Constructible functions on 2−dimensional analytic manifolds

ISABELLE BONNARD^{1} and FEDERICA PIERONI^{2} 1 Max-Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany ibonnard@mpim-bonn.mpg.de 2 Dipartamento di Matematica L. Tonelli, Via F. Buonarroti 2, 56127 Pisa, Italy pieroni@mail.dm.unipi.it

Abstract: We give a characterization of sums of signs of global analytic functions on a real analytic manifold M of dimension two. We use it to study analytically constructible functions. On the real spectrum of the field of meromorphic functions on M, our characterization shows that unbounded orderings are not relevant to recognize signatures of quadratic forms.

Mathematics Subject Classifications (2000): 14P15

Key words: global semianalytic sets, constructible functions, real spectrum.

Introduction.

Let M be a paracompact real analytic manifold. A function φ : M → ℤ is said to be semianalytically constructible if it is constant on each element of a finite partition of M into global semianalytic sets.

The aim of this paper is to give, if dim M = 2, a characterization of the semianalytically constructible functions which are finite sums of signs of global analytic functions. We prove that in dimension two, the obstructions for a semianalytically constructible function φ to be a sum of signs of global analytic functions lie in the boundaries of the φ^{−1}(n) for n ∈ ℤ. We do it in two steps: first we reduce our problem to the characterization of functions which are generically sums of signs of analytic functions, and then we give such a characterization. We get it as a consequence of a characterization of principal open sets, i.e. sets of the form {x ∈ M | f(x) > 0} where f is a global analytic function on M.

These results are similar to the ones for sums of signs of polynomials in [B1], and for sums of signs of Nash functions in [B2], except that here we do not need any hypothesis of compactness. Indeed, φ is a sum of signs of global analytic functions on M if and only if it is true on each compact subset of M. Compare with [BP] for similar results.

Our motivation is to study analytically constructible functions, which are defined as linear combinations, with integer coefficients, of Euler characteristic of fibres of proper analytic morphisms. We show that in general, the two classes of analytically constructible and semianalytically constructible functions are not contained in each other, even if M is compact of dimension 2. However, we prove that, if dim M = 2, the intersection between these two classes coincides exactly with the class of constructible functions which are sums of signs of global analytic functions on M. Similar results exist in the algebraic case, cf. [CK], [PS], and in the Nash case for a compact set, cf. [B2]. We also prove that the compactness assumption is not necessary in dimension 2 for the Nash case.

Assume that M is connected of dimension 2, and denote by ℳ(M) the field of meromorphic functions on M. Artin-Lang property gives a correspondence between semianalytically constructible functions on M (considered generically) and constructible functions on the real spectrum of ℳ(M), i.e. functions Spec_{r}ℳ(M) → ℤ which are continuous with respect to the constructible topology of Spec_{r}ℳ(M) and the discrete topology of ℤ. Sums of signs of global analytic functions correspond to signatures of quadratic forms over Spec_{r}ℳ(M). This allows us to bound the minimal number of analytic functions we need to write generically a sum of signs of analytic functions. A characterization of signatures of quadratic forms over the real spectrum is given by the Representation Theorem of Becker and Bröcker. We prove that, when working on ℳ(M), it is not necessary to consider unbounded orderings in this characterization.

The paper is organized as follows. In the first section, we define the functions we work on, and we give some results in dimension one and two. The second section is devoted to a characterization of principal open sets (in any dimension), and a characterization of sums of signs of analytic functions in dimension two. In the third part, we study analytically constructible and Nash constructible functions in dimension two. The fourth part deals with quadratic forms over the real spectrum.

We want to thank Francesca Acquistapace, Carlos Andradas and Fabrizio Broglia for their help.

1 Definitions and first results

1.1 Definitions and notations

Let M be a paracompact real analytic manifold. We denote by 𝒪(M) the ring of global analytic functions on M. If M is connected, we denote by ℳ(M) the field of meromorphic functions on M, i.e. the quotient field of 𝒪(M).

We will say that a property is true generically on M if it is true on M outside a global analytic subset of M of strictly smaller dimension. We will write =_{gen} for an equality holding generically.

A subset S ⊂ M is called global semianalytic if it can be written in the form

S = ⋃_{i=1}^{p}{x ∈ M | f_{i}(x) = 0, g_{i,1}(x) > 0, …, g_{i,r_{i}}(x) > 0}

where f_{i}, g_{i,j} ∈ 𝒪(M). It is said to be basic open if S = {x ∈ M | g_{1}(x) > 0, …, g_{r}(x) > 0} with g_{i} ∈ 𝒪(M), and to be principal open if S = {x ∈ M | g(x) > 0} with g ∈ 𝒪(M).

The stability index of M is the smallest integer s such that every basic open set is an intersection of at most s principal open sets, if such an integer exists. Otherwise, this stability index is ∞.

We define semianalytically constructible functions on M as functions φ : M → ℤ which admit a presentation as a finite sum

φ = ∑_{i} m_{i}1_{X_{i}}, (1.1)

where the m_{i}’s are integers, the X_{i}’s are global semianalytic subsets of M and where 1_{X_{i}} denotes the characteristic function of the subset X_{i}. In other words, a function φ : M → ℤ is semianalytically constructible if it is constant on each element of a finite partition of M into global semianalytic sets. More generally, if S ⊂ M is a semianalytic subset of M, we say that φ : S → ℤ is semianalytically constructible on S if φ is the restriction to S of a semianalytically constructible function on M.

Remark 1.1. Our definition of semianalytically constructible functions can be compared to the definition of semialgebraically constructible functions in the algebraic setting. If M ⊂ ℝ^{N} is an affine algebraic (or semialgebraic) set, a function φ : M → ℤ is semialgebraically constructible if it admits a presentation (1.1) where the X_{i}’s are semialgebraic subsets of M, cf. [MP1].

Note also that in our definition, the adjective “semianalytically” is used to stress the difference with the classical definition of constructible function in the analytic setting, cf. [S], [KS]. There, a function φ : M → ℤ on a real analytic manifold M is said to be constructible if for each m ∈ ℤ the set φ^{−1}(m) is subanalytic and the family {φ^{−1}(m)}_{m∈ℤ} is locally finite. These constructible functions form a wider class than our semianalytically constructible functions.

Let X ⊂ ℝ^{k} be a real analytic set. We follow [MP1] and [S]. If φ is a semianalytically constructible function on X with compact support, there is a presentation (1.1) where the X_{i}’s are global compact semianalytic subsets of X, by [ABR, Th. 7.2]. Then, the Euler integral of φ on X is defined by

∫_{X} φ = ∑_{i} m_{i}χ(X_{i}),

where χ denotes the Euler characteristic. This number depends only on φ.

