of Section 2 (see Proposition 5.3). Any subset of a closed ◁^{n}-complex (resp. closed cellular complex) is a ◁^{n}-complex (resp. a cellular complex. As usually we call them monoplexes if they form a tree with respect to the specialization order.

10The condition ∂^{i}Y A is a cell means that on Y, μ̄_{A} and ν̄_{A} either take values in K^{×} or are constant.

When A is a ◁^{n}-complex of cells mod G, for every Y ∈ Â and every cells A, B in A such that B meets \overline{A}, there is an integer α ∈ \{0, 1\} and a semi-algebraic function h_{B,A} : B̂ → K such that for every (x, t) in B:

t − c̄_{A}(x) = U_{n}(x, t)h_{B,A}(x)^{α}(t − c_{B}(x))^{1−α}.

An inner ◁^{n}-system for A is the data of one function h_{B,A} as above for every possible A, B ∈ A.

Proposition 5.3. Let A be a closed ◁^{n}-complex of cells mod G. Then A is a closed complex of sets. Moreover, for every A, B ∈ A and every Z ∈ Â if B meets ∂_{Z}^{0}A then B = ∂_{Z}^{0}A = Gr c̄_{A|Z}.

Proof: By assumption the socle of every cell A in A is relatively open and pure dimensional. Thanks to the restrictions we made on the bounds in our definition of presented cells, it follows that A is also relatively open and pure dimensional.

In order to show that A is a partition, let A, B be two cells in A which are not disjoint and let X = Â. Both B̂ and X belong to Â and are not disjoint, hence B̂ = X. Since B meets A ⊆ \overline{A}, by assumption B is contained in ∂_{X}^{i}A with i = tp B. But then ∂_{X}^{i}A meets A, hence obviously is equal to A. So B ⊆ A, and equality holds by symmetry.

Now let A be any cell in A and X = Â. Since ⋃ A is closed, every point of \overline{A} belongs to a unique B ∈ A. Since B meets \overline{A}, by assumption B ⊆ ∂_{Y}^{i} A with Y = B̂ and i = tp B. In particular B ⊆ \overline{A}, which proves that \overline{A} is a union of cells in A (hence so is ∂A since A is a partition and ∂A is disjoint from A). This proves that A is closed complex of sets.

The last point follows. Indeed, if B meets ∂_{Z}^{0}A ⊆ \overline{A} then it is contained in ∂_{Y}^{i} A for some i ∈ {0, 1}, with Y = B̂. In particular ∂_{Z}^{0}A meets ∂_{Y}^{i} A. They are two pieces of a partition of \overline{A} (see Remark 5.1) hence ∂_{Y}^{0} A = ∂_{Z}^{i}A. Therefore Y = Z and i = 0, so B ⊆ ∂_{Z}^{0}A. That is, B is of type 0 and c_{B} = c̄_{A} on B̂ = Z, so B = Gr c̄_{A|Z} = ∂_{Z}^{0}A.

Proposition 5.4. Let A be a finite family of largely continuous fitting cells mod G and n ≥ 1 an integer. There exists a ◁^{n}–complex D of cells mod G refining A such that D ◁^{n} A.

In the next section we will prove that one can even require that D is a cellular monoplex mod G.

Proof: The proof goes by induction on d = dim ⋃ Â. If a ◁^{n}-complex D^{+} is found which proves the result for a family A^{+} of cells mod G containing A^{+} then obviously the family D of cells in D^{+} contained in ⋃ A proves the result for A. Thus, enlarging A if necessary, we can assume that ⋃ A and ⋃ Â are closed. By Denef’s Theorem 2.11 and Remark 5.2 there is a finite family B of largely continuous fitting cells mod G refining A such that B ◁^{n} A. Replacing A by this refinement if necessary we can also assume that A is a partition.

If B is any vertical refinement of A then obviously B◁^{n} A. Thus, by taking if necessary a finite partition X refining Â and replacing A by the corresponding vertical refinement (that is the family of all cells A ∩ (X × K) with A ∈ A and X ∈ X contained in Â), we can assume that Â = X is a partition. By the same argument we can assume as well that for every A ∈ A and every X ∈ X contained in ∂X, the restrictions of μ̄_{A} and ν̄_{A} to X take values in K^{×} or are constant, hence ∂_{X}^{0}A (resp. ∂_{X}^{1}A) is a cell with socle X whenever it is non-empty. By Remark 2.16 we can even assume that it is a complex of pure dimensional sets. Let X_{d} be the family of X ∈ X with dimension d. Note that every X ∈ X_{d} is open in ⋃ X because X is a complex and dim ⋃ X = d.

For every X ∈ X_{d} let A_{X} be the family of cells in A with socle X. For every cell A ∈ A_{X} of type 1 such that ν_{A} = 0, Gr c_{A} is contained in \overline{A} hence in ⋃ A_{X} since ⋃ A is closed and Â is a partition. It may happen that Gr c_{A} does not belong to A. With Proposition 5.3 in view we have to remedy this. Every point (x, c_{A}(x)) in Gr c_{A} belongs to some cell B in A_{X}. This cell must be of type 0 otherwise the fiber B_{x} = {t ∈ K : (x, t) ∈ B} would be open, hence it would contain a neighbourhood V of c_{A}(x) and so {x} × V would be contained in B and meet A, which implies that B ⊆ A since A_{X} is partition, in contradiction with the fact B meets Gr c_{A}. So there is a finite partition Y_{A} of X in semi-algebraic pieces Y on each of which there is a unique cell B ∈ A_{X} of type 0 whose center coincides with c_{A} on Y. Repeating the same argument for every A ∈ A_{X} and every X ∈ X_{d} gives a finite partition Y of ⋃ X_{d} finer then every such Y_{A}. Let X′ be a complex of pure dimensional semi-algebraic sets refining X ∪ Y. Replacing if necessary A by the vertical refinement defined by X′, we can then assume from now on that for every X in X_{d} and every A ∈ A with socle X, if ν_{A} = 0 then Gr c_{A} belongs to A.

Let A_{d} = {A ∈ A : Â ∈ X_{d}} and B be the union of A \ A_{d} and of the family of non-empty ∂_{Y}^{i} A for i ∈ {0, 1}, A ∈ A_{d} and Y ∈ X \ X_{d}. Clearly dim ⋃ B̂ < d so the induction hypothesis gives a ◁^{n}–complex C of cells mod G refining B such that C ◁^{n} B. A fortiori C ◁^{n} (A \ A_{d}) because the latter is contained in B. So if we let D = C ∪ A_{d}, then D refines A and D ◁^{n} A. It only remains to check that D is a ◁^{n}-complex, and first that D̂ is a complex of sets.

Note that Â_{d} = X_{d} hence D̂ = Ĉ ∪ Â_{d} = Ĉ ∪ X_{d} is a partition, and every set in D̂ is pure dimensional and relatively open (by induction hypothesis for Ĉ and by construction for X_{d}). For every X ∈ D̂, we have to prove that ∂X is a union of sets in Ĉ∪ X_{d}. If X ∈ Ĉ this is clear because Ĉ is a complex. Otherwise X ∈ X_{d} hence ∂X is a union of sets in X (because X is a complex). All these sets have dimension < d = dim X hence belong to X \ X_{d}. But C refines B, which contains A \ A_{d}, whose socle is X \ X_{d}, hence Ĉ refines X \ X_{d}. Thus ∂X is also the union of sets in Ĉ, hence of D̂.

Now let D, E ∈ D be such that E meets \overline{D}, let X = D̂ and Y = Ê. By construction ∂_{Y}^{0} D and ∂_{Y}^{1} D are cells (if non-empty) and cover \overline{D} ∩ (Z × K). So there is i ∈ {0, 1} such that ∂_{Y}^{i} D is a cell which meets E. We have to prove that E ◁^{n} ∂_{X}^{i}D. Note that Y meets the socle of \overline{D}, which is contained in \overline{X}, hence Y = X or Y ⊆ ∂X because D̂ is a complex. So, if dim X < d then also dim Y < d hence D, E ∈ C. In that case E ◁^{n} ∂_{X}^{j}D because C is a ◁^{n}–complex. Thus we can assume that dim X = d, that is D ∈ A_{d}. We know that Y = X or Y ⊆ ∂X. In the first case Y = X hence ∂_{X}^{i}D ∈ A_{d} ⊆ D by construction, so E = ∂_{X}^{i}D because D is a partition. In the second case Y ∈ Ĉ hence E ∈ C. Now Y is contained in some Z ∈ X \ X_{d} because Ĉ refines X \ X_{d}, and E meets ∂_{Z}^{i}D. By construction ∂_{Z}^{i}D belongs to B. Since C ◁^{n} B it follows that E ◁^{n} ∂_{Z}^{i}D hence a fortiori E ◁^{n} ∂_{Y}^{i} D because E ⊆ Y × K and ∂_{Y}^{i} D = ∂_{Z}^{i}D ∩ (Y × K).

Before entering in more complicated constructions, let us mention here two elementary properties of fitting cells which will be of some use latter.

Proposition 5.5. Let A ⊆ K^{m+1} be a cell mod G of type 1. Then:

• μ_{A} is a fitting bound if and only if μ_{A} = ∞ or vμ_{A}(Â) ⊆ vG_{A}.

• ν_{A} is a fitting bound if and only if ν_{A} = 0 or vν_{A}(Â) ⊆ vG_{A}).

Proof: The case where μ_{A} = ∞ being trivial, we can omit it. If μ_{A} ≠ ∞ is a fitting bound then obviously vμ_{A}(Â) ⊆ vG_{A} because v(t − c_{A}(x)) ∈ vG_{A} for every (x, t) ∈ A. Conversely assume that vμ_{A}(Â) ⊆ vG_{A}. Let x be any element of Â. We have to prove that |μ_{A}(x)| = max{|d| : d ∈ D_{x}} where D_{x} = {t − c_{A}(x) : (x, t) ∈ A}. D_{x} is bounded since μ_{A} ≠ ∞, hence by Corollary 2.7 it contains an element d of maximal norm. By construction |d| ≤ |μ_{A}(x)|. Assume for a contradiction that |d| < |μ_{A}(x)|, that is v(d/μ_{A}(x)) > 0. By construction v(d) and vμ_{A}(x) belong to vG_{A} = vλ_{A} + vG hence v(d/μ_{A}(x)) ∈ vG = N_{0}Z. Thus v(d/μ_{A}(x)) ≥ N_{0}, that is |d| ≤ |π^{N_{0}}μ_{A}(x)|. Pick any g ∈ G such that v(g) = N_{0} and let t′ = c_{A}(x) + d/g. We have t′ − c_{A}(x) = d/g ∈ G_{A}, |ν_{A}(x)| ≤ |d| ≤ |d/g| and |d/g| ≤ |μ_{A}(x)|, hence (x, t′) ∈ A. So t′ − c_{A}(x) ∈ D_{x} and |d| < |t′ − c_{A}(x)|, a contradiction. The proof for ν_{A} is similar and left to the reader.

Proposition 5.6. For every fitting cell A mod Q_{N_{0},M_{0}} in K^{m+1}, if A ⊆ R^{m+1} then vμ_{A} ≥ −M_{0}.

Since A ⊆ R^{m+1}, one may naively expect that |μ_{A}| ≤ 1, that is vμ_{A} ≥ 0. The presented cell A = (−π^{−M_{0}}, π^{−M_{0}}, π^{−M_{0}}, Q_{N_{0},M_{0}}) is a counterexample in K: it is contained in R (it is actually equal to R) and vμ_{A} = −M_{0} < 0.

Proof: Assume the contrary, that is vμ_{A}(x) < −M_{0} for some x ∈ Â. Since A is a fitting cell there is t ∈ K such that (x, t) ∈ A and v(t−c_{A}(x)) = vμ_{A}(x). Since A ⊆ R^{m+1}, t ∈ R hence v(t − c_{A}(x)) < 0 = v(t) implies that vc_{A}(x) = v(t − c_{A}(x)) = vμ_{A}(x). So there are a ∈ R and g ∈ Q_{N,M} such that c_{A}(x) = aπ^{M_{0}+1} and t − c_{A}(x) = λ_{A}g. In particular v(λ_{A}g) = v(t − c_{A}(x)) = vμ_{A}(x) < −M_{0} so π^{M_{0}}λ_{A}g ∉ R. Now let t′ = t + π^{M_{0}}λ_{A}g, then t′ ∉ R since t ∈ R and π^{M_{0}}λ_{A}g ∉ R. On the other hand 1 + π^{M_{0}} ∈ Q_{N_{0},M_{0}} and

t′ − c_{A}(x) = t − c_{A}(x) + π^{M_{0}}λ_{A}g = λ_{A}g + +π^{M_{0}}λ_{A}g = λ_{A}(1 + π^{M_{0}})g.

So t′ − c_{A}(x) ∈ λ_{A}Q_{N_{0},M_{0}} and v(t′ − c_{A}(x)) = v(λ_{A}(1 + π^{M_{0}})g) = v(λ_{A}g) = vμ_{A}(x). Thus (x, t′) ∈ A, a contradiction since t′ ∉ R and A ⊆ R^{m+1}.

6 Cellular monoplexes

We keep as in Section 5 a semi-algebraic clopen subgroup G of K^{×} with finite index, and N_{0} ≥ 1 an integer such that vG = N_{0}Z. Lemma 6.1 below (together with Lemma 7.10) is the technical heart of this paper. All this section is devoted to its proof.