If f : Y → X is a proper analytic map from a real analytic set Y ⊂ ℝ^{l}, and φ is semianalytically constructible on Y, the pushforward of φ by f is the constructible function f_{∗}φ on X defined by

∀x ∈ X, f_{∗}φ(x) = ∫_{Y} φ1_{f^{−1}(x)}.

The function f_{∗}φ does not need to be semianalytically constructible (cf. Examples 3.1 and 3.2).

We define analytically constructible functions following [MP2]:

Definition 1.2. A function φ : X → ℤ is analytically constructible if it admits a presentation as a finite sum

φ = ∑ m_{i}f_{i∗}1_{Z_{i}} (1.2)

where the m_{i}’s are integers, the Z_{i}’s are real analytic sets, and each f_{i} : Z_{i} → X is a proper analytic map. Then, for x ∈ X, we have φ(x) = ∑ m_{i}χ(f_{i}^{−1}(x)).

Remark 1.3. Assume that X is an affine real algebraic set. A function φ : X → ℤ is algebraically constructible if it admits a presentation (1.2) where the Z_{i}’s are real algebraic sets, and the f_{i}’s are proper polynomial maps. The function φ is said to be Nash constructible if it admits a presentation (1.2) where the Z_{i}’s are connected components of real algebraic sets, and the f_{i}’s are proper polynomial maps.

Algebraically constructible and Nash constructible functions have been introduced by McCrory and Parusiński in [MP1] to study the topology of real algebraic sets and semialgebraic arc-symmetric sets respectively.

Algebraically constructible functions coincide with sums of signs of polynomials, see [CK] or [PS]. Similarly, Nash constructible functions on X are sums of signs of semialgebraic arc-analytic functions if X is compact or dim X = 2, cf. [B2] and Section 3. (A function on an analytic set is arc-analytic if it becomes analytic when composed with any analytic arc on the set, cf. [K].)

Note that the algebraic setting and the analytic one are different: the class of algebraically constructible functions is included in the class of the Nash constructible ones, and all these functions are semialgebraically constructible.

1.2 First results in dimension 1 and 2

Let M be a paracompact real analytic manifold. If f ∈ 𝒪(M), the function sign f : M → ℤ is defined as sign f = 1_{{f>0}} −1_{{f<0}}, so it is semianalytically constructible. But there are semianalytically constructible functions that can not be written as sums of signs of global analytic functions. For instance, a sum of signs of global analytic functions on M has to be generically constant mod 2 on each connected component of M (since the sign of a global analytic function is so). In dimension one, this condition is sufficient for a semianalytically constructible function to be a sum of signs of global analytic functions. More precisely:

Proposition 1.4. Let W ⊂ M be an analytic set of dimension one, and let φ : W → ℤ be a semianalytically constructible function. Then φ is a finite sum of signs of global analytic functions on W if and only if φ is generically constant mod 2 on each irreducible component of W of dimension one.

Proof. The “only if” part is clear. For the “if” part, assume first W to be irreducible. Changing φ into φ − 1 if necessary, we may assume that φ is generically even. Let r be the half of the maximal positive generic value of φ (or 0 if φ is generically non positive), and let −s be the half of the minimal negative generic value of φ (or 0 if φ is generically non negative). Set X_{j} = ∪_{n≥j}Int_{W} (φ^{−1}(2n)) for j = 1, …, r and Y_{k} = ∪_{n≥k}Int_{W} (φ^{−1}(−2n)) for k = 1, …, s, where Int_{W} means the interior taken in W. There exists a discrete set of points D ⊂ W such that φ = 21_{X_{1}} + … + 21_{X_{r}} − (21_{Y_{1}} + … + 21_{Y_{s}}) on W \ D.

By [DC-A, Prop. 4.3], any open semianalytic subset of W is global principal. So, for j = 1, …, r and k = 1, …, s, there exist f_{j}, f′_{k} ∈ 𝒪(W) such that X_{j} = {x ∈ W |f_{j}(x) > 0} and Y_{k} = {x ∈ W |f′_{k}(x) > 0}. Then we have

φ = (1 + sign f_{1}) + … + (1 + sign f_{r}) − (1 + sign f′_{1}) − … − (1 + sign f′_{s}),

on W \D′, where D′ = D ∪⋃_{i}{f_{i} = 0}∪⋃_{k}{f′_{k} = 0} ⊂ W is a discrete set of points. So φ is generically a sum of signs of global analytic functions on W.

From now on we do not assume that W is irreducible. Let W = ∪_{i∈I}W_{i} be the decomposition of W into irreducible components. Denote I′ = {i ∈ I | dim W_{i} = 1} and W′ = ∪_{i∈I′}W_{i}. By the irreducible case, we know that for each i ∈ I′, there are f_{1}^{i}, …, f_{r_{i}}^{i} ∈ 𝒪(W_{i}) such that φ_{|_{W_{i}}} =_{gen} sign f_{1}^{i} + … + sign f_{r_{i}}^{i} on W_{i}. Since φ assumes only a finite number of values, we can assume that all r_{i}’s are equal, say r_{i} = r for any i ∈ I′. For i ∈ I′, let t_{i} ∈ 𝒪(W) be a positive equation for \overline{W \ W_{i}}. Then φ_{|_{W_{i}}} =_{gen} sign (t_{i}f_{1}^{i}) + … + sign (t_{i}f_{r}^{i}) on W_{i}.

For j = 1, …, r, set g_{j} = ∑_{i∈I} f_{j}^{i}t_{i}. These are global analytic functions on W since the sum is locally finite. We have φ =_{gen} sign g_{1} + … + sign g_{r} on W. Then, we can write φ − ∑_{j=1}^{r} sign g_{j} = k_{1}1_{D_{1}} + … + k_{m}1_{D_{m}}, where k_{1}, …, k_{m} are integers, and D_{1}, …, D_{m} are discrete subsets of W. For l = 1, …, m, let h_{l} ∈ 𝒪(W) be a positive equation of D_{l}. Then, it is easy to check that φ = ∑_{j=1}^{r} sign g_{j} + k_{1}(1 − sign h_{1}) + … + k_{m}(1 − sign h_{m}).

Remark 1.5. Since W is coherent, it follows from Cartan’s Theorem B that φ is the restriction to W of a sum of signs of global analytic functions on M if and only if it is a sum of signs of global analytic functions on W.

Remark 1.6. If the dimension of M is ≥ 2, there are semianalytically constructible functions which are generically constant mod 2, but which are not generically sums of signs of global analytic functions. For instance, the function 21_{{x>0,y>0}} is not generically a sum of signs of global analytic functions on ℝ^{2}, since if the sign of a global analytic function changes crossing {y = 0, x < 0}, it changes also crossing {y = 0, x > 0}.

Proposition 1.4 solves completely the problem for semianalytically constructible functions on an analytic set of dimension one. We come now to the case of dimension two.

We first prove an analogous in dimension two of the previous proposition. We will use this result in the following section.

Proposition 1.7. Assume that the dimension of M is two. Let φ : M → ℤ be a semianalytically constructible function which is generically constant mod 4. Then φ is generically a sum of signs of global analytic functions on M.

Proof. Let S ⊂ M be a global semianalytic subset of M, we have to prove that 41_{S} is generically a finite sum of signs of global analytic functions on M. By [DC-A], the set S is generically the union of two basic open global semianalytic sets S_{1} and S_{2}. Then 1_{S} =_{gen} 1_{S_{1}} + 1_{S_{2}} − 1_{S_{1}∩S_{2}}. Since a finite intersection of basic open sets is still basic open, we can assume that S is basic open. Again by [DC-A], the stability index of M is two. So there exist f_{1}, f_{2} ∈ 𝒪(M) such that S = {x ∈ M | f_{1}(x) > 0, f_{2}(x) > 0}. Then we have 41_{S} =_{gen} sign f_{1} + sign f_{2} + sign (f_{1}f_{2}) + 1.

Remark 1.8. The value of the stability index of M is not known if dim M > 2. So we do not know if, in dimension n, a semianalytically constructible function which is generically constant mod 2^{n} is generically a sum of signs of analytic functions.

Proposition 1.4 shows that, in dimension one, a semianalytically constructible function is a sum of signs of global analytic functions if and only if this is true generically. But in dimension ≥ 2, there are semianalytically constructible functions which are sums of signs of global analytic functions only generically. In Proposition 1.9, we characterize, for a manifold of dimension two, the “smallest” set outside which a semianalytically constructible function that is generically a sum of signs, is a sum of signs.

Let M be of any dimension. If S ⊂ M is a semianalytic set, we denote by ∂_{Z}S the Zariski closure of the Euclidean boundary ∂S of S, i.e. the smallest global analytic subset of M containing ∂S. If S is global, then ∂S is contained in the union of the zero sets of the functions appearing in a description of S, so the set ∂_{Z}S is proper. Note that, if dim M > 2, it is not known if ∂S is still a global semianalytic set.

Proposition 1.9. Assume that the dimension of M is two. Let φ : M → ℤ be a semianalytically constructible function which is generically a sum of signs of global analytic functions on M. Let X be the union of the irreducible components of dimension one of ⋃_{n∈ℤ} ∂_{Z}(φ^{−1}(n)) on which φ is not generically constant mod 2. Then, there exist f_{1}, …, f_{k} ∈ 𝒪(M) such that φ_{|_{M\X}} = (sign f_{1} + … + sign f_{k})_{|_{M\X}}.

Proof. By hypothesis we can write φ_{|_{M\W}} = (sign f_{1} + … + sign f_{k})_{|_{M\W}} where f_{1}, …, f_{k} ∈ 𝒪(M) and W ⊂ M is an analytic set of dimension one. We can always suppose (sign f_{1} + … + sign f_{k})_{|_{W}} = 0. Write W = ⋃_{i∈I} W_{i} ∪ D, where D is a discrete set, the W_{i}’s are distinct irreducible analytic sets of dimension one, and ⋃_{i} W_{i} ∩ D = ∅.

Set W′ = ∪_{i∈I′}W_{i} where I′ = {i ∈ I|φ_{|_{W_{i}}}is generically constant mod 2}. Note that for each i ∈ I \ I′, the component W_{i} is contained in ⋃_{n∈ℤ} ∂_{Z}(φ^{−1}(n)). Indeed, let i ∈ I such that W_{i} ⊄ ⋃_{n∈ℤ} ∂_{Z}(φ^{−1}(n)). Then W_{i} ∩⋃_{n∈ℤ} ∂_{Z}(φ^{−1}(n)) is a discrete subset of W_{i}, and outside this subset the parity of φ_{|_{W_{i}}} is the generic parity of φ on the connected component of M containing W_{i}. So i belongs to I′, and we have X = ∪_{i∈I\I′}W_{i}.

By Proposition 1.4 and Remark 1.5 there exist s_{1}, …, s_{r} ∈ 𝒪(M) such that φ_{|_{W′∪D}} = (sign s_{1} + … + sign s_{r})_{|_{W′∪D}}. Let t ∈ 𝒪(M) be a positive equation for W′ ∪ D. Then φ_{|_{M\X}} = (sign f_{1} + … + sign f_{k} − sign ts_{1} − … − sign ts_{r} + sign s_{1} + … + sign s_{r})_{|_{M\X}}.

Corollary 1.10. Assume dim M = 2. Let φ : M → ℤ be a semianalytically constructible function, which is generically a sum of signs of global analytic functions on M. Then, φ is a sum of signs of global analytic functions everywhere on M if and only if φ is generically constant mod 2 on each global irreducible component of dimension one of ⋃_{n∈ℤ} ∂_{Z}(φ^{−1}(n)).

2 The characterizations

2.1 A criterion for principality

Let M be a paracompact real analytic manifold of dimension n, and let S ⊂ M be a global semianalytic subset. Denote S^{∗} = Int(Adh(S)). This is a semianalytic set, although its global character is known only in dimension one and two by [C]. Note that ∂S^{∗} ⊂ ∂S, so ∂_{Z}S^{∗} has codimension one in M. We define a wall of S as an irreducible component of ∂_{Z}S^{∗} of codimension one in M. The notion of wall is used in [ABV], [1] and [BP] to study basicness and separation of semialgebraic and global semianalytic sets.

Definition 2.1. Let W be a wall of S. We say that W is even if dim(W ∩S^{∗}) = n−1 or if dim (W ∩ (M \ S)^{∗}) = n − 1, and we say that W is odd if dim (W ∩ Adh(S^{∗}) ∩ Adh((M \ S)^{∗})) = n − 1, cf. [1], [AB]. (The dimension of a semianalytic set T is the maximum integer d such that T contains regular points of dimension d.)

Proposition 2.2. Let S ⊂ M be a global semianalytic set. Then S\∂_{Z}S is principal open if and only if no wall of S is simultaneously odd and even.

Proof. Assume S \ ∂_{Z}S is principal open, i.e. there exists f ∈ 𝒪(M) such that S \ ∂_{Z}S = {f > 0}. Let W be wall of S. Denote by t ∈ 𝒪(M) an uniformizer of 𝒪(M)_{ℐ(W)}, which is a discrete valuation ring of rank one by [BP, Prop.1.5]. Then f = t^{k}g for some integer k and g ∈ 𝒪(M) not vanishing identically on W. The number k is even (respectively odd) if W is so, and W cannot be simultaneously odd and even.