Lemma 6.1. Assume T_{m}. Let A be a finite set of bounded, largely continuous, fitting cells mod G in K^{m+1}. Let F_{0} be a finite family of definable functions with domains in Â. Let n, N ≥ 1 be a pair of integers. For some integers e, M > 2v(e) which can be made arbitrarily large (in the sense of footnote 2), there is a tuple (V, φ, D, F_{D}) such that:

• D is a cellular monoplex mod G refining A such that D ◁^{n} A.

• F_{D} is a ◁^{n}-system for (D, A).

• (V, φ) is a triangulation of^{11} F_{0} ∪ F_{D} ∪ CB(D) with parameters (n, N, e, M), such that D̂ = φ(V).

Note that, in order to obtain this result, it does not suffice to find a continuous monoplex D of well presented cells mod G refining A such that D ◁^{n} A, and then to select an arbitrary ◁^{n}-system F_{D} for (D, A) and to apply T_{m} to F_{0}∪F_{D}∪CB(D). Indeed, this will give a triangulation (V, φ) of F_{0}∪F_{D}∪CB(D). But φ(V) will then be a refinement of D̂, not D̂ itself. It is then tempting to refine vertically D, that is to replace D by the family E of cells D ∩ (φ(V) × K) for D ∈ D and V ∈ V such that φ(V) ⊆ D̂. This ensures that Ê = φ(V) and E is a cellular complex such that E ◁^{n} A. But E is no longer a monoplex.

In order to break this vicious circle we have to build V, D and F_{D} simultaneously. The remaining of this section is devoted to this construction. It is divided in three parts: (6.a) preparation, (6.b) vertical refinement, (6.c) horizontal refinement.

6.a Preparation

Given a family X of subsets of K^{m}, we let F_{0|X} denote the family of all the restrictions f_{|X} with f ∈ F_{0} and X ∈ X contained in the domain of f. By the same argument as in the beginning of the proof of Proposition 5.4, we can assume that ⋃ A is closed. Finally, replacing if necessary A by a refinement D given by Proposition 5.4 and F_{0} by F_{0|X} with X = D̂, we are reduced to the case where A is a closed ◁^{n}–complex of bounded cells mod G. Enlarging F_{0} if necessary, we can, and will, assume that it contains CB(A) and an inner ◁^{n}-system for A. For some integers e, M > 2v(e) which can be made arbitrarily large, T_{m} gives a triangulation (S, φ) of F_{0} with parameters (n, N, e, M). For every A ∈ A we let S_{A} = φ^{−1}(Â).

Since ⋃ A is bounded and closed in K^{m+1}, its image ⋃ Â by the coordinate projection is closed in K^{m} by Theorem 2.6. Now φ is a homeomorphism from ⨄ S to ⋃ Â, hence S is closed by Remark 2.19.

Let A′ be the family of cells A ∩ (φ(S) × K) for A ∈ A and S ∈ S such that φ(S) ⊆ Â, and let F′_{0} = F_{0|φ(S)}. Since every cell in A′ has the same center and bounds as the unique cell in A which contains it, clearly A′ is still a closed ◁^{n}–complex, F′_{0} contains CB(A′) and an inner ◁^{n}-system for A′, and (S, φ) is still a triangulation of F′_{0}. Thus, replacing (A, F_{0}) by (A′, F′_{0}) if necessary, we can assume that φ(S) = Â, that is S_{A} ∈ S for every A ∈ A.

A preparation for (S, φ, A, F_{0}) is a tuple (T , B, F_{B}) such that:

11Recall that CB(D) denotes the family of center and bounds of the cells in D.

(P1) T is a simplicial subcomplex of S. We let S_{|T} = {S ∈ S : S ⊆ ⨄ T }, and A_{|T} be the family of cells A ∈ A such that S_{A} ∈ S_{|T}. Note that:

• T is closed because ⨄ T is closed in ⨄ S.

• By Remark 2.19 it follows that the image by φ of ⋃ T , that is the socle of A_{|T}, is closed too.

• Hence A_{|T} is closed because ⋃ A_{|T} is the inverse image of its socle by the (continuous) coordinate projection of ⋃ A onto ⋃ Â.

(P2) B is a cellular monoplex mod G refining A_{|T} such that φ(T ) = B̂. For every B ∈ B we let T_{B} = φ^{−1}(B̂). Note that B is closed because ⋃ B = ⋃ A_{|T}.

(P3) B ◁^{n} A_{|T} and F_{B} is a ◁^{n}-system for (B, A_{|T}).

(P4) T together with the restriction of φ to ⨄ T , which we will denote φ_{|T}, is a triangulation of F_{B} ∪ CB(B) with parameters (n, N, e, M). Note that, since T refines S_{|T} and (S, φ) is a triangulation of F_{0}, (T , φ_{|T}) is also a triangulation of F_{0|X} with X = φ(T ).

Remark 6.2. Obviously (∅, ∅, ∅) is preparation for (S, φ, A, F_{0}). Given an arbitrary preparation (T , B, F_{B}) for (S, φ, A, F_{0}) such that ⋃ T ≠ ⋃ S, and S a minimal element in S \S_{|T}, it suffices to build from it a preparation (U, C, F_{C}) such that ⨄ U = ⨄ T ∪ S. Indeed, S_{|U} contains one more element of S than S_{|T} thus, starting from (∅, ∅, ∅) and repeating the process inductively we will finally get a preparation (V, D, F_{D}) such that ⨄ V = ⨄ S, hence A_{|V} = A. (P4) then implies that (V, φ) is a triangulation of F_{0} ∪ F_{B} ∪ CB(B) with parameters (n, N, e, M). So the tuple (V, φ, D, F_{D}) satisfies the conclusion of Lemma 6.1, which finishes the proof.

So from now on, let (T , B, F_{B}) be a given preparation for (S, φ, A, F_{0}) such that ⨄ T ≠ ⨄ S. Let S be a minimal element in S \ S_{|T} and A_{S} = {A ∈ A : Â = φ(S)}. The minimality of S ensures that every proper face of S belongs to S_{|T}, hence ⨄ S ∪ S and ⋃(A_{|T} ∪ A_{S}) are closed.

Claim 6.3. Let A be a cell of type 1 in A_{S}, T a simplex in T contained in \overline{S}, and Y = φ(T). If ν̄_{A} = 0 on Y then Gr c̄_{A|Y} = ∂_{Y}^{0} A belongs to B. If moreover μ̄_{A} ≠ 0 on Y then ∂_{Y}^{1} A is covered by the cells in B that it meets, and among them there is a unique cell B_{T}^{1} whose closure meets ∂_{Y}^{0} A. More precisely:

B_{T}^{1} = (c̄_{A|Y} , 0, μ_{B_{T}^{1}} , G_{A})

and either |μ_{B_{T}^{1}} | = |μ̄_{A}| on Y, or |μ_{B_{T}^{1}} | ≤ |π^{N_{0}} μ̄_{A}| on Y. In particular the closure of B_{T}^{1} contains ∂_{Y}^{0} A.

Proof: Note first that for every i ∈ \{0, 1\}, ∂_{Y}^{i} A is contained in \overline{A} hence in ⋃ A since it is closed by assumption. Every cell in A which meets ∂_{Y}^{i} A is contained in it since A is a ◁^{n}-complex, and belongs to A_{|T} (otherwise its socle would not meet Y since Y ⊆ ⋃ φ(T )). Since B refines A_{|T} it follows that ∂_{Y}^{i} A is the union of the cells B in B which it contains.

In particular, if ν̄_{A} = 0 on A then ∂_{Y}^{0} A ≠ ∅ hence it contains a cell B ∈ B. Necessarily B is of type 0 since ∂_{Y}^{0} A is so, and thus B = ∂_{Y}^{0} A since they have the same socle Y. This proves the first point.

For the second point, since ν̄_{A} = 0 ≠ μ̄_{A} on Y both ∂_{Y}^{0} A and ∂_{Y}^{1} A are non-empty. Now ∂_{Y}^{0} A is contained in the closure of ∂_{Y}^{1} A, which is the union of the closure of the cells in B contained in ∂_{Y}^{1} A. Hence necessarily the closure of at least one of them, say B, meets ∂_{Y}^{0} A.

B̂ meets Y and both of them belong to B̂ so B̂ = Y. Since \overline{B} ∩ (Y × K) meets ∂_{Y}^{0} A and B ⊆ ∂_{Y}^{1} A is disjoint from ∂_{Y}^{0} A, B must be of type 1 with ν_{B} = 0 because otherwise B would be closed in Y ×K. It follows that \overline{B} ∩(Y ×B) is the union of B and ∂_{Y}^{0} B, and the latter meets ∂_{Y}^{0} A. By the first point ∂_{Y}^{0} A ∈ B. By Proposition 5.3 applied to B, ∂_{Y}^{0} B ∈ B. Thus ∂_{Y}^{0} B = ∂_{Y}^{0} A, in particular they have the same center so c_{B} = c̄_{A|Y}. Pick any (x, t) ∈ B, so that t−c_{B}(x) ∈ G_{B}. B is contained in ∂_{Y}^{1} A hence t − c̄_{A}(x) ∈ G_{A}. Since c_{B}(x) = c̄_{A}(x) it follows that G_{B} ∩ G_{A} ≠ ∅ hence G_{A} = G_{B}.

This proves that B = (c̄_{A|Y} , 0, μ_{B}, G_{A}). The uniqueness of B follows. Indeed if B′ is any cell in B contained in ∂_{Y}^{1} A whose closure meets ∂_{Y}^{0} A, the same argument shows that B′ = (c̄_{A|Y} , 0, μ_{B′}, G_{A}). This implies that for any t′ ∈ K such that t′ − c̄_{A}(x) is small enough and belongs to G_{A}, the point (x, t′) will belong both to B and B′, so B = B′.

If |μ_{B}| = |μ̄_{A|Y} | we are done, so let us assume the contrary. Then |μ_{B}(x)| ≠ |μ̄_{A}(x)| for some x ∈ Y. B is a fitting cell so let t ∈ K be such that (x, t) ∈ B and |t − c_{B}(x)| = |μ_{B}(x)|. We have |t − c̄_{A}(x)| ≤ |μ̄_{A}(x)| because (x, t) ∈ ∂_{Y}^{1} A, so |μ_{B}(x)| < |μ̄_{A}(x)|. We are going to show that |μ_{B}| < μ̄_{A}| on Y. Since ∂_{Y}^{1} A is a fitting cell it follows that ∂_{Y}^{1} A is not contained in B, so there is at least one other cell C in B contained in ∂_{Y}^{1} A. Now Ĉ is contained in Y and both of them belong to B̂ so B̂ = Y. For each y in Y fix t_{y} in K such that (y, t_{y}) ∈ C. Since C is contained in ∂_{Y}^{1} A we have:

0 ≤ |t_{y} − c̄_{A}(y)| ≤ |μ̄_{A}(y)| and t_{y} − c̄_{A}(y) ∈ G_{A}

Necessarily |μ_{B}(y)| < |t_{y} − c̄_{A}(y)| because otherwise (y, t_{y}) would belong both to C and B, a contradiction. Hence a fortiori |μ_{B}(y)| < |μ̄_{A}(y)|. By Proposition 5.5 this implies that |μ_{B}(y)| ≤ |π^{N_{0}} μ̄_{A}(y)| because B and A are fitting cells mod G, vG = N_{0}Z and G_{B} = G_{A}. So |μ_{B}| ≤ |π^{N_{0}} μ̄_{A|Y} | in that case, which proves our claim.

We can start now our construction of a preparation (U, C, F_{C}) for (S, φ, A, F_{0}) such that ⨄ U = ⨄ T ∪ S. We are going to refine A_{S} twice. First “vertically”, according to the image by φ of a certain partition of S which, together with T , forms a simplicial subcomplex U of S refining S_{|T} ∪ {S} (Claim 6.5). Then “horizontally” by enlarging the cells in B contained in the closure of ⋃ A_{S} in such a way that the family of these new cells, together with B, forms a cellular monoplex C mod G refining A_{|T} ∪A_{S} = A_{|U} such that C ◁^{n}A_{|U}. The point of the construction is to ensure that C comes with a ◁^{n}-system F_{C} for (C, A_{|U}) such that (U, C, F_{C}) is a preparation for (S, φ, A, F_{0}).

6.b Vertical refinement

Let T_{S} be the list of proper faces of S. We first deal with the case when S is not closed, that is T_{S} ≠ ∅. For every A in A_{S} let:

(c_{A}^{∘}, ν_{A}^{∘}, μ_{A}^{∘}) = (c_{A} ∘ φ, ν_{A} ∘ φ, μ_{A} ∘ φ)_{|S}

For every T ∈ T_{S} and every A ∈ A_{S} let Φ_{T,A}(t, ε) be the formula saying that (t, ε) ∈ T × R^{∗} and that one of the following conditions hold, with n_{1} = max(n, 1 + 2vN):

(A1)_{t,ε}: ν̄_{A}^{∘}(t) ≠ 0 and for every s ∈ S such that ∥s − t∥ ≤ |ε|:

|c_{A}^{∘}(s) − c̄_{A}^{∘}(t)| ≤ |π^{n_{1}}ν̄_{A}^{∘}(t)|
and |ν_{A}^{∘}(s)| = |ν̄_{A}^{∘}(t)| and |μ_{A}^{∘}(s)| = |μ̄_{A}^{∘}(t)|

(A2)_{t,ε}: ν̄_{A}^{∘}(t) = 0, μ̄_{A}^{∘}(t) ≠ 0 and for every s ∈ S such that ∥s − t∥ ≤ |ε|:

|c_{A}^{∘}(s) − c̄_{A}^{∘}(t)| ≤ |π^{n_{1}−N_{0}}μ_{B}^{∘}(t)|

and |ν_{A}^{∘}(s)| ≤ |μ_{B}^{∘}(t)| ≤ |μ̄_{A}^{∘}(t)| = |μ_{A}^{∘}(s)| where B is the cell B_{T}^{1} given by Claim 6.3.