Since S\∂_{Z}S is principal open if and only if it is principal open in each component of M, it enough to prove the converse implication if M is connected. Assume that no wall is simultaneously odd and even. Write ∂_{Z}S = Y ∪Z where Y is the union of the odd walls, and Z is a global analytic set. If f ∈ 𝒪(M) is such that S\∂_{Z}S = {f > 0}, then f has to vanish on Y and also on the components of Z which are adherent to S \ ∂_{Z}S. We will construct an f such that its zero set coincides exactly with ∂_{Z}S.

Let p ∈ 𝒪(M) be a positive equation for Z. Denote by ℐ the ideal sheaf induced by ℐ(Y). Note that it is enough to prove that ℐ is principal. Indeed if f ∈ 𝒪(M) generates ℐ(Y), then, by construction we get S \ ∂_{Z}S = {pf > 0} or S \ ∂_{Z}S = {pf < 0}.

By [BP, Lemma 1.2], the ideal sheaf ℐ is locally principal, so for each x ∈ M we can choose a generator h_{x} of ℐ_{x}. Fix x ∈ M, we consider the analytic components of the analytic set germ induced by ∂_{Z}S at x. The definition of odd and even walls still makes sense, and by the hypothesis no wall of S_{x} can be simultaneously odd and even. Note that the components of codimension one of Y_{x} are odd. So the analytic function germ h_{x}p_{x} is such that (S\∂_{Z}S)_{x} = {h_{x}p_{x} > 0} or (S\∂_{Z}S)_{x} = {h_{x}p_{x} < 0}. Changing h_{x} into −h_{x} if necessary, we can assume that (S \ ∂_{Z}S)_{x} = {h_{x}p_{x} > 0}. So we obtain local generators {h_{i}} for ℐ on a locally finite open covering {U_{i}} of M such that h_{i_{|U_{i}∩U_{j}}} and h_{j_{|U_{i}∩U_{j}}} have the same sign, i.e the cocycle {h_{i}/h_{j}} is zero in H^{1}(M, ℤ_{2}). Now we get the thesis by general theory of sheaves.

The exponential map and the associated usual exact sequence

0 → 𝒪_{M} → 𝒪_{M}^{∗} → 𝒪_{M}^{∗}/𝒪_{M}^{+} = ℤ_{2} → 0

induce an isomorphism between H^{1}(M, ℤ_{2}) and H^{1}(M, 𝒪_{M}^{∗}), so the cocycle {h_{i}/h_{j}} is zero also in H^{1}(M, 𝒪_{M}^{∗}).

Then, using the exact sequence

0 → 𝒪_{M}^{∗} → ℳ_{M}^{∗} → ℳ_{M}^{∗}/𝒪_{M}^{∗} → 0,

we see that the global section x ↦ h_{i,x} of the sheaf of the divisors lies in the image of the map Γ(M, ℳ_{M}^{∗}) → Γ(M, ℳ_{M}^{∗}/𝒪_{M}^{∗}). This means that there exists a non zero global meromorphic function f on M such that f_{x}𝒪^{∗} = h_{i,x}𝒪^{∗}. In particular, f lies in 𝒪(M) since f_{x} lies in 𝒪_{M,x} for each x ∈ M, and f generates ℐ(Y). The proof is done.

Remark 2.3. Proposition 2.2 shows in particular that S is generically principal if and only if S \ ∂_{Z}S is principal open. It shows also that S is generically principal if and only if for each compact set K ⊂ M, the set S ∩ K is generically principal.

2.2 A characterization of sums of signs in dimension 2

Let M be a paracompact real analytic manifold of dimension 2, and let φ : M → ℤ be a semianalytically constructible function. We want to know if φ is a finite sum of signs of global analytic functions. By Corollary 1.10 we can reduce ourselves to study a generic problem. A wall of the function φ is a wall of a φ^{−1}(n) for n ∈ ℤ.

Let W ⊂ M be a wall of φ, and denote X_{n} = φ^{−1}(n) for n ∈ ℤ. Only a finite number of X_{n} are non empty. Then, we can define the function ∂_{W} φ : W → ℤ[1/2] in this way:

∂_{W} φ = ∑_{n∈ℤ} \frac{n}{2}1_{∂(X_{n}^{∗})∩W} + n1_{X_{n}^{∗}∩W} .

The function ∂_{W} φ is constant on each element of a finite semianalytic partition of W, but a priori it is not integer-valued. If φ is generically constant mod 2 on each connected component of M, then ∂_{W} φ is generically semianalytically constructible.

Let x be a point in W ∩ Reg ( ⋃_{n} ∂_{Z}X_{n}). Then φ is constant on each of the two connected components of (M \ W)_{x}, and ∂_{W} φ(x) coincides with the media of the values of φ on these two components.

In the algebraic or Nash context, the functions ∂_{W} φ have been used to study algebraically constructible and Nash constructible functions defined in Remark 1.3 (cf. [B1], [B2]).

Remark 2.4. If M is embedded in some ℝ^{k}, and φ is a semianalytically constructible function on M, the link of φ is the constructible function Λφ on M defined by

∀x ∈ M, (Λφ)(x) = ∫_{M} φ1_{S(x,ε)},

where ε > 0 is small enough, and S(x, ε) denotes the sphere with center x and radius ε in M, cf. [MP1]. This value depends only on φ and x.

It can be checked that we have ∂_{W} φ =_{gen} (φ − \frac{1}{2}Λφ)_{|_{W}}.

Theorem 2.5. Let M be a paracompact real analytic manifold of dimension two. A semianalytically constructible function φ : M → ℤ is generically a sum of signs of global analytic functions if and only if, for each wall W of φ, the function ∂_{W} φ : W → ℤ is generically integer-valued and generically constant mod 2.

Proof. Assume there exist f_{1}, …, f_{r} ∈ 𝒪(M) such that φ =_{gen} sign f_{1} + … + sign f_{r} on M. Let t ∈ 𝒪(M) be an uniformizer of 𝒪(M)_{ℐ(W)}, which is a discrete valuation ring of rank one by [BP]. For i = 1, …, r, we can write f_{i} = t^{k_{i}}g_{i} where k_{i} is an integer and g_{i} ∈ 𝒪(M) is not identically zero on W. Then on W we have

∂_{W} φ =_{gen} ∑_{i | k_{i} even} sign g_{i}.