(A3)_{t}: ν̄_{A}^{∘}(t) = μ̄_{A}^{∘}(t) = 0.

Let Φ(t, ε) be the conjunction of the (finitely many) Φ_{T,A}(t, ε)’s as T ranges over T_{S} and A over A_{S}. Finally let Ψ(t, ε) be the formula saying that that |ε| is maximal among the elements ε′ in R^{∗} such that K ⊧ Φ(t, ε′). Obviously Ψ(t, ε) implies Φ(t, ε).

By continuity of the center and bounds of A, for every t ∈ T there exists ε_{t,T,A} ∈ R^{∗} such that K ⊧ Φ_{T,A}(t, ε_{t,T,A}). Hence for every t ∈ ∂S there is ε ∈ R^{∗} such that K ⊧ Φ(t, ε) (every ε ∈ R^{∗} such that |ε| ≤ |ε_{t,T,A}| for every (T, A) ∈ T_{S} × A_{S} is a solution). For every t ∈ ∂S, the set E_{t} of elements ε of R^{∗} such that K ⊧ Φ(t, ε) is semi-algebraic, bounded and non-empty. So by Corollary 2.7 there is ε_{t} ∈ E_{t} such that |ε_{t}| is maximal in |E_{t}|, that is K ⊧ Ψ(t, ε_{t}). Theorem 2.8 then gives a semi-algebraic function ε : ∂S → R^{∗} such that K ⊧ Ψ(t, ε(t)) for every t ∈ ∂S, hence a fortiori:

∀t ∈ ∂S, K ⊧ Φ(t, ε(t)).  (13)

Claim 6.4. Let ε : ∂S → R^{∗} be the semi-algebraic function defined above. Then the restriction of |ε| to every proper face T of S is continuous.

Proof: Note first that if K ⊧ Φ_{T,A}(t, ε′) for some t ∈ T and ε′ ∈ R^{∗} then K ⊧ Φ_{T,A}(t′, ε′) for every t′ ∈ T ∩ B where B = B(t, ε′) is the ball with center t and radius ε′.

Indeed, assume for example that ν̄_{A}^{∘}(t) ≠ 0, hence (A1)_{t,ε′} holds true. It asserts that for every s ∈ S ∩ B

|c_{A}^{∘}(s) − c̄_{A}^{∘}(t)| ≤ |π^{n_{1}} ν̄_{A}(t)|  (14)

and |ν_{A}(s)| = |ν̄_{A}^{∘}(t)| and |μ_{A}(s)| = |μ̄_{A}^{∘}(t)|. Now T ∩ B ⊆ \overline{S ∩ B} hence, as s tends in S ∩ B to any given t′ ∈ T ∩ B we get

|c_{A}^{∘}(t′) − c̄_{A}^{∘}(t)| ≤ |π^{n_{1}} ν̄_{A}(t)|  (15)

and |ν_{A}(t′)| = |ν̄_{A}^{∘}(t)| and |μ_{A}(t′)| = |μ̄_{A}^{∘}(t)|. By combining (14) and (15) with the triangle inequality we obtain that for every s ∈ S ∩ B

|c_{A}^{∘}(s) − c̄_{A}^{∘}(t′)| ≤ |π^{n_{1}} ν̄_{A}(t)| = |π^{n_{1}} ν̄_{A}(t′)|

and |μ_{A}(s)| = |μ̄_{A}^{∘}(t′)|, that is (A1)_{t′,ε′}.

Assume now that μ̄_{A}^{∘}(t) = 0, hence ν̄_{A}^{∘}(t) = 0, that is (A3)_{t} holds true. Then φ(T) ∈ Â and μ̄_{A}(φ(t)) = 0 imply that μ̄_{A} = 0 on φ(T) because A is a closed ◁^{n}–complex (see foonote 10). So μ̄_{A}^{∘} = ν̄_{A}^{∘} = 0 on T, and (A3)_{t′} follows.

The intermediate case (A2)_{t,ε} where ν̄_{A}^{∘}(t) = 0 and μ̄_{A}^{∘}(t) ≠ 0 is similar, and left to the reader.

Now it follows that if K ⊧ Ψ(t, ε′) and ∥t′ − t∥ ≤ |ε′|, then K ⊧ Ψ(t, ε′′) if and only if |ε′| = |ε′′|. So |ε(t)| = |ε(t′)| for every t, t′ ∈ T such that ∥t − t′∥ ≤ |ε(t)|. Thus |ε| is locally constant, hence continuous on T.

Theorem 2.17 applies to S, T_{S} and the function ε. It gives a partition U_{S} of S such that U_{S} ∪ T_{S} is a simplicial complex, for each T ∈ T_{S} there is a unique U ∈ U_{S} with facet T, and for every u ∈ U:

∥u − π_{U}(u)∥ ≤ |ε(π_{U}(u))|  (16)

where π_{U} is the coordinate projection of U onto T (see Remark 2.15). On φ(U) let σ_{U} = φ ∘ π_{U} ∘ φ^{−1}. This is a continuous retraction of φ(U) onto φ(T).

For every U ∈ U_{S} and every A ∈ A_{S} let

A_{U} = A ∩ (φ(U) × K).  (17)

Let T_{U} = ∅ if U is closed, and T_{U} ∈ T_{S} be the facet of U otherwise. Finally let U = U_{S} ∪ T_{S}.

Claim 6.5. With the notation above, U is a simplicial subcomplex of S refining S_{|T} ∪ {S} and containing T . For every U ⊆ S in U and every A ∈ A_{S}, A_{U} is a largely continuous fitting cell mod G. Moreover if U is not closed then:

1. If |ν̄_{A}| ≠ 0 on φ(T_{U}), then for every x ∈ Â_{U}:

|c_{A}(x) − c̄_{A}(σ_{U}(x))| ≤ |π^{n_{1}} ν̄_{A}(σ_{U}(x))|  (18)
|ν_{A}(x)| = |ν̄_{A}(σ_{U}(x))| and |μ_{A}(x)| = |μ̄_{A}(σ_{U}(x))|  (19)

2. If |ν̄_{A}| = 0 < |μ̄_{A}| on φ(T_{U}), then for every x ∈ Â_{U}:

|c_{A}(x) − c̄_{A}(σ_{U}(x))| ≤ |π^{n_{1}−N_{0}}μ_{B}(σ_{U}(x))|  (20)
|ν_{A}(x)| ≤ |μ_{B}(σ_{U}(x))| ≤ |μ̄_{A}(σ_{U}(x))| = |μ_{A}(x)|  (21)

where B is the cell B_{T}^{1} given by Claim 6.3.

Proof: By construction U is clearly a simplicial complex refining T ∪ {S}, hence refining S_{|T} ∪ {S} since T refines S_{|T}. For every U ⊆ S in U and every A ∈ A_{S}, A_{U} is a largely continuous fitting cell mod G by (17), because A is so. If moreover U is not closed let T = T_{U} ∈ T_{S} be its facet, let x be any element of Â_{U} = φ(U), s = φ^{−1}(x) ∈ U and t = π_{U}(s) ∈ T, where π_{U} is the coordinate projection of U onto T (see Remark 2.15). Note that σ_{U}(x) = φ ∘ π_{U}(s) = φ(t) hence c̄_{A}(σ_{U}(x)) = c̄_{A}^{∘}(t), and similarly for ν̄_{A}(σ_{U}(x)) and μ̄_{A}(σ_{U}(x)). By (13) we have K ⊧ Φ(t, ε(t)).

If |ν̄_{A}| ≠ 0 on φ(T) then ν̄_{A}^{∘}(t) = ν̄_{A}(σ_{U}(x)) ≠ 0 hence Φ(t, ε(t)) says that (A1)_{t,ε(t)} holds for t. By (16), ∥s − t∥ ≤ |ε(t)| so (18) and (19) follow from (A1)_{t,ε(t)}. Similarly, if |ν̄_{A}| = 0 < |μ̄_{A}| on φ(T) then (20) and (21) follow from (A2)_{t,ε(t)}.

This finishes the construction of the vertical refinement of A_{S} if S is not closed. When S is closed we simply take U = S_{|T} ∪ {S}. Claim 6.5 holds true in this case too, for the trivial reason that there is no non-closed U ⊆ S in U.

Remark 6.6. For every U ∈ U_{S}, if ν_{A_{U}} = 0 then Gr c_{A_{U}} = B_{U} for some B ∈ A_{S}. Indeed ν_{A|φ(U)} = ν_{A_{U}} = 0 implies that ν_{A} = 0 (thanks to our definition of presented cells) hence Gr c_{A} = ∂_{φ(S)}^{0}A belongs to A: it is contained in \overline{A}, hence in ⋃ A since the latter is closed, in particular it meets at least one cell B in A, and the last point of Proposition 5.3 then gives that B = Gr c_{A}. Thus B = Gr c_{A} ∈ A_{S}, and clearly Gr c_{A_{U}} = B_{U}.

6.c Horizontal refinement

For every A ∈ A_{S} we are going to construct for each U ∈ U_{S} a partition E_{A,U} of A_{U}, and for each E in E_{A,U} a semi-algebraic function h_{E,A_{U}} : φ(U) → K such that:

(Pres) Ê = φ(U) = Â_{U} and E is a largely continuous fitting cell mod G.

(Fron) One of the following holds:

(∂1) ∂E = ∅.
(∂2) ∂E = \overline{Gr c_{E}} and Gr c_{E} ∈ E_{C,U} for some C ∈ A_{S}.

(∂3) ∂E = \overline{B} for some B ∈ B, in which case U is not closed, B̂ = φ(T_{U})

and:

(c_{B}, ν_{B}, μ_{B}) = (c̄_{E}, ν̄_{E}, μ̄_{E})_{|φ(T_{U})}.

(Out) E ◁^{n} A_{U} and h_{E,A_{U}} is a ◁^{n}-transition for (E, A_{U}).

(Mon) c_{E} ∘φ_{|U}, μ_{E} ∘φ_{|U}, ν_{E} ∘φ_{|U} and h_{E,A_{U}} ∘φ_{|U} are N-monomial mod U_{e,n}.

This last construction will finish the proof of Lemma 6.1. Indeed, assuming that it is done, let C be the union of B and all the cells E in E_{A,U} for A ∈ A_{S} and U ∈ U_{S}. Let F_{C} be the union of the family of the corresponding functions h_{E,A_{U}} and of F_{B}. By Claim 6.5, U is a simplicial subcomplex of S such that ⨄ U = ⨄ T ∪ S. The assumption (P2) for B, together with (Pres) and (Fron), give that C is a cellular monoplex mod G refining A_{|T} ∪ A_{S} and that φ(U) = Ĉ. The assumption (P3) for B and F_{B}, together with (Out) above, give that C ◁^{n} A_{|U} and F_{C} is a ◁^{n}-system for (C, A_{|U}). Finally the assumption (P4) for (T , φ_{|T}) together with (Mon) ensure that (U, φ_{|U}) is a triangulation of F_{C} ∪ CB(C) with parameters (n, N, e, M). So (U, C, F_{C}) is a preparation of (S, φ, A, F_{0}), and since ⨄ U = ⨄ T ∪ S we conclude by Remark 6.2.

So let A ∈ A_{S} and U ∈ U_{S} be fixed once for all in the remaining.

Remark 6.7. Recall that (S, φ) is a triangulation of F_{0}, and F_{0} contains CB(A). In particular c_{A} ∘φ_{|S} is N-monomial mod U_{e,n} hence a fortiori c_{A} ∘φ_{|U} is so. By (17) c_{A_{U}} = c_{A|φ(U)} hence c_{A_{U}} ∘ φ_{|U} = c_{A} ∘ φ_{|U}. Thus c_{A_{U}} ∘ φ_{|U} is N-monomial mod U_{e,n}, and so are μ_{A_{U}} ∘ φ_{|U} and ν_{A_{U}} ∘ φ_{|U} by the same argument.

Let us first assume that U is closed. We distinguish two elementary cases.

Case 1.1: μ_{A_{U}} = 0 or ν_{A_{U}} ≠ 0. Then A_{U} is closed. We let E_{A,U} = {A_{U}} and h_{A_{U} ,A_{U}} = 1. (Pres), (∂1), (Out) and (Mon) are obvious (using Remark 6.7 for the latter).

Case 1.2: 0 = |ν_{A_{U}} | < |μ_{A_{U}} |. We let E_{A,U} = {A_{U}} and h_{A_{U} ,A_{U}} = 1. Again (Pres), (Out) and (Mon) are obvious (same as Case 1.1). Moreover ∂A_{U} = Gr c_{A_{U}} = Gr c_{A}, which belongs to A_{S} by Remark 6.6. If we let B = Gr c_{A}, we have B_{U} = Gr c_{A_{U}} and μ_{B_{U}} = 0 hence E_{B,U} = {B_{U}} by the previous case. So ∂A_{U} = Gr c_{A} and Gr c_{A} ∈ E_{B,U}, which proves that (∂2) holds true.