We now prove the converse implication. Assume that on each wall W of φ, the function ∂_{W} φ : W → ℤ is generically integer-valued and generically constant mod 2. We first prove that φ is generically constant mod 2 on each connected component of M. Let φ̄ : M → {0, 1} be a semianalytically constructible function such that φ − φ̄ is generically even. If φ̄ had a wall W, then ∂_{W} φ̄ would be generically 1/2 on a non empty open subset of W. As ∂_{W} (φ − φ̄) is generically integer-valued and as we have ∂_{W} φ =_{gen} ∂_{W} (φ−φ̄)+∂_{W} φ̄, it would mean that ∂_{W} φ is not generically integer-valued on W. So φ̄ has no wall, it is generically constant on each connected component of M, and φ is generically constant mod 2 on each connected component. We can write φ = φ_{1} + φ_{2} + 1_{C} where φ_{1}, φ_{2} : M → ℤ are semianalytically constructible functions such that φ_{1} is generically constant mod 4, φ_{2} takes only the two values 1 and −1 and C is a union of connected components of M.

By Proposition 1.7, we only have to prove that the function φ_{2} is generically a sum of signs of analytic functions. It is enough to show that S = φ_{2}^{−1}(1) is generically principal. Indeed, if S =_{gen} {f > 0}, then φ_{2} =_{gen} sign f. By Proposition 2.2 we know that S is generically principal if and only if no wall of S is simultaneously odd and even. Let W be a wall of S. It is also a wall of φ. Remark that ∂_{W} φ_{2} =_{gen} 0 on W ∩ Adh(S^{∗})∩ Adh((M \ S)^{∗}), and ∂_{W} φ_{2} =_{gen} 1 (respectively −1) on W ∩ S^{∗} (respectively on W ∩ (M \ S)^{∗}). As we have ∂_{W} φ_{2} =_{gen} ∂_{W} φ − ∂_{W} φ_{1} if W ⊄ C and ∂_{W} φ_{2} =_{gen} ∂_{W} φ − ∂_{W} φ_{1} + 1 if W ⊂ C, the function ∂_{W} φ_{2} is generically constant mod 2 on W, so the wall W of S cannot be simultaneously odd and even. The proof is done.

Corollary 2.6. Let M be a paracompact real analytic manifold of dimension two. A semianalytically constructible function φ : M → ℤ is a sum of signs of global analytic functions if and only if, for each compact set K ⊂ M, the function φ_{|_{K}} is the restriction to K of a finite sum of signs of global analytic functions on M.

3 Analytically constructible functions in dimension two

3.1 The analytic case

Let X be a real analytic subset of ℝ^{k}, and let φ be an analytically constructible function on X (cf. Definition 1.2). As remarked before, φ is not necessarily semianalytically constructible, even in dimension 1.

Example 3.1. Set X = ⋃_{n≥0} \{(x, y) ∈ ℝ^{2}|x − n = \frac{y^{2}}{2^{n}} \}. The set X is a locally finite union of global analytic subsets of ℝ^{2}, so it is global analytic. Let π : X → ℝ be the projection on the first component: π(x, y) = x. Then, π_{∗}1_{X} : ℝ → ℤ is not semianalytically constructible since it is not finitely valued. Indeed, for each n ∈ ℕ, we have π_{∗}1_{X}(x) = 2n for any x ∈]n − 1, n[.

Note that if X ⊂ ℝ^{k} is a compact analytic set, a constructible function on X has to be finitely valued, i.e. it has to be constant on a finite partition of X into subanalytic sets. If dim X = 1, any subanalytic set is semianalytic and any semianalytic set is global. So, any constructible function on a compact analytic curve is semianalytically constructible.

But if dim X > 1, the compactness assumption is not sufficient: even if X is compact and smooth, there are analytically constructible functions on X which are not semianalytically constructible. The reason of this fact is the presence of compact semianalytic subsets of X which are not global.

Example 3.2. Set Z = \{(x, y) ∈ ℝ^{2}|(x − \frac{3}{2})^{2} + y^{2} = \frac{1}{4}\} and let f : Z → ℝ^{2} be defined by f(x, y) = (x, sin \frac{1}{x}). This is an analytic morphism since the circle Z does not intersect the y-axis and it is proper since Z is compact. The values of the function φ = f_{∗}1_{Z} : ℝ^{2} → ℤ are:

φ(x, y) = \{ \begin{matrix}1 & if (x, y) = (1, sin 1) or (x, y) = (2, sin \frac{1}{2}), \\ 2 & if 1 < x < 2 and y = sin \frac{1}{x}, \\ 0 & else. \end{matrix}

The function φ is finitely valued, but it is not semianalytically constructible, not even generically. Indeed, the set φ^{−1}(2) is not global. We have φ^{−1}(2) = {(x, y) ∈ ℝ^{2}|1 < x < 2, y = sin \frac{1}{x}\}, and any global analytic function on ℝ^{2} vanishing on φ^{−1}(2) has to be identically zero. So the Zariski closure of φ^{−1}(2) coincides with the whole ℝ^{2}.

Let g : ℝ^{2} → S^{2} be the one-point compactification, then (g ∘ f)_{∗}1_{Z} : S^{2} → ℤ is an example of analytically constructible function on a two-dimensional compact analytic manifold which is not semianalytically constructible.

Vice-versa, it is easy to find examples of semianalytically constructible functions which are not analytically constructible. For instance, analytically constructible functions must verify some parity conditions, see Lemma 3.3. We prove that, on a two-dimensional analytic manifold, a semianalytically constructible function is analytically constructible if and only if it is a sum of signs of global analytic functions (and equivalently an analytically constructible function is semianalytically constructible if and only if it is such a sum of signs).

Lemma 3.3. Let X be a real analytic subset of ℝ^{k}, and let φ be an analytically constructible function on X.

1. If Y ⊂ ℝ^{l} is a real analytic set, and f : Y → X is an analytic map, then the function φ ∘ f is analytically constructible on Y.

2. If X is an irreducible curve (maybe singular), then φ is generically constant mod 2 on X.

3. The function \frac{1}{2}Λφ is analytically constructible on X.

4. If f is an analytic function on X, then sign f is analytically constructible on X.

Proof. 1. This point can be proved as for algebraically constructible and Nash constructible functions in [MP1] and [B2].

2. There exists π : X′ → X a locally finite sequence of blowings-up of points such that X′ is a non singular connected curve. The function φ ∘ π is analytically constructible on X′ by point 1., therefore by [MP1, Prop.5.1] it is generically constant mod 2 on X′, and so is φ on X. (Here generically means outside a discrete set of points.)

3. We can copy the proof of Theorem 2.5 in [MP1]. The main element of this proof is the desingularization of analytic sets, which is possible in our context using global blowings-up by [H].

4. Denote Y = {(x, t) ∈ X × ℝ | f(x) = t^{2}} ⊂ ℝ^{k} × ℝ, and let π : Y → X be the projection on the first component. Then we have signf = π_{∗}1_{Y} − 1_{X}, and sign f is analytically constructible.