These cases being solved, we assume in the remaining that U is not closed. Recall that T_{U} is then the facet of U and belongs to T . By construction ∂φ(U) = φ(∂U) = φ(\overline{T_{U}}) = \overline{φ(T_{U})}. For the convenience of the reader, each of the following cases is illustrated by a geometric representation of its conditions (almost like if we were dealing with a cell A over a real closed field, except that the vertical intervals representing the fibres of A over Â can be clopen). In this figures A_{U} is represented by a gray area in K^{2}, its bounds by dotted lines, its socle φ(U) by the horizontal axe, ∂φ(U) = φ(T_{U}) by a dot on the left bound of φ(U), and \overline{A} ∩ (φ(T_{U}) × K) by a thick line or dot on the vertical axe above φ(T_{U}).

Case 2.1: |μ̄_{A_{U}} | = 0 on φ(T_{U}).

\begin{matrix}• \\ • \\ c_{A} \\ E = A_{U} \\ or \\ • \\ • \\ c_{A} \\ E = A_{U} \end{matrix}

We let E_{A,U} = {A_{U}} and h_{A_{U} ,A_{U}} = 1. (Pres), (Out) and (Mon) are obvious as in the previous cases.

• Sub-case 2.1.a: ν_{A_{U}} ≠ 0 or μ_{A_{U}} = ν_{A_{U}} = 0. Then ∂A_{U} is the closure of Gr c̄_{A_{U} |φ(T_{U})} = Gr c̄_{A|φ(T_{U})}. The latter belongs to B by Claim 6.3, which proves (∂3).

• Sub-case 2.1.b: ν_{A_{U}} = 0 ≠ μ_{A_{U}}. Then ∂A_{U} is the closure of Gr c_{A_{U}}. By Remark 6.6, there is a cell C ∈ A_{S} such that C_{U} = Gr c_{A_{U}}. Then μ_{C_{U}} = 0 (because A_{U} is a fitting cell) hence E_{C,U} = {C_{U}} by the previous sub-case. So ∂A_{U} = \overline{Gr c_{A_{U}}} and Gr c_{A_{U}} ∈ E_{C,U}, which proves that (∂2) holds true.

Case 2.2: 0 < |ν̄_{A}| on φ(T_{U}).

• c_{A} E_{B_{1}} E_{B_{3}} c_{B_{2}} ∘ σ_{U} = E_{B_{2}} B_{1} •B_{2} B_{3}

In this case, by Claim 6.3, ∂_{φ(T_{U})}^{1}A = Ā ∩ (φ(T_{U}) × K) is the union of the

cells B ∈ B which it contains. For every such B, B̂ = φ(T_{U}) (because B̂ = φ(T ) and B̂ ⊆ φ(T_{U})) and we let:

E_{B} = (c_{B} ∘ σ_{U}, ν_{B} ∘ σ_{U}, μ_{B} ∘ σ_{U}, G_{B})

These E_{B}’s form a family E_{A,U} of two by two disjoint largely continuous cells because the various cells B involved are so and:

(x, t) ∈ E_{B} ⟺ x ∈ Â_{U} and (σ_{U}(x), t) ∈ B.  (22)

Each E_{B} has socle φ(U) = Â_{U} and for every x ∈ φ(U), σ_{U}(x) belongs to φ(T_{U}) = B̂. If B is of type 0, then so is E_{B} and μ_{B}(σ_{U}(x)) = 0 (because B is a fitting cell of type 0) hence E_{B} is a fitting cell. If B is of type 1, then μ_{B}(σ_{U}(x)) ∈ vG_{B} by Proposition 5.5 (because B is a fitting cell of type 1). That is μ_{E_{B}}(x) ∈ G_{E_{B}} hence μ_{E_{B}} is a fitting bound by Proposition 5.5. Similarly ν_{E_{B}} is a fitting bound, so E_{B} is a fitting cell. This proves (Pres), and one can easily derive from (22) that ∂E_{B} = \overline{B} so that (∂3) holds true. Note also that c_{E_{B}} ∘ φ_{|U} is N-monomial mod U_{e,n} because c_{B} ∘ φ_{|T_{U}} is so and c_{E_{B}} ∘ φ_{|U} = c_{B} ∘ σ_{U} ∘ φ_{|U} = c_{B} ∘ φ ∘ π_{U} = c_{B} ∘ φ_{|T_{U}}. The same reasoning applies to ν_{E_{B}} and μ_{E_{B}}. So the next claim finishes to prove that E_{A,U} is a partition of A_{U} and that (out), (Mon) hold true.

Claim 6.8. E_{B} ◁^{n} A_{U} and there is a semi-algebraic ◁^{n}-transition h_{E_{B},A_{U}} for (E_{B}, A_{U}) such that h_{E_{B},A_{U}} ∘ φ_{|U} is N-monomial mod U_{e,n}.

Proof: For every (x, t) in E_{B}, let us prove that (x, t) belongs to A_{U}. Since x ∈ Â_{U} it suffices to prove that (x, t) ∈ A. By construction (σ_{U}(x), t) belongs to B hence to ∂_{φ(T_{U})}^{1}A so:

|ν̄_{A}(σ_{U}(x))| ≤ |t − c̄_{A}(σ_{U}(x))| ≤ |μ̄_{A}(σ_{U}(x))| and t − c̄_{A}(σ_{U}(x)) ∈ G_{A} (23)

By (19) |ν_{A}(x)| = |ν̄_{A}(σ_{U}(x))| and |μ_{A}(x)| = |μ̄_{A}(σ_{U}(x))|. Moreover by (18):

|(t − c_{A}(x)) − (t − c̄_{A}(σ_{U}(x)))| = |c_{A}(x) − c̄_{A}(σ_{U}(x)|
≤ |π^{n_{1}} ν̄_{A}(σ_{U}(x))|  (24)
< |t − c̄_{A}(σ_{U}(x))|

Thus |t − c_{A}(x)| = |t − c̄_{A}(σ_{U}(x))| and by (23):

|ν_{A}(x))| ≤ |t − c_{A}(x)| ≤ |μ_{A}(x)|

Moreover by (24):

|\frac{t − c_{A}(x)}{t − c̄_{A}(σ_{U}(x))} − 1| ≤ |π^{n_{1}} \frac{ν̄_{A}(σ_{U}(x))}{t − c̄_{A}(σ_{U}(x)}| ≤ |π^{n_{1}} |  (25)

n_{1} > 2vN hence 1 + π^{n_{1}}R ⊆ P_{N} by Hensel’s lemma. Since t − c̄_{A}(σ_{U}(x)) ∈ G_{A} by (23) and G_{A} ∈ K^{×}/P_{N}^{×}, it follows that t − c_{A}(x) ∈ G_{A}. So (x, t) ∈ A which proves that E_{B} ⊆ A.

It remains to check that E_{B} ◁^{n}A_{U}, and to find a ◁^{n}-transition for (E_{B}, A_{U}). For every (x, t) ∈ E_{B} let:

ω_{B}(x, t) = π^{−n}(\frac{t − c_{A}(x)}{t − c̄_{A}(σ_{U}(x))} − 1)

By (25) ω_{B} takes values in π^{n_{1}−n}R hence in R since n_{1} ≥ n, thus for every (x, t) ∈ E_{B}:

t − c_{A}(x) = U_{n}(x, t)(t − c̄_{A}(σ_{U}(x)))  (26)

with U_{n} = 1 + π^{n}ω_{B} in this case. We have B ⊆ ∂_{φ(T_{U})}^{1}A and by (P3) B ◁^{n} A_{|T}. Since A is a closed ◁^{n}-complex this implies that for some A′ ∈ A we have B ◁^{n} A′ ◁^{n} ∂_{φ(T_{U})}^{1}A. Let h_{0} ∈ F_{0} be a ◁^{n}-transition function for (A′, ∂_{φ(T_{U})}^{1}A), and h_{1} ∈ F_{B} a ◁^{n}-transition function for (B, A′). Then for some α_{0}, α_{1} ∈ {0, 1} and every (x′, t′) in B we have

t′ − c̄_{A}(x′) = U_{n}(x′, t′)h_{0}^{α_{0}}(x′)(t′ − c_{A′}(x′))^{1−α_{0}}

and

t′ − c_{A′}(x′) = U_{n}(x′, t′)h_{1}^{α_{1}}(x′)(t′ − c_{B}(x′))^{1−α_{1}}

hence t′ − c̄_{A}(x′) = U_{n}(x′, t′)h(x′)^{α}(t′ − c_{B}(x′))^{1−α} with h = h_{0}^{1−α_{0}}h_{1}^{(1−α_{0})α_{1}} and α = α_{0} + α_{1} − α_{0}α_{1}. So h is a ◁^{n}-transition function for (B, ∂_{φ(T_{U})}^{1}A). Moreover h_{0} ∘ φ_{|T_{U}} and h_{1} ∘ φ_{|T_{U}} are N-monomial mod U_{e,n} by (P4), hence so is h ∘ φ_{|T_{U}}. For every (x, t) in E_{B}, (σ_{U}(x), t) ∈ B so

t − c̄_{A}(σ_{U}(x)) = U_{n}(x, t)h(σ_{U}(x))^{α}[t − c_{B}(σ_{U}(x))]^{1−α}.

Combining this with (26) and the definition of c_{E_{B}} = c_{B} ∘ σ_{U} we get

t − c_{A}(x) = U_{n}(x, t)h(σ_{U}(x))^{α}[t − c_{E_{B}}(x)]^{1−α}.

So E_{B} ◁^{n}A_{U} and h∘σ_{U} is a ◁^{n}-transition for (E_{B}, A_{U}). Moreover h∘σ_{U} ∘φ_{|U} = h ∘ φ ∘ π_{U} by definition of σ_{U}. The coordinate projection π_{U} of U onto T_{U} is obviously 1-monomial, and h ∘ φ_{|T_{U}} is N-monomial mod U_{e,n} by construction. So h ∘ σ_{U} ∘ φ is also N-monomial mod U_{e,n} and we can take h_{E_{B},A_{U}} = h ∘ σ_{U}.

Case 2.3: 0 = |ν̄_{A}| < |μ̄_{A}| on φ(T_{U}) and ν_{A} ≠ 0.

• c_{A} E D μ_{B_{1}} ∘ σ_{U} = μ_{E} •B_{0} B_{1} … B_{3}, B_{2}

Let B_{0} = B_{T_{U}}^{0} and B_{1} = B_{T_{U}}^{1} the two cells in B given by claim 6.3. Let:

E = (c_{A}, ν_{A}, μ_{B_{1}} ∘ σ_{U}, G_{A})_{|φ(U)}

If |μ_{B_{1}}| = |μ̄_{A}| on φ(T_{U}) then |μ_{B_{1}} ∘ σ_{U}| = |μ_{A}| on φ(U) by (21). Thus E and A_{U} have the same underlying set. In this case we let E_{A,U} = {E} and properties (Pres), (Mon), (∂3) are trivially true. So is (out), using Remark 6.7 for c_{A} ∘φ_{|U}, ν_{A} ∘ φ_{|U}, and (P4) for μ_{B_{1}} ∘ σ_{U} ∘ φ_{|U} = μ_{B_{1}} ∘ φ_{|T_{U}}.

Otherwise |μ_{B_{1}}| < |μ̄_{A}| on φ(T_{U}) by Claim 6.3 and we let:

D = (c_{A}, π^{−N_{0}}μ_{B_{1}} ∘ σ_{U}, μ_{A}, G_{A})_{|φ(U)}

|μ_{B_{1}}| ≤ |π^{N_{0}} μ̄_{A}| on φ(T_{U}) by Claim 6.3, |μ̄_{A} ∘ σ_{U}| = |μ_{A}| on φ(U) by (21), so |ν_{D}| = |π^{−N_{0}}μ_{B_{1}} ∘ σ_{U}| ≤ |μ̄_{A} ∘ σ_{U}| ≤ |μ_{A}| = |μ_{D}| on φ(U). Moreover A is a fitting cell hence for every x ∈ φ(U) there is t ∈ K such that (x, t) ∈ A and |t − c_{A}(x)| = |μ_{A}(x)|, so (x, t) ∈ D. Thus D is indeed a cell, with socle φ(U). It is actually a largely continuous cell, and μ_{D} = μ_{A} is a fitting bound. Let us check that ν_{D} = π^{−N_{0}}μ_{B_{1}} ∘σ_{U} is a fitting bound too. B_{1} is a fitting cell of type 1 with socle φ(T_{U}) hence μ_{B_{1}}(φ(T_{U})) ⊆ vG_{B_{1}} by Proposition 5.5. But G_{B_{1}} = G_{A} by Claim 6.3, G_{A} = G_{D} and φ(T_{U}) = σ_{U}(φ(U)) by construction, and N_{0} ∈ vG so ν_{D}(φ(U)) ⊆ vG_{D}. Thus ν_{D} is indeed a fitting bound by Proposition 5.5.

Clearly A_{U} is the disjoint union of E and D. Moreover the cells in B contained in D̄ ∩ (φ(T_{U}) × K) are exactly those contained in Ā ∩ (φ(T_{U}) × K) except B_{0} and B_{1}. Thus the construction that we have done for A_{U} in case 2.2 applies to D because ν̄_{D} = π^{−N_{0}}μ_{B_{1}} ≠ 0 on φ(T_{U}) and because the analogues of conditions (18) and (19) that we used for A_{U} in case 2.2 hold for D in the present case. Indeed by (20) we have

|c_{A_{U}} (x) − c̄_{A_{U}} (σ_{U}(x))| ≤ |π^{n_{1}−N_{0}} μ̄_{B_{1}}(σ_{U}(x))|.