Proposition 3.4. Let X ⊂ ℝ^{k} be an analytic curve, not necessarily smooth. Let φ be a semianalytically constructible function on X, i.e. a finitely valued constructible function on X. The function φ is analytically constructible if and only if it is a sum of signs of global analytic functions on X.

Proof. By Proposition 1.4, the function φ is a sum of signs of global analytic functions on X if and only if it is generically constant mod 2 on each irreducible component of X of dimension 1. The same condition is necessary (and then, sufficient) for φ to be analytically constructible.

Theorem 3.5. Let M be a paracompact real analytic manifold of dimension two. A semianalytically constructible function φ : M → ℤ is analytically constructible if and only if it is a sum of signs of global analytic functions on M.

Proof. The “if” part is given by point 4. in Lemma 3.3. For the “only if” part, assume that φ is analytically constructible. We use Theorem 2.5 and Corollary 1.10 to prove that φ is a sum of signs. Let W be a wall of φ. As analytically constructible functions are preserved by the operator \frac{1}{2}Λ, the function ∂_{W} φ is generically analytically constructible by Remark 2.4. In particular, ∂_{W} φ is generically integer-valued and generically constant mod 2 on W. By Theorem 2.5, the function φ is generically a sum of signs of global analytic functions on M.

Let W′ be an irreducible component of dimension one of ⋃_{n∈ℤ} ∂_{Z}(φ^{−1}(n)). The function φ_{|_{W′}} is analytically constructible on W′, so it is generically constant mod 2 on W′. Using Corollary 1.10, we conclude that φ is a sum of signs of global analytic functions on M.

Remark 3.6. The result of Theorem 3.5 is no more true if dim M ≥ 3, or if we work on a singular analytic set, cf. the examples in Remark 3.5 in [B2].

3.2 The Nash case

In this section M is an affine Nash manifold, i.e. a semialgebraic analytic submanifold of some ℝ^{k}. We denote by N (M) the ring of Nash functions on M, i.e. analytic and semialgebraic functions on M. Let S ⊂ M be a semialgebraic set. The Nashwalls of S are defined in [B2] as the Nash-irreducible components of codimension one of the Nash-closure of the Euclidean boundary of S^{∗}. By [CS], a global analytic set that is Nash, and Nash-irreducible, is also irreducible as a global analytic set. So the Nash-walls of S coincide with the (analytic) walls of S we use in this paper.

Theorem 3.7. Let M ⊂ ℝ^{k} be a Nash manifold and let S ⊂ M be a semialgebraic set such that no wall of S is simultaneously odd and even. Then S \ ∂_{Z}S is Nash principal open, i.e. S \ ∂_{Z}S = {f > 0} with f ∈ N (M). In particular, a semialgebraic set is Nash principal open if and only if it is principal open as a global semianalytic set.

Proof. By Proposition 2.2, we know that there exists f ∈ 𝒪(M) such that S \∂_{Z}S = {f > 0}. Write ∂_{Z}S = Y ∪Z where Y is the union of the odd components of ∂_{Z}S of codimension one in M and Z is a Nash set. By the proof of Proposition 2.2, we can assume that f = hp where p ∈ N (M) is a positive equation for Z and h ∈ 𝒪(M) is a generator for the principal ideal ℐ(Y)𝒪(M). Being Y a Nash set, we can find g ∈ N (M) a positive equation of Y. Then we have g ∈ ℐ(Y)𝒪(M), and in particular we can write g = hq with q ∈ 𝒪(M). By the factorization conjecture proved in [CS], we can find ϕ_{1}, ϕ_{2} ∈ 𝒪(M) strictly positive analytic functions, and ψ_{1}, ψ_{2} Nash functions on M, such that ϕ_{1}ϕ_{2} = 1, and such that h = ψ_{1}ϕ_{1} and q = ψ_{2}ϕ_{2}. The function ψ_{1}p is Nash on M and sign ψ_{1} = sign h on M. Then, S \ ∂_{Z}S = {ψ_{1}p > 0} is Nash principal.

We can use this result to study Nash constructible functions. A semialgebraic set S ⊂ M is arc-symmetric if for each analytic arc γ :] − 1, 1[→ M such that γ(] − 1, 0[) ⊂ S, we have γ(] − 1, 1[) ⊂ S (cf. [K]).

Corollary 3.8. Let φ : M → ℤ be a generically Nash constructible function which assumes only the values 1 and −1. Suppose that the walls of φ are irreducible as semialgebraic arc-symmetric sets. Then, there is f ∈ N (M) such that φ is generically sign f.

Proof. We want to prove that φ^{−1}(1) is generically Nash principal. By Theorem 3.7, it is enough to show that no wall of φ^{−1}(1) is simultaneously odd and even. Assume that a wall W of φ^{−1}(1) is simultaneously odd and even. This would mean that ∂_{W} φ is not generically constant mod 2 on the irreducible arc-symmetric semialgebraic set W. So ∂_{W} φ would not be generically Nash constructible on W by [B2, Prop.1.5]. This contradicts the generic Nash constructibility of φ, since Nash constructible functions are preserved by the operator \frac{1}{2}Λ.

Theorem 3.9. Let M be an affine Nash manifold of dimension two. A function φ : M → ℤ is Nash constructible if and only if it is a sum of signs of Nash functions.

Proof. We can copy the proofs of Theorem 2.5 and Theorem 3.5, since the stability index of N (M) is two, and since an irreducible analytic curve is irreducible as an arc-symmetric set (cf. [K]).

Again, this result is false in general for a Nash manifold of dimension ≥ 3 or a singular set, see the examples of [B2, Rk.3.5].

Repeating the proof of Theorem 3.6 in [B2], we get:

Corollary 3.10. Let X ⊂ ℝ^{k} be a semialgebraic arc-symmetric set of dimension two. A function φ : X → ℤ is Nash constructible if and only if φ is a sum of signs of semialgebraic arc-analytic functions on X.

4 Constructible functions and quadratic forms

4.1 Recalls about real spectrum and Artin-Lang property

Let M be a paracompact real connected analytic manifold. The real spectrum Spec_{r}K of a field K is the set of the orderings of the field K. In this section we want to translate the criterion of Theorem 2.5 in terms of the behaviour of fans in the real spectrum of ℳ(M).

On the real spectrum, the characterization of signatures of quadratic forms and many others problems, as for instance basicness and separability for constructible sets, can be solved in terms of finite spaces of orderings. In the real spectrum of the field of rational functions on an irreducible real algebraic set, the relevant information lies in some particular finite spaces of orderings, that are attached to discrete valuation rings, and therefore behave specially well with respect to geometry (see for instance [1]). Thanks to Artin-Lang property that acts as a translator between constructible sets and semialgebraic sets, we can read the statements proved in the real spectrum as geometric results.