This is just condition (18) for D since c_{D} = c_{A_{U}} and ν_{D} = π^{−N_{0}}μ_{B_{1}}. Moreover condition (19) for D is:

|ν_{D}| = |ν̄_{D} ∘ σ_{U}| and |μ_{D}| = |μ̄_{D} ∘ σ_{U}|

The first equality is true by definition of ν_{D} as π^{−N_{0}}μ_{B_{1}} ∘ σ_{U}. The second one is true because μ_{D} = μ_{A} and because of (21).

So the construction of Case 2.2 gives a partition E′ of D and for each E′ ∈ E′ a semi-algebraic function^{12} h_{E′,A_{U}} : φ(U) → K satisfying conditions (Pres), (∂3), (out) and (Mon). Since E also has these properties (with h_{E,A_{U}} = 1 since c_{E} = c_{A} on φ(U)) we can take E_{A,U} = {E} ∪ E′.

Case 2.4: μ̄_{A} ≠ 0 on φ(T_{U}) and ν_{A} = 0.

• ; c_{A} ; \begin{matrix}E \\ D \\ μ = ν_{D} \end{matrix} ; \begin{matrix}• \\ B_{0} \\ … B_{2}, B_{1} \end{matrix}

12Case 2.2 applied to D actually gives for each E′ ∈ E′ a ◁^{n}-transition h_{E′,D} for (E′, D).

But D ⊆ A_{U} and c_{D} = c_{A_{u}} so h_{E′,D} is also a ◁^{n}-transition for (E′, A_{U}) and we can set h_{E′,A_{U}} = h_{E′,D}.

Let again B_{1} = B_{T_{U}}^{1} be the cell given by claim 6.3. We are going to split A_{U} in two cells E and D to which previous cases apply. In order to do so, choose any i ∈ Supp U \ Supp T_{U}. For every u ∈ U let ξ_{i}(u) = u_{i}, the i-th coordinate of u. Clearly ξ_{i} is largely continuous and ξ̄_{i} = 0 on ∂U = T̄_{U}. So the function:

μ = (ξ_{i} ∘ φ^{−1})^{N}.(μ_{B_{1}} ∘ σ_{U})

is largely continuous on Â_{U} = φ(U) and μ̄ = 0 on φ(T_{U}), hence also on \overline{φ(T_{U})} = ∂φ(U). Note that μ ∘ φ_{|U} is N-monomial mod U_{e,n}. Let:

E = (c_{A}, 0, π^{N_{0}}μ, G_{A})_{|φ(U)}
D = (c_{A}, μ, μ_{A}, G_{A})_{|φ(U)}

E and D are largely continuous fitting cells mod G which define a partition of A_{U}. (Here we use that A is a fitting cell: for every x ∈ φ(U) there is t ∈ K such that (x, t) ∈ A and |t − c_{A}(x)| = |μ_{A}(x)| so (x, t) ∈ D, which proves that D is really a cell. That D, E are fitting cells and A_{U} = E ∪ D then follows from Proposition 5.5.) In particular E satisfies condition (Pres). Since ν_{E} = 0 and μ̄_{E} = π^{N_{0}} μ̄ = 0 on ∂φ(U), we have ∂E = \overline{Gr c_{E}}. By Remark 6.6, Gr c_{A_{U}} = C_{U} for some C ∈ A_{S}, and by Sub-case 2.1.1 applied to C_{U}, Gr c_{A_{U}} ∈ E_{C,U}. This proves (∂2) for E since c_{E} = c_{A_{U}}. Let h_{E,A_{U}} = 1, this is a ◁^{n}-transition for (E, A_{U}) since they have the same center, so E satisfies (out). It also satisfies (Mon), thanks to Remark 6.7 for c_{E} = c_{A_{U}} and because μ_{E} ∘ φ_{|U} = π^{N_{0}}μ ∘ φ_{|U} is N-monomial mod U_{e,n}.

Case 2.3 applies to D because ν_{D} = μ ≠ 0, |ν̄_{D}| = |μ̄| = 0 on φ(T_{U}) and |μ̄_{D}| = |μ̄_{A}| ≠ 0 on φ(T_{U}), and because the analogues of conditions (20) and (21) that we used for A_{U} in case 2.3 hold for D in the present case. Indeed (20) holds for D because it holds for A_{U}, and because D and A_{U} have the same center. Condition (21) for D is:

|ν_{D}| ≤ |μ_{B_{1}} ∘ σ_{U}| ≤ |μ̄_{D} ∘ σ_{U}| = |μ_{D}|

The first inequality is true because |ν_{D}| = |μ| ≤ |μ_{B_{1}} ∘ σ_{U}| by construction, the second one is true by claim 6.3 and because μ_{D} = μ_{A}, and the last equality is true because it is true for A_{U} by (21) and because μ_{D} = μ_{A_{U}} = μ_{A|φ(U)}.

So the construction of case 2.3 gives a partition E′ of D and for each E′ ∈ E′ a semi-algebraic function^{13} h_{E′,A_{U}} : φ(U) → K satisfying conditions (Pres), (Fron), (out) and (Mon). Since E also has these properties we can take E_{A,U} = {E} ∪ E′.

7 Cartesian morphisms

Let A be a cellular monoplex mod G such that ⋃ A is a closed subset of R^{m+1}. Let (U, ψ) be a triangulation of CB(A) with parameters (n, N, e, M) such that for every A ∈ A, ψ^{−1}(A) ∈ U (we will denote it U_{A}). Note that this is essentially the data given by the conclusion of Lemma 6.1. The aim of this section is to build a triangulation (S, φ) of A with the same parameters (n, N, e, M), together with a continuous projection Φ : ⨄ S → ⨄ U such that the following diagram is commutative.

13Same remark as in footnote 12.

\xymatrix{⋃ A \ar@{->>}[d] & ⨄ S \ar[l]_{φ} \ar@{->>}[d]^{Φ} \\ _{⋃ Â} & ⨄ U \ar[l]_{ψ}}

In order to do so we will make the assumption that G = Q_{N,M′} with M′ = M + v(N) and M > v(N). In addition we temporarily assume that A is a rooted tree, and U a simplicial complex in D^{M}R^{q_{1}} for some q_{1}. We keep these data and assumptions until the end of this section, where we finally state our result in a more precise and slightly more general form.

The construction is done below through a series of claims whose guiding lines is the following. We first book some place F_{H(A)}(D^{M}R^{q_{A}}) for each A ∈ A. So we consider A and U as trees, ordered by specialization, and build a pair of trees of finite subsets of N^{∗} ordered by inclusion, H = (H(A))_{A∈A} and P = (P (A))_{A∈A}, such that H is isomorphic to A and P is isomorphic to U (Claim 7.3. Then we allocate this place F_{H(A)}(D^{M}R^{q_{A}}) to a simplex S_{A} which we build in it, together with a semi-algebraic isomorphism φ_{A} and a semi-algebraic projection Φ_{A} such that the following diagram is commutative (Claim 7.7).

\xymatrix{A \ar@{->>}[d] & S_{A} \ar[l]_{φ_{A}} \ar@{->>}[d]^{Φ_{A}} \\ Â & U_{A} \ar[l]_{ψ_{|U_{A}}}}

It is then the shape of the trees H and P which controls how the various pieces glue together, so that S = (S_{A})_{A∈A} is a simplicial complex (Claim 7.8) and the resulting maps φ, Φ are continuous on ⨄ S (Claims 7.5 and 7.9).

Claim 7.1. The faces of U_{A} are exactly the sets U_{B} with B ≤ A in A.

Proof: Let B ≤ A in A, Y = B̂ and i = tp B. Then, with the notation of Section 5, B = ∂_{Y}^{i} A because A is a cellular complex. Since A is bounded, the socle of \overline{A} is closed hence Y must be contained in it. Since ψ^{−1}(Y) = U_{B}, it follows that U_{B} is a face of U_{A}. Conversely for every face V of U_{A}, the set B = ∂_{ψ(V)}^{0}A (resp. B = ∂_{ψ(V)}^{1}A) is non-empty if ν̄_{A|Y} = 0 (resp. μ̄_{A|Y} ≠ 0) hence belongs to A. One of these two cases necessarily happens (because |ν̄_{A}| ≤ |μ̄_{A}| on Y), which gives B ∈ A such that U_{B} = V.

Claim 7.2. Given any two cells B ≤ A in A, B < A if and only if either U_{B} < U_{A} or tp B < tp A. In particular if B is the predecessor of A in A then either U_{B} is the facet of U_{A}, or U_{B} = U_{A}, in which case tp B = 0 and tp A = 1

Proof: Recall that B = ∂_{Y}^{j} A with Y = B̂ = ψ(U_{B}) and j = tp B. In particular A = ∂_{X}^{i}A with X = Â and i = tp A. Thus B ≠ A if and only if U_{B} ≠ U_{A} or tp B ≠ tp A. Since U_{B} ≤ U_{A} by the previous claim, and obviously tp B ≤ tp A (otherwise ∂_{Y}^{j} A = ∅) this proves the equivalence. In particular if U_{B} = U_{A} then tp B < tp A hence tp B = 0 and tp A = 1.

If B is the predecessor of A in A and U_{B} ≠ U_{A}, then U_{B} < U_{A} by Claim 7.1. Let V be the facet of U_{A}. Then U_{B} ≤ V < A hence B ≤ ∂_{ψ(V)}^{j}A < A. On the

other hand B is the predecessor of A in A, hence B = ∂_{ψ(V)}^{j}A. So B̂ = ψ(V) and finally U_{B} = V.

Given a strictly increasing map σ : I → J with I ⊆ [[1, r]] and J ⊆ [[1, s]], we let [σ] : K^{s} → K^{r} be defined by [σ](y) = u where u_{i} = y_{σ(i)} if i ∈ I, and u_{i} = 0 otherwise. We say that an application f : S ⊆ K^{r} → K^{s} is a Cartesian map if for every I ⊆ [[1, r]] the restriction of f to S ∩ F_{I}(K^{r}) is of that form, that is if there is J ⊆ [[1, s]] and a strictly increasing map σ : I → J such that f(y) = [σ](y) for every y ∈ S with support I. If X is the disjoint union of finitely many sets X_{k} ⊆ K^{r_{k}} for various k, then a Cartesian map on X is simply the data of a Cartesian map on each X_{k}. A Cartesian morphism is a continuous Cartesian map.

Claim 7.3. There exists a pair of applications H, P from A to P(N^{∗}) such that H is strictly increasing and for every B ≤ A in A:

(C0) If tp A = 0 then H(A) = P (A).

(C1) If tp A = 1 then H(A) = P (A) ∪ {r_{A}} for some r_{A} > max P (A).

(C2) Card P (A) = Card(Supp U_{A}).

(C3) P (B) = H(B) ∩ P (A) (in particular P is increasing and P (B) ⊆ H(B)).

(C4) If σ_{A} : Supp U_{A} → P (A) denotes the increasing bijection given by (C2) then σ_{A}(Supp U_{B}) = P (B).

Remark 7.4. Since σ_{A} and σ_{B} are strictly increasing, (C4) implies that σ_{A}(i) = σ_{B}(i) for every i ∈ Supp U_{B}.

Proof: The construction goes by induction in Card A. For the root A of A we let P (A) = Supp U_{A}, H(A) = P (A) if tp A = 0, and H(A) = P (A) ∪ {q_{1} + 1} if tp A = 1 (recall that U is a simplicial complex in D^{M}R^{q_{1}}). If A = {A} we are done. Otherwise let A be a maximal element of A and apply the induction hypothesis to A \ {A}. This defines P (A′), H(A′) for every A′ ∈ A \ {A} so that H is strictly increasing on A \ {A} and properties (C0) to (C4) hold true for every B′ ≤ A′ in A \ {A}.

Let B be the predecessor of A in A and k = Card(Supp U_{A} \ Supp U_{B}) + 1. For every A′ ∈ A \ {A} let P_{k}(A′) = {ki}_{i∈P (A′)} and H_{k}(A′) = {ki}_{i∈H(A′)}. Clearly P_{k} and H_{k} inherit all the properties of P and H. Thus, replacing if necessary P and H by P_{k} and H_{k} we can assume that H(A′) ⊆ kN^{∗} for every A′ ∈ A \ {A}.

Let q′ be the maximum of the integers in all these sets H(A′). We have to define P (A) and H(A) so that the resulting maps P, H satisfy (C0) to (C4) for every B′ ≤ A′ in A, H(B′) ⊆ H(A′) and H(B′) ≠ H(A′) if B′ ≠ A′. By induction hypothesis it suffices to check these properties when A′ = A and B′ = B.

We are going to build σ_{A} first, and then let P (A) = σ_{A}(Supp U_{A}). Let j_{1} < ⋯ < j_{r} be an enumeration of Supp U_{B}. Let j_{0} = 0 and j_{r+1} = q′ + 1. For every i ∈ Supp U_{A} there is a unique l ∈ [[0, r]] such that j_{l} ≤ i < j_{l+1}. We then let σ_{A}(i) = σ_{B}(j_{l}) + i − j_{l} (if j_{l} = j_{0} = 0 we let σ_{B}(j_{l}) = 0 in this definition). Note that j_{l} + k ≤ j_{l+1} and σ_{B}(j_{l+1}) ∈ P (B) ⊆ kN^{∗} hence

σ_{A}(j_{l}) ≤ σ_{A}(i) < σ_{B}(j_{l}) + j_{l+1} − j_{l} ≤ σ_{B}(j_{l}) + k ≤ σ_{B}(j_{l+1}).

It follows immediately that σ_{A} is strictly increasing. Let P (A) = σ_{A}(Supp U_{A}), by construction (C2) and (C4) hold true, P (A) ∩ kN^{∗} = P (B) and P (B) is strictly contained in P (A) except if Supp U_{A} = Supp U_{B}. Note also that in any case q′ + k ∉ H(B) ∪ P (A). In order to define H(A) we distinguish four cases, given by by Claim 7.2.

Case 1: tp A = 0, hence tp B = 0 and U_{B} is the facet of A. In particular Supp U_{B} is strictly contained in Supp U_{A}, hence so is P (B) in P (A). By induction hypothesis (C0), H(B) = P (B). Let H(A) = P (A), then H(B) ⊆ H(A), H(B) ≠ H(A) and (C0), (C3) are obvious.

Case 2: tp A = 1, tp B = 0 and U_{B} = U_{A}. Then P (B) = P (A) by construction, and P (B) = H(B) by induction hypothesis (C0). Let H(A) = H(B) ∪ {q′ + k}, then H(B) ⊆ H(A), H(B) ≠ H(A) and (C1) are obvious because q′ + k ∉ H(B), and H(B) ∩ P (A) = P (B) ∩ P (A) = P (B) which proves (C3).

Case 3: tp A = 1, tp B = 0 and U_{B} is the facet of U_{A}. We let H(A) = P (A) ∪ {q′ + 1}. By induction hypothesis (C0) H(B) = P (B). By construction P (B) ⊆ P (A) ⊆ H(A). So H(B) ⊆ H(A), H(B) ≠ H(A) and (C1) are obvious because q′ + k ∉ H(B) ∪ P (A). As in Case 2, H(B) ∩ P (A) = P (B) ∩ P (A) = P (B) which proves (C3).

Case 4: tp A = tp B = 1 and U_{B} is the facet of U_{A}. By induction hypothesis (C1), P (B) is strictly contained in P (A). Let H(A) = P (A) ∪ H(B). Then H(B) ⊆ H(A), H(B) ≠ H(A) because H(A) \ H(B) = P (A) \ kN^{∗} = P (A) \ P (B) ≠ ∅, H(A) ∩ P (B) = (P (A) ∩ P (B)) ∪ (H(B) ∩ P (B)) = P (B) ∪ P (B) = P (B) which proves (C3), and (C1) follows because then H(A) \ P (A) = H(B) \ P (A) = H(B) \ P (B) is a singleton by induction hypothesis (C1).

With the notation of Claim 7.3, let q_{2} be the maximal element of ⋃_{A∈A} H(A) and S^{†} = {F_{H(A)}(D^{M}R^{q_{2}})}_{A∈A}. For every A ∈ A let Φ_{A} = [σ_{A}] : F_{H(A)}(D^{M}R^{q_{2}}) → D^{M}R^{q_{1}}. Finally let Φ : ⋃ S^{†} → D^{M}R^{q_{1}} be the resulting Cartesian map.

Claim 7.5. Φ is continuous, hence a Cartesian morphism.

Proof: We have to show that for every T ≤ S in S^{†} and every z ∈ T, Φ(y) tends to Φ(z) as y tends to z in S. By construction there are A, B in A such that H(A) = Supp S, H(B) = Supp T, Φ(y) = [σ_{A}](y) and Φ(z) = [σ_{B}](z). Since [σ_{A}] is obviously continuous, it tends to [σ_{A}](z) so we have to prove that [σ_{A}](z) = [σ_{B}](z). Let u = [σ_{A}](z) and u′ = [σ_{B}](z). Recall that u, u′ ∈ D^{M}R^{q_{1}} and for every i ∈ [[1, q_{1}]], u_{i} = z_{σ_{A}(i)} if i ∈ Supp U_{A}, u_{i} = 0 otherwise, u′_{i} = z_{σ_{B}(i)} if i ∈ Supp U_{B}, and u′_{i} = 0 otherwise.

Since T ≤ S we have Supp T ≤ Supp S, that is H(B) ≤ H(A), hence B ≤ A since H is strictly increasing. In particular Supp U_{B} ⊆ Supp U_{A} hence for every i ∈ [[1, q_{1}]], we have u_{i} = u′_{i} = 0 if i ∉ Supp U_{A}, and by Remark 7.4 z_{σ_{A}}(i) = z_{σ_{B}}(i) if i ∈ Supp U_{B}, that is u_{i} = u′_{i} in this case too. The remaining case occurs when i ∈ Supp U_{A} \ Supp U_{B}, so that u_{i} = z_{σ_{A}(i)} and u′_{i} = 0. We have to prove that z_{σ_{A}(i)} = 0, that is σ_{A}(i) ∉ Supp z. By (C4) and the assumption on i, σ_{A}(i) ∈ P (A) \ P (B). By (C3), P (A) \ P (B) = P (A) \ H(B). So σ_{A}(i) ∉ H(B), and we are done since Supp z = Supp T = H(B).

For every A ∈ A, μ_{A} ∘ ψ is N-monomial mod U_{e,n} so there are ζ ∈ K and some integers β_{i,A} for i ∈ Supp U_{A} such that for every u ∈ U_{A}

μ_{A} ∘ ψ(u) = U_{e,n}(u) · ζ · ∏_{i∈Supp U_{A}}u_{i}^{Nβ_{i,A}}.

If μ_{A} ≠ 0 then vμ_{A}(Â) = vG_{A} = vλ_{A} + NZ by Proposition 5.5, and by the above displayed equation v(ζ) ≡ v(λ_{A}) [N]. So there is β_{0,A} ∈ Z such that v(ζ) = v(λ_{A}) + Nβ_{0,A}. Let μ_{A}^{v} : vU_{A} → Z be defined by^{14} μ_{A}^{v}(a) = M′ + β_{0,A} + ∑_{i∈Supp U_{A}} β_{i,A}a_{i}. If μ_{A} = 0 then we let μ_{A}(a) = +∞ for every a ∈ vU_{A}. Define ν_{A}^{v} accordingly. By construction, for every u ∈ U_{A} we have

vμ_{A}(ψ(u)) = vλ_{A} + Nμ_{A}^{v}(vu) − NM′  (27)
vν_{A}(ψ(u)) = vλ_{A} + Nν_{A}^{v}(vu) − NM′  (28)

In particular μ_{A}^{v} (resp. ν_{A}^{v}) is uniquely determined by μ_{A} (resp. ν_{A}), even if the coefficients β_{i,A} are not.

Remark 7.6. Since A is a fitting cell mod Q_{N,M′} contained in R, vμ_{A}+M′ ≥ 0 by Proposition 5.6. On the other hand 0 ≤ vλ_{A} ≤ N − 1 (see Section 2). So, for every u ∈ U_{A} we have by (27):

μ_{A}^{v}(vu) = vμ_{A}(ψ(u)) + NM′ − vλ_{A}
≥ −M′ + NM′ − (N − 1)
= (N − 1)(M′ − 1) ≥ 0

Let S_{A} ⊆ D^{M}R^{q_{2}} be defined as follows.

• If tp A = 0, S_{A} is the set of y ∈ F_{H(A)}(D^{M}R^{q_{2}}) = F_{P (A)}(D^{M}R^{q_{2}}) such that Φ(y) ∈ U_{A}.

• If tp A = 1, S_{A} is the set of y ∈ F_{H(A)}(D^{M}R^{q_{2}}) such that Φ(y) ∈ U_{A} and μ_{A}^{v}(vΦ(y)) ≤ vy_{r_{A}} ≤ ν_{A}^{v}(vΦ(y)).

In both cases, for every y ∈ S_{A} let

φ_{A}(y) = (ψ ∘ Φ(y), c_{A}(ψ ∘ Φ(y)) + π^{−NM′}λ_{A}y_{r_{A}}^{N})

where r_{A} = max H(A) (if H(A) = ∅, which happens when A is a point, then r_{A} is not defined but in that case tp A = 0, hence λ_{A} = 0 and we can let λ_{A}y_{r_{A}}^{N} = 0 by convention).

14We remind the reader that A is a cell mod Q_{N,M′} with M′ = M + v(N).

Claim 7.7. Φ(S_{A}) = U_{A} and φ_{A} is a bijection from S_{A} to A.

Proof: If tp A = 0 the result is trivial because in that case H(A) = P (A) hence the restriction of Φ to F_{H(A)}(D^{M}R^{q_{2}}) is a bijection onto F_{Supp U_{A}}(D^{M}R^{q_{1}}). So from now we assume that tp A = 1, hence H(A) = P (A) ∪ {r_{A}} and r_{A} ∉ P (A) by (C1).

Let y, y′ ∈ S_{A} be such that φ_{A}(y) = φ_{A}(y′). Then ψ(Φ(y)) = ψ(Φ(y′)) and

c_{A}(ψ ∘ Φ(y)) + π^{−NM′}λ_{A}y_{r_{A}}^{N} = c_{A}(ψ ∘ Φ(y′)) + π^{−NM′}λ_{A}y_{r_{A}}^{′N}

These two equations imply that y_{r_{A}}^{N} = y_{r_{A}}^{′N}. Since y_{r_{A}}, y′_{r_{A}} ∈ D^{M}R = Q_{1,M} ∩ R and M > v(N) it follows that y_{r_{A}} = y′_{r_{A}} by Lemma 2.9. On the other hand ψ(Φ(y)) = ψ(Φ(y′)) implies Φ(y) = Φ(y′) (because ψ is one-to-one), that is y_{i} = y′_{i} for every i ∈ P (A) (because Φ(y) = [σ_{A}](y) by construction). Thus y_{i} = y′_{i} for every i ∈ P (A) ∪ {r_{A}} = H(A), that is y = y′ since Supp y = Supp y′ = H(A). This proves that φ_{A} is one-to-one.

Let us check now that A ⊆ φ_{A}(S_{A}). Pick any (x, t) ∈ A, let u = ψ^{−1}(x) and δ = t − c_{A}(x). Since δ ∈ λ_{A}Q_{N,M′} and π^{NM′} ∈ Q_{N,M′} we have π^{NM′}δ ∈ λ_{A}Q_{N,M′}. Recall that M′ = M+v(N), by Lemma 2.9 there is a unique z ∈ Q_{1,M} such that π^{NM′}δ = λ_{A}z^{N}, hence t = c_{A}(x) + π^{−NM′}λ_{A}z^{N}. On the other hand we have vμ_{A}(ψ(u)) = vμ_{A}(x) ≤ vδ so by (27)

vz = \frac{v(π^{NM′}δ/λ_{A})}{N} ≥ \frac{NM′ + vμ_{A}(ψ(u)) − vλ_{A}}{N} = μ_{A}^{v}(vu).

In particular vz ≥ 0 by Remark 7.6 so z ∈ Q_{1,M} ∩ R = D^{M}R. Similarly vz ≤ ν_{A}^{v}(vu) by (28). Let y ∈ D^{M}R^{q_{2}} be such that y_{i} = u_{σ_{A}(i)} if i ∈ P (A), y_{i} = z if i = r_{A}, y_{i} = 0 otherwise. Then y ∈ F_{H(A)}(D^{M}R^{q_{2}}), Φ(y) = [σ_{A}](y) = u and μ_{A}^{v}(vu) ≤ vy_{r_{A}} ≤ ν_{A}^{v}(vu) since y_{r_{A}} = z, so y belongs to S_{A}. By construction φ_{A}(y) = (x, t) ∈ A, which proves that A ⊆ φ_{A}(S_{A}).

We turn now to Φ(S_{A}). For every u ∈ U_{A}, ψ(u) ∈ Â so there is (x, t) ∈ A such that u = ψ^{−1}(x). The above construction gives y ∈ S_{A} such that φ_{A}(y) = (x, t). In particular ψ ∘ Φ(y) = x, so Φ(y) = ψ^{−1}(x) = u, which proves that Φ(S_{A}) ⊆ U_{A}. Since Φ(S_{A}) ⊆ U_{A} by definition of S_{A} we get that Φ(S_{A}) = U_{A}.

It only remains to show that φ_{A}(S_{A}) ⊆ A. Pick any y ∈ S_{A}, let (x, t) = φ_{A}(y) and δ = t − c_{A}(x) = π^{−NM′}λ_{A}y_{r_{A}}^{N}. Since Φ(y) ∈ Φ(S_{A}) = U_{A}, we have x = ψ(Φ(y)) ∈ ψ(U_{A}) = Â. Since y_{r_{A}} ∈ D^{M}R = Q_{1,M} ∩ R, by Lemma 2.9 y_{r_{A}}^{N} ∈ Q_{N,M+v(N)} = Q_{N,M′}. Hence δ = π^{−NM′}λ_{A}y_{r_{A}}^{N} belongs to λ_{A}Q_{N,M′}. We have μ_{A}^{v}(vΦ(y)) ≤ vy_{r_{A}} by definition of S_{A}, so by (27)

vμ_{A}(ψ(Φ(y))) = vλ_{A} + Nμ_{A}^{v}(vΦ(y)) − NM′ ≤ vλ_{A} + Nvy_{r_{A}} − NM′.

The left hand side is equal to vμ_{A}(x). For the right hand side we have

vλ_{A} + Nvy_{r_{A}} − NM′ = v(π^{−NM′}λ_{A}y_{r_{A}}^{N}) = vδ.

So vμ_{A}(x) ≤ vδ, that is |δ| ≤ |μ_{A}(x)|. Similarly |ν_{A}(x)| ≤ |δ| hence (x, t) ∈ A.