In the global analytic case, in general we cannot use the same technique. For instance, Artin-Lang property is known only for Spec_{r}ℳ(M) if dim M ≤ 2, and it is known that this property does not hold for 𝒪(M) even in dimension one. Another problem is the lackness of a natural compactification (this is one among the reasons why many problems are still open also in the Nash setting).

In this section we assume that the dimension of M is two, so Artin-Lang property holds for Spec_{r}ℳ(M) by [C].

Let K be a field. A constructible subset of Spec_{r}K is a set of the form ∪_{i=1}^{p}{σ ∈ Spec_{r}K | σ(g_{i,1}) > 0, …, σ(g_{i,j_{i}}) > 0} with g_{i,j} ∈ K. The constructible subsets of Spec_{r}K form a basis for the constructible topology of Spec_{r}K.

A constructible function on the real spectrum of K is defined as a function Spec_{r}K → ℤ, continuous with respect to the constructible topology of Spec_{r}K and the discrete topology of ℤ. Note that such a function can assume only a finite number of values, since Spec_{r}K is compact with respect to the constructible topology.

Assume K = ℳ(M). The so called tilde map associates to a global semianalytic set S ⊂ M the constructible set S̃ ⊂ Spec_{r}ℳ(M) with the same description. Artin-Lang property means that this map is well defined, and that if S is a global semianalytic set such that S̃ is non empty, then S is non empty.

A semianalytically constructible function φ = ∑_{i} n_{i}1_{S_{i}} on M induces a constructible function φ̃ = ∑_{i} n_{i}1S̃_{i} on Spec_{r}ℳ(M). If we change the presentation of φ, by Artin-Lang property we get the same function φ̃ on Spec_{r}ℳ(M). So the function φ̃ is well-defined.

Two semianalytically constructible functions induce the same function on the real spectrum of ℳ(M) if and only if they are generically equal. We get a “generic” correspondence between semianalytically constructible functions on M and constructible functions on the real spectrum Spec_{r}ℳ(M). Artin-Lang property implies directly that a semianalytically constructible function φ : M → ℤ is generically the sum of the signs of the global analytic functions f_{1}, …, f_{r} on M if and only if the associated function φ̃ on Spec_{r}ℳ(M) is the signature of the quadratic form ⟨f_{1}, …, f_{r}⟩ over Spec_{r}ℳ(M) .

4.2 A bound of the number of functions

Let M be a paracompact real connected analytic manifold of dimension 2, and let φ : M → ℤ be a semianalytically constructible function. Assume that φ is a sum of signs of global analytic functions. We are interested in the minimal number n(φ) of functions necessary to write φ generically, that is

n(φ) = min {r ∈ ℕ | ∃f_{1}, …, f_{r} ∈ 𝒪(M), φ =_{gen} sign f_{1} + … + sign f_{r}}.

Denote by m(φ) the maximal generic value of | φ |. Clearly we have n(φ) ≥ m(φ).

If W is an irreducible analytic curve in M, and ψ : W → ℤ is a sum of signs of analytic functions on W, we can define n(ψ) and m(ψ) in the same way as before and we have n(ψ) = m(ψ). This follows from the proof of Proposition 1.4.

The correspondence between sums of signs of global analytic functions and signatures of quadratic forms allows us to prove the following:

Proposition 4.1. Let M be a paracompact real analytic manifold of dimension 2, and let φ : M → ℤ be a sum of signs of global analytic functions on M. Let k ∈ ℕ, δ ∈ ℤ be such that the set of the generic values of φ is contained in the interval [δ − k, δ + k]. Then we have

n(φ) ≤ \{ \begin{matrix}2k + |δ| if k is even, \\ 2k − 1 + |δ| if k is odd. \end{matrix}

Proof. We can assume that M is connected. By Proposition 3 in [B1], there is a quadratic form over Spec_{r}ℳ(M) representing φ̃ and of dimension at most 2^{s−1}k +|δ| if k is even, 2^{s−1}(k − 1) + 1 + |δ| if k is odd, where s is the stability index of ℳ(M). By [DC-A], we know that s = 2.

Example 4.2. If φ : M → [−1, 1] is generically a sum of signs of analytic functions, then it is generically the sign of an analytic function.

Remark 4.3. The bound of Proposition 4.1 is sharp. Indeed, assume k is even and δ ≥ 0, and consider φ : ℝ^{2} → ℤ defined by φ(x, y) = δ − k if x < 0 and y < 0, and φ(x, y) = δ + k else. Then φ is a sum of signs of 2k+ | δ | analytic functions on ℝ^{2} and not less, by an argument of [B1, §5.2]. Similarly, if k is odd and δ ≥ 0, the function φ : ℝ^{2} → ℤ defined by φ(x, y) = δ + k if y > 0, φ(x, y) = δ + k − 2 if x > 0 and y < 0, and φ(x, y) = δ − k else, is the sum of the signs of 2k − 1 + |δ| analytic functions, and not less. The proof is similar if δ < 0.

Let W be a wall of φ, and let t ∈ 𝒪(M) be an uniformizer of 𝒪(M)_{ℐ(W)}. Then we define ∂_{W}^{t} φ = ∂_{W} ((sign t)φ). If t′ ∈ 𝒪(M) is another uniformizer of 𝒪(M)_{ℐ(W)}, we have ∂_{W}^{t′} φ = (sign (tt′))∂_{W}^{t} φ, and n(∂_{W}^{t} φ) = n(∂_{W}^{t′} φ). So we will use ∂_{W}^{t} φ without making precise the chosen uniformizer t. Write φ = ∑_{i=1}^{n(φ)}sign f_{i}. For i = 1, …, n(φ), write f_{i} = t^{k_{i}}g_{i} with k_{i} ∈ ℕ and g_{i} ∈ 𝒪(M) \ ℐ(W). Then we have ∂_{W} φ = ∑_{i|k_{i} even}sign g_{i} and ∂_{W}^{t} φ = ∑_{i|k_{i} odd}sign g_{i}. So we get n(φ) ≥ n(∂_{W} φ) + n(∂_{W}^{t} φ).

If δ = 0 and k = 2h is even, we have a characterization of the functions which reach the bound. Note that, for such a function φ, neither φ^{−1}(2h) nor φ^{−1}(−2h) are generically empty. Indeed, if one of these two sets is generically empty, then the values of φ are generically in the interval [δ − (2h − 1), δ + (2h − 1)] with δ = 1 or −1, and Proposition 4.1 gives that n(φ) ≤ 2(2h − 1) − 1+ | δ |= 4h − 2.

We say that two global semianalytic subsets A and B of M are generically separable if there exists a non zero analytic function f ∈ 𝒪(M) such that f_{|_{A}} ≥ 0 and f_{|_{B}} ≤ 0.