Claim 7.8. S_{A} is a simplex in D^{M}R^{q_{2}}, whose faces are exactly the sets S_{B} with B ≤ A in A.

Proof: Let q = Card P (A) and q′ = Card H(A). Let τ_{A} (resp. τ′_{A}) be the strictly increasing map from P (A) to [[1, q]] (resp. from H(A) to [[1, q′]]). By construction and by Claim 7.5 the following diagram is commutative (vertical arrows are the natural coordinate projections).

\xymatrix{D^{M}R^{q′} \ar[r]^{[τ′_{A}]} \ar[d] & \overline{F_{H(a)}(D^{M}R^{q_{2}})} \ar[dr]^{[σ_{A}]=Φ} \ar[d] \\ D^{M}R^{q} \ar[r]^{[τ_{A}]} & \overline{F_{P (a)}(D^{M}R^{q_{2}})} \ar[r]^{[σ_{A}]=Φ} & \overline{F_{Supp U_{A}}(D^{M}R^{q_{1}})}}

The horizontal arrows in this diagram are isomorphisms. All of them are obtained simply by renumbering the coordinates, hence they preserve the faces and the property of being a simplex. It will then be convenient here to identify isomorphic spaces, hence to consider U_{A} ⊆ D^{M}R^{q} and S_{A} ⊆ D^{M}R^{q′}. Since Φ(S_{A}) = U_{A} by Claim 7.7, after this identification U_{A} is just the image of S_{A} by the coordinate projection of D^{M}R^{q′} to D^{M}R^{q}. Since H(A) = Supp S_{A} we identify also H(A) with [[1, q′]], and P (A) with [[1, q]].

If tp A = 0 then H(A) = P (A), q′ = q and the vertical arrows are identity maps. Thus S_{A} identifies with U_{A}. In particular S_{A} is a simplex. Every B ≤ A is also of type 0 and S_{B} identifies to U_{B}. The conclusion follows by Claim 7.1.

From now on, let us assume that tp A = 1. Then q′ = q + 1 hence U_{A} is just the socle of S_{A}. Similarly, U_{B} is the socle of S_{B} for every B ≤ A (if tp B = 0 we have S_{B} = U_{B} × {0}). By construction S_{A} is the inverse image of vS_{A} by the valuation (restricted to D^{M}R^{q+1}) and

vS_{A} = {a ∈ Z^{q+1} : â ∈ vU_{A} and μ_{A}^{v}(â) ≤ a_{q+1} ≤ ν_{A}^{v}(â)}.

Since μ_{A} ∘ ψ and ν_{A} ∘ ψ are largely continuous on U_{A}, (27) and (28) imply that μ_{A}^{v} is largely continuous on vU_{A}. They are affine maps by definition. Since 0 ≤ μ_{A}^{v} by Remark 7.6, and μ_{A}^{v} ≤ ν_{A}^{v} because |ν_{A}| ≤ |μ_{A}|, it follows that vS_{A} is a polytope in Γ^{q+1}. We are going to check that its faces are exactly the sets vS_{B} for B ≤ A in A. This will finish the proof since S_{A} will then have the expected faces, which implies that S_{A} is a simplex because these faces form a chain by specialisation (because A is a tree).

Step 1. Let B ≤ A in A, then B = ∂_{Y}^{i} A with Y = B̂ and i = tp B. Let J = H(B) ⊆ H(A) = [[1, q + 1]] and Ĵ = P (A) = J \ {q + 1}. Since (μ_{B}, ν_{B}) = (μ̄_{A}, ν̄_{A})_{|Y}, if tp B = 1 we have by construction

vS_{B} = {a ∈ F_{J}(Γ^{q+1}) : â ∈ vU_{B} and μ̄_{A}^{v}(â) ≤ a_{q+1} ≤ ν̄_{A}^{v}(â)}.  (29)

This remains true also if tp B = 0 because in that case q + 1 ∉ J and ν̄_{A}^{v} = +∞ on vU_{B} (because ν̄_{A} = ν_{B} = 0 on Y) so the right hand side is just vU_{B} ×{+∞}, that is vS_{B} (because S_{B} = U_{B} × {0} when tp B = 0). In both cases we also have vU_{B} = FĴ(U_{A}), because vU_{B} is a face of U_{A} by Claim 7.1 and Supp vU_{B} = P (B) = Ĵ. So the description of vS_{B} given by (29) coincides with the description of F_{J}(vS_{A}) given by Proposition 2.13, which proves that vS_{B} = F_{J}(vS_{A}).

Step 2. Conversely let F_{J}(vS_{A}) ≠ ∅ be a face of vS_{A}, for some J ⊆ [[1, q + 1]], and let Ĵ = J \ {q + 1}. By Proposition 2.13 the socle of F_{J}(vS_{A}) is FĴ(vU_{A}) (because vU_{A} is the socle of vS_{A}) and two cases can happen: q + 1 ∈ J and μ̄_{A}^{v} < +∞ on FĴ(vU_{A}), or q + 1 ∉ J and ν̄_{A}^{v} = +∞ on FĴ(vU_{A}). In both cases

F_{J}(vS_{A}) = {a ∈ F_{J}(Γ^{q+1}) : â ∈ FĴ(vU_{A}) and μ̄_{A}^{v}(â) ≤ a_{q+1} ≤ ν̄_{A}^{v}(â)}. (30)

Since FĴ(vU_{A}) is a face of vU_{A}, by Claim 7.1 there is C ≤ A in A such that FĴ(vU_{A}) = vU_{C}. Let Y = Ĉ = ψ(U_{C}).

If q + 1 ∉ J then by Proposition 2.13, ν̄_{A}^{v} = +∞ on FĴ(vU_{A}) = vU_{C}. That is ν̄_{A} = 0 on Y = ψ(U_{C}), hence ∂_{Y}^{0} A ∈ A. Let B = ∂_{Y}^{0} A and apply Step 1 to B. Since J = Ĵ is the support of vU_{C} = vU_{B} and of S_{B} (because tp B = 0), we deduce from (29) and (30) that vS_{B} = F_{J}(S_{A}).

If q + 1 ∈ J then by Proposition 2.13, μ̄_{A}^{v} ≠ +∞ on FĴ(vU_{A}) = vU_{C}. That is μ̄_{A} ≠ 0 on Y = ψ(U_{C}), hence ∂_{Y}^{1} A ∈ A. Let B = ∂_{Y}^{1} A and apply Step 1 to B. Since Ĵ is the support of vU_{C} = vU_{B} and J = Ĵ∪ {q + 1} is the support of S_{B} (because tp B = 1), we deduce from (29) and (30) that vS_{B} = F_{J}(S_{A}).

Finally let φ : ⋃ S → ⋃ A be defined by φ_{|A} = φ_{A} on each S_{A}.

Claim 7.9. φ is a homeomorphism from ⋃ S to ⋃ A.

Proof: We already know by Claim 7.7 that φ is a bijection from ⋃ S to ⋃ A. It follows from Claim 7.8 that ⋃ S is closed, and it is obviously bounded. Thus by Theorem 2.6 it suffices to show that φ is continuous. Since each φ_{A} is obviously continuous on S_{A}, we only have to prove that for every z ∈ ∂S_{A} and y ∈ S_{A}, φ_{A}(y) tends to φ(z) as y tends to S_{A}. By Claim 7.8 there is B ≤ A in A such that z ∈ S_{B}, hence φ(z) = φ_{B}(z). By Claim 7.5, ψ ∘ Φ(y) tends to ψ ∘ Φ(z). By Claim 7.7, ψ ∘ Φ(y) ∈ Â and ψ ∘ Φ(z) ∈ B̂ hence c_{A}(ψ ∘ Φ(y)) tends to c̄_{A}(ψ ∘ Φ(z)), which is equal to c_{B}(ψ ∘ Φ(z)) since c̄_{A|B̂} = c_{B}. Thus it only remains to check that λ_{A}y_{r_{A}}^{N} tends to λ_{B}z_{r_{B}}^{N}.

If tp A = 0 then also tp B = 0 hence and we are done, since λ_{A}y_{r_{A}}^{N} = 0 = λ_{B}z_{r_{B}}^{N}. If tp B = 1 then λ_{B} = λ_{A} (because A is a cellular complex) and r_{A} = r_{B} (because by (C1) and (C3), H(B) ≠ P (B) = H(B)∩P (A) implies that H(B) is not contained in P (A), hence r_{A} ∈ H(B) since H(B) ⊆ H(A) = P (A) ∪ {r_{A}}). Hence obviously λ_{A}y_{r_{A}}^{N} tends to λ_{A}z_{r_{A}}^{N} = λ_{B}z_{r_{B}}^{N} in that case. Finally if tp A = 1 and tp B = 0 then r_{A} ∉ H(B) (because by (C0) and (C3), H(B) = P (B) ⊆ P (A)), hence z_{r_{A}} = 0 since Supp z = Supp S_{B} = H(B). Thus λ_{A}y_{r_{A}}^{N} tends to λ_{A}z_{R_{A}}^{N} = 0, which proves the result because λ_{B}z_{r_{B}}^{N} = 0 since tp B = 0.

We can recap now our construction and state the result which was the aim of this section.

Lemma 7.10. Let A be a cellular monoplex mod Q_{N,M+v(M)}^{×} such that ⋃ A is a closed subset of R^{m+1}. Let (U, ψ) be a triangulation of CB(A) with parameters (n, N, e, M) such that M > v(N) and for every A ∈ A, ψ^{−1}(A) ∈ U (let us denote it U_{A}). Then there exists a simplicial complex S of index M, a Cartesian morphism Φ : ⨄ S → ⨄ U and a semi-algebraic homeomorphism φ : ⨄ S → ⋃ A such that for every A ∈ A, φ^{−1}(A) ∈ S (let us denote it S_{A}) and for every y ∈ S_{A}

φ(y) = (ψ ∘ Φ(y), c_{A}(ψ ∘ Φ(y)) + π^{−NM′}λ_{A}y_{r_{A}}^{N})

where^{15} r_{A} = max(Supp S_{A}).

Proof: Let (A_{k})_{1≤k≤r} be the list of minimal elements in A, and for each k let A_{k} be the family of elements in A greater than A_{k}. This is a rooted, cellular monoplex mod Q_{N,M+v(N)}^{×}. For every A ∈ A_{k}, \overline{A} is the union of the cells B ≤ A in A since A is a cellular complex and ⋃ A is closed. All these cells B belong to A_{k} hence ⋃ A_{k} is closed. Since ⋃ A\⋃ A_{k} is the union of the finitely many other A_{l} it is closed, hence ⋃ A_{k} is clopen in ⋃ A. Let U_{k} = {ψ^{−1}(Â) : A ∈ A_{k}}, this is a lower subset of U with smallest element ψ^{−1}(Â_{k}) hence a rooted simplicial complex in D^{M_{1}}R^{q_{1},k} for some q_{1,k}. Finally let ψ_{k} be the restriction of ψ to ⋃ U_{k}.

Claims 7.1 to 7.9 apply to (U_{k}, ψ, A_{k}) and give a simplicial complex S_{k} in D^{M}R^{q_{2},k} for some q_{2,k}, a Cartesian morphism Φ_{k} : ⋃ S_{k} → ⋃ U_{k} and a semi-algebraic homeomorphism φ_{k} : ⨄ S_{k} → ⋃ A_{k} satisfying all the required properties. Since each ⋃ A_{k} is clopen in ⋃ A, and each ⋃ U_{k} is clopen in ⨄ U, the conclusion follows by taking for U the family {U_{k}}_{1≤k≤r} and for Φ (resp. φ) the map obtained by glueing together the various Φ_{k} (resp. φ_{k}).

8 Triangulation

We have come up to the moment when we can show that T_{m} ⇒ T_{m+1}. As T_{0} is rather obvious, this will finish the proof of T_{m} for every m.

Theorem 8.1. Assume T_{m}. Let (θ_{i} : A_{i} ⊆ K^{m+1} → K)_{i∈I} be a finite family of semi-algebraic functions, and n, N ≥ 1 be any integers. Then for some integers e, M ≥ 1 which can be chosen arbitrarily large (in the sense of footnote 2), there exists a simplicial complex T of index M and a semi-algebraic homeomorphism φ : ⨄ T → ⋃_{i∈I} A_{i} such that for every i in I:

1. {φ(T) : T ∈ T and φ(T) ⊆ A_{i}} is a partition of A_{i}.

2. ∀T ∈ T such that φ(T) ⊆ A_{i}, θ_{i} ∘ φ_{|T} is N-monomial mod U_{e,n}.

Proof: By using the same partition of K^{m+1} as in the proof of Lemma 3.3 we are reduced to the case where each A_{i} is contained in R^{m+1}. We can also extend each θ_{i} to R^{m+1} by an arbitrary value, and add to this family the indicator functions of each A_{i} inside R^{m+1}, hence assume that all these functions have domain R^{m+1}, which is closed and bounded. Let e_{∗} ≥ 1 and M_{∗} ≥ 1 be any integers.

Theorem 4.6 applies to (θ_{i})_{i∈I}. It gives an integer e_{0} ≥ 1, a tuple η ∈ R^{m}, a linear automorphism u_{η}(x, t) = (x + tη, t) of K^{m+1} (note that u_{η}(R^{m+1}) = R^{m+1} since η ∈ R^{m+1}), a pair of integers N_{0} ≥ 1 and M_{0} > 2v(e_{0}) such that e_{0}N divides N_{0}, and a finite family A of largely continuous cells mod Q_{N_{0},M_{0}}^{×} partitioning u_{η}^{−1}(R^{m+1}) = R^{m+1} such that for every i ∈ I, every A ∈ A and every (x, t) ∈ A

θ_{i} ∘ u_{η}(x, t) = U_{e_{0},n}(x, t)h_{i,A}(x)[λ_{A}^{−1}(t − c_{A}(x))]^{\frac{α_{i,A}}{e_{0}}}  (31)

15If Supp S_{A} = ∅ then r_{A} is not defined but in that case S_{A} is a point, hence so is A so λ_{A} = 0 and we can let λ_{A}y_{r_{A}}^{N} = 0 by convention.

where h_{i,A} : Â → K is a semi-algebraic function and α_{i,A} ∈ Z.

Let n_{1} = max(1 + 2v(e_{0}), n + v(e_{0})), Lemma 6.1 applied to A and the family F_{0} = {h_{i,A} : i ∈ I, A ∈ A} gives a pair of integers e_{1} ≥ 1 and M_{1} > 2v(e_{1}), a cellular monoplex B mod Q_{N_{0},M_{0}} refining A such that B ◁^{n_{1}} A, a ◁^{n_{1}}-system F_{1} for (B, A), and a triangulation (U, ψ) of F_{0} ∪ F_{1} ∪ CB(B) with parameters (n_{1}, N_{0}, e_{1}, M_{1}). Moreover e_{1}, M_{1} can be chosen arbitrarily large, in the sense of footnote 2, so we can require that e_{∗} divides e_{1} and M_{1} ≥ M_{∗}, and that M_{1} ≥ M_{0} − v(N_{0}) and M_{1} > v(N_{0}) ≥ v(e_{0}).

Q_{N_{0},M_{1}+v(N_{0})}^{×} is a subgroup of Q_{N_{0},M_{0}}^{×} (because M_{1}+v(N_{0}) ≥ M_{0}) with finite index. Hence every cell in B is the disjoint union of finitely many cells C mod Q_{N_{0},M_{1}+v(N_{0})}^{×} with the same socle and bounds as B. Since vQ_{N_{0},M_{1}+v(N_{0})}^{×} = N_{0}Z = vQ_{N_{0},M_{0}}^{×}, these cells C are still fitting cells by Proposition 5.5. One easily sees that they form a cellular monoplex C refining A such that C ◁^{n_{1}} A and F_{1} is a ◁^{n_{1}}-system for (C, A). Moreover CB(C) = CB(B) and Ĉ = B̂ so (U, ψ) is a triangulation of B(C) with parameters (n_{1}, N_{0}, e_{1}, M_{1}) such that ψ^{−1}(Ĉ) ∈ U for every C ∈ C.

Since M_{1} > v(N_{0}), Lemma 7.10 applies to C and (U, ψ). It gives a simplicial complex T of index M_{1}, a Cartesian morphism Φ : ⨄ T → ⨄ U and a semialgebraic homeomorphism φ : ⨄ T → ⋃ C such that φ^{−1} maps each C in C onto some T in T , and for every y in T

φ(y) = (ψ ∘ Φ(y), c_{C}(ψ ∘ Φ(y)) + π^{−NM′}λ_{C}y_{r_{C}}^{N_{0}})  (32)

where^{16} r_{C} = max(Supp T). Let φ_{η} = u_{η} ∘φ, this is a semi-algebraic homeomorphism from ⨄ T to R^{m+1}. We are going to check that θ_{i} ∘ φ_{η|T} is N-monomial mod U_{e_{0}e_{1},n} for every i ∈ I and every T ∈ T . This will prove the result, with e = e_{0}e_{1} and M = M_{1}.

So pick any T ∈ T , let C = φ(T) and r_{C} be as above. There is a unique B ∈ B containing C, a unique A ∈ A containing B. For every (x, t) ∈ C let δ_{C}(x, t) = t − c_{C}(x). Let δ_{A} and δ_{B} be defined accordingly. Note that δ_{C} = δ_{B} on C because C has the same center as B by construction. For every y ∈ T, by (31) and (32) we have

θ_{i} ∘ φ_{η}(y) = U_{e_{0},n}(φ(y))h_{i,A}(ψ ∘ Φ(y))[λ_{A}^{−1}δ_{A}(φ(y))]^{\frac{α_{i,A}}{e_{0}}} .

We have U_{e_{0},n}(φ(y)) ∈ U_{e_{0},n} ⊆ U_{e_{0}e_{1},n} so the factor U_{e_{0},n}(φ(y)) can be replaced by U_{e_{0}e_{1},n}(y). Recalling that (V, ψ) is a triangulation of F_{0}∪F_{1} with parameters (n_{1}, N_{0}, e_{1}, M_{1}), that Φ is a Cartesian morphism and h_{i,A} ∈ F_{0}, we get that the second factor h_{i,A}(ψ ∘ Φ(y)) = h_{i,A} ∘ ψ(Φ(y)) is N_{0}-monomial mod U_{e_{1},n_{1}} hence a fortiori N-monomial mod U_{e_{0}e_{1},n} since N divides N_{0} and n_{1} ≥ n. So it only remains to prove that the last factor [λ_{A}^{−1}δ_{A} ∘ φ_{|T}]^{α_{i,A}/e_{0}} is N-monomial mod U_{e_{0}e_{1},n}. It suffices to prove it for [λ_{A}^{−1}δ_{A} ∘ φ]^{1/e_{0}}.

We can assume that tp A = 1 otherwise λ_{A}^{−1}δ_{A} = 1 and the result is trivial (see Remark 4.7). Recall that C ◁^{n_{1}} A and F_{1} is a ◁^{n}-system for (C, A). For every (x, t) ∈ C we then have

t − c_{A}(x) = U_{n_{1}}(x, t)h_{C,A}(x)^{β}(t − c_{C}(x))^{1−β}

16See footnote 15.

with h_{C,A} ∈ F_{1} and β ∈ {0, 1} (depending on A, C). So by (32) we have

δ_{A}(φ(y)) = U_{n_{1}}(φ(y))h_{C,A}(ψ ∘ Φ(y))^{β}(π^{−NM}λ_{C}y_{r_{C}}^{N_{0}})^{1−β}.  (33)

(V, ψ) is a triangulation of F_{1} with parameters (n_{1}, N_{0}, e_{1}, M_{1}) hence h_{C,A}(ψ ∘ Φ(y)) is N_{0}-monomial mod U_{e_{1},n_{1}}. So (33) implies that δ_{A} ∘φ_{|T} is N_{0}-monomial mod U_{e_{1},n_{1}}, hence so is λ_{A}^{−1}δ_{A} ∘ φ_{|T}. Let χ : T → U_{e_{1}} and g : T → K be semialgebraic functions that for every y ∈ T

λ_{A}^{−1}δ_{A} ∘ φ(y) = χ(y)U_{n_{1}}(y)ζg(y) and g(y) = ∏_{1≤i≤q} y_{i}^{α_{i}N_{0}}

with ζ ∈ K, α_{1}, …, α_{q} ∈ Z. Let k = N_{0}/(e_{0}N), by construction e_{0}N divides N_{0} hence k ∈ N^{∗}. Since T ⊆ D^{M_{1}}R^{q′}, each y_{i} ∈ D^{M_{1}}R ⊆ Q_{1,M_{1}} ⊆ Q_{1,v(e_{0})+1} (because M_{1} > v(e_{0})) hence y_{i}^{e_{0}} ∈ Q_{e_{0},2v(e_{0})+1}. A fortiori y^{α_{i}N_{0}} = y^{e_{0}Nkα_{i}} belongs to Q_{e_{0},2v(e_{0})+1} hence g takes values in Q_{e_{0},2v(e_{0})+1} and g^{1/e_{0}} is N-monomial:

(g(y))^{\frac{1}{e_{0}}} = ( ∏_{1≤i≤q} y_{i}^{e_{0}Nkα_{i}})^{\frac{1}{e_{0}}} = ∏_{1≤i≤q} y_{i}^{Nkα_{i}}

But λ_{A}^{−1}δ_{A} also takes values in Q_{e_{0},2v(e_{0})+1} because δ_{A}(x, t) ∈ λ_{A}Q_{N_{0},M_{0}} for every (x, t) ∈ A, and Q_{N_{0},M_{0}} ⊆ Q_{e_{0},2v(e_{0})+1} since e_{0} divides N_{0} and M_{0} > 2v(e_{0}). Thus (λ_{A}^{−1}δ_{A} ∘ φ_{|T})/g = U_{n_{1}}ζχ takes values in Q_{e_{0},2v(e_{0})+1} as well. So does the factor U_{n_{1}} since n_{1} > 2v(e_{0}). Hence finally ζξ(y) ∈ Q_{e_{0},2v(e_{0})+1} for every y ∈ T, so (ζχ)^{1/e_{0}} is well defined. Note that ζ^{e_{1}} = ζ^{e_{1}}χ^{e_{1}} = [(ζχ)^{1/e_{0}}]^{e_{0}e_{1}} hence ζ^{e_{1}} ∈ P_{e_{0}e_{1}}. Pick any η ∈ K such that ζ^{e_{1}} = η^{e_{0}e_{1}}, and for every y ∈ T let χ′(y) = (ζχ(y))^{1/e_{0}}/η. This is a semi-algebraic function taking values in U_{e_{0}e_{1}} because

[(ζχ)^{1/e_{0}}]^{e_{0}e_{1}} = ζ^{e_{1}} = η^{e_{0}e_{1}}.

By Remark 2.10, U_{n_{1}}^{1/e_{0}} = U_{n_{1}−v(e_{0})} because n_{1} > 2v(e_{0}), and by definition χ′U_{n_{1}−v(e_{0})} = U_{e_{0}e_{1},n_{1}−v(e_{0})}. Altogether this gives that

[λ_{A}^{−1}δ_{A} ∘ φ_{|T} ]^{\frac{1}{e_{0}}} = Un_{1}^{\frac{1}{e_{0}}} (ζχ)^{\frac{1}{e_{0}}} g^{\frac{1}{e_{0}}}
= χ′U_{n_{1}−v(e_{0})}((ζχ)^{\frac{1}{e_{0}}} /χ′i)g^{\frac{1}{e_{0}}}
= U_{e_{0}e_{1},n_{1}−v(e_{0})}ηg^{\frac{1}{e_{0}}}

Thus [λ_{A}^{−1}δ_{A} ∘ φ]^{1/e_{0}} is N-monomial mod U_{e_{0}e_{1},n_{1}−v(e_{0})} (because g^{\frac{1}{e_{0}}} is so). It is a fortiori N-monomial mod U_{e_{0}e_{1},n} since n_{1} − v(e_{0}) ≥ n by construction.

9 Open problems

Almost every proof in this paper remains valid for subanalytic sets and maps, and more generally for definable sets and maps in any p-optimal structure over a p-adically closed field having the Extreme Value Property (see [DH15]). Unfortunately there is one exception: the argument given in Theorem 4.6 to build a cell decomposition with largely continuous centers and bounds, by means of small deformations, only works for semi-algebraic sets. This is the single obstruction for a generalization to other structures, and there is good hope that a work-around can be found in the future.

Question 9.1. How to extend the p-adic triangulation to richer structures, such as the subanalytic structure?

The triangulation sheds some new light on the problem of classifying p-adic semi-algebraic sets up to semi-algebraic homeomorphisms, but does not solve it. Indeed we do not know at the moment when two simplicial complexes are semi-algebraically homeomorphic. A necessary condition is obviously that they have the same dimension.

Question 9.2. Given two semi-algebraic sets A ⊆ K^{m} and B ⊆ K^{n}, how to decide if they are semi-algebraically homeomorphic?

When A, B are both closed, bounded, without isolated point and of the same dimension it is not so obvious to find an obstruction for them to be semialgebraically homeomorphic. If they have dimension 1 then Clucker’s construction in [Clu01] actually shows that they are semi-algebraically homeomorphic. Here is a new criterion involving higher dimensions. Let C(A) denote the ring of continuous semi-algebraic functions from A to K, and L(A) the lattice of closed semi-algebraic subsets of A. Now let L^{∘}(A) be the subscaled lattice (in the sense of [Dar06]) of L(A) generated by A. This lattice contains the pure dimensional components of A, the pure dimensional components of their intersections, the closure of every boolean combination of such (inside A) and so on. But contrary to L(A) it is a finite lattice. Let C(B), L(B) and L^{∘}(B) be defined accordingly. If A and B are semi-algebraically homeomorphic, then C(A) and C(B) are isomorphic, hence L(A) and L(B) (which encode the topology of the spectra of C(A) and C(B) by means of Stone’s duality) are isomorphic, and this isomorphism maps L^{∘}(A) to L^{∘}(B). This leads us to a more precise question, if not a conjecture.

Question 9.3. Let A ⊆ K^{m} and B ⊆ K^{n} be closed, bounded, semi-algebraic sets without isolated points, such that dim A = dim B and L^{∘}(A) ≃ L^{∘}(B). Are they semi-algebraically homeomorphic?

If A and B are pure dimensional then L^{∘}(A) ≃ L^{∘}(B) follows from the other assumptions in Question 9.3. Hence it might be sufficient to answer the next weaker question.

Question 9.4. Are any two closed, bounded, semi-algebraic sets with the same pure dimension ≥ 1 semi-algebraically homeomorphic?

Of course a positive answer to the above questions would solve the next conjecture.

Conjecture 9.5. There are only countably many different homeomorphism classes of semi-algebraic sets over K.

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