Proposition 4.4. Let φ : M → [−2h, 2h] be a sum of signs of global analytic functions, with h ∈ ℕ. Then we have n(φ) = 4h if and only if the two sets φ^{−1}(2h) and φ^{−1}(−2h) are not generically separable.

To prove Proposition 4.4 we need the following theorem about separation of global semianalytic sets.

Theorem 4.5. Let M be a 2-dimensional analytic manifold and let A, B ⊂ M be global semianalytic subsets. Let π : M′ → M be a locally finite sequence of blowingsup of points such that the walls of π^{−1}(A) and π^{−1}(B) are non singular and with normal crossing intersections. Then, π^{−1}(A) and π^{−1}(B) are generically separable in M′ if and only if A and B are generically separable in M.

The proof of the theorem will follow from this lemma.

Lemma 4.6. Let M be a paracompact connected real analytic manifold of dimension two and let π : M′ → M be a locally finite sequence of blowings-up of points.

1. If X′ is an analytic curve in M′ not contained in the exceptional divisor, then π(X′) is an analytic curve in M.

2. If Y is an analytic irreducible curve in M, then its strict transform Y′ is still irreducible in M′.

Proof. 1. Since the dimension of M is two, π(X′) is a semianalytic subset of dimension one by [L]. Then, it is an analytic curve if and only if it is arc-symmetric. Let γ : (−ε, ε) → M be an analytic arc such that γ(0, ε) ⊂ π(X′), we want to prove that γ(−ε, ε) is contained in π(X′). We can assume, up to take a smaller ε, that γ(0, ε) ⊂ M\D, where D is a discrete set of points such that π : M′\π^{−1}(D) → M\D is an isomorphism. Let γ′ : (−ε, ε) → M′ be the analytic arc such that π ∘ γ′ = γ. Since γ(0, ε) is contained in π(X′) \ D, we have γ′(0, ε) ⊂ X′. As X′ is a curve, this implies that γ′(−ε, ε) ⊂ X′.

2. Assume Y′ = Y′_{1} ∪ Y′_{2}, where both Y′_{1} and Y′_{2} are analytic curves, we have to prove that Y′ = Y′_{1} or Y′ = Y′_{2}. Since Y′ does not contain the exceptional divisor, neither Y′_{1} nor Y′_{2} contains it. So, by point 1., the sets π(Y′_{1}) and π(Y′_{2}) are analytic curves such that Y = π(Y′_{1}) ∪ π(Y′_{2}). This implies π(Y′_{1}) = Y or π(Y′_{2}) = Y. Since π is an isomorphism M′ \ π^{−1}(D) → M \ D, we get the thesis.

Corollary 4.7. Let π be as in Lemma 4.6. If S′ is a global semianalytic subset of M′, then π(S′) is a global analytic subset of M.

Proof. Since π(S′) is semianalytic ([L]), by [CA, Lemma 3.1] and [CA, Prop. 3.3], it is enough to prove that the exists a proper analytic set in M containing ∂π(S′). We have ∂π(S′) ⊂ π(∂S′) ⊂ π(∂_{Z}S′), where π(∂_{Z}S′) is a proper analytic set by Lemma 4.6.

Proof of Theorem 4.5 . If A and B are generically separated by f ∈ 𝒪(M) then, f ∘ π ∈ 𝒪(M′) generically separates π^{−1}(A) and π^{−1}(B).

Let us prove the converse implication. Let f ∈ 𝒪(M′) generically separating π^{−1}(A) and π^{−1}(B). Denote S′ = {f > 0}. By Corollary 4.7, the set π(S′) is a global semianalytic subset of M. We prove that it is generically principal open, i.e. there exists g ∈ 𝒪(M) such that π(S′) =_{gen} {g > 0}. Then, by construction, g generically separates A from B.

By Proposition 2.2, it is enough to prove that no global irreducible component of dimension one of ∂_{Z}π(S′) is simultaneously odd and even. Assume that there exists such a component, say W. Then, by Lemma 4.6, its strict transform is an irreducible component of codimension one of ∂_{Z}S′, and it is simultaneously odd and even with respect to S′; this contradicts the principality of S′. The proof is done.

Proof of Proposition 4.4. Assume that φ^{−1}(2h) and φ^{−1}(−2h) are not generically separable, and that the walls of φ are non singular with normal crossing intersections. Then, by [BP, Th. 4.2], there is a wall W of φ which is simultaneously odd and even for φ^{−1}(2h) and φ^{−1}(−2h), i.e. dim (W ∩ \overline{φ^{−1}(2h)^{∗}} ∩ \overline{φ^{−1}(−2h)^{∗}}) = 1 and dim (W ∩ (φ^{−1}(2h)^{∗} ∪ φ^{−1}(−2h)^{∗})) = 1. On the odd part of W, the generic value of ∂_{W} φ is 0, and the generic values of ∂_{W}^{t} φ are 2h and −2h. On the even part of W, the generic values of ∂_{W} φ are 2h and −2h, and the generic value of ∂_{W}^{t} φ is 0. So we have n(∂_{W} φ) = n(∂_{W}^{t} φ) = 2h, and n(φ) ≥ 4h. As by Proposition 4.1 we have n(φ) ≤ 4h, we conclude that n(φ) = 4h.

We assume now that φ^{−1}(2h) and φ^{−1}(−2h) are not generically separable, but now we do not make any hypothesis about the walls. There exists π : M′ → M a locally finite sequence of blowings-up of points such that the walls of the function φ ∘ π are non singular with normal crossing intersections. Theorem 4.5 assures us that (φ ∘ π)^{−1}(2h) and (φ ∘ π)^{−1}(−2h) are not generically separable. So we have n(φ ∘ π) = 4h. Note that n(φ) ≥ n(φ ∘ π). Indeed, if we can write φ = ∑_{i=1}^{r}sign f_{i} with f_{i} ∈ 𝒪(M), then we have φ ∘ π = ∑_{i=1}^{r}sign (f_{i} ∘ π) with f_{i} ∘ π ∈ 𝒪(M′). So we get n(φ) ≥ 4h, and by Proposition 4.1 we have n(φ) = 4h.

Conversely, assume that φ^{−1}(2h) and φ^{−1}(−2h) are generically separable. Let f ∈ 𝒪(M) be a non zero function such that f_{|_{φ^{−1}(2h)}} ≥ 0 and f_{|_{φ^{−1}(−2h)}} ≤ 0. We consider the function ψ = φ−sign f : M → [−2h + 1, 2h − 1]. By Proposition 4.1, we have n(φ) ≤ n(ψ) + 1 ≤ 4h − 2.

Corollary 4.8. Let φ : M → [−2h, 2h] be a sum of signs of global analytic functions, with non singular walls with normal crossing intersections. Then we have n(φ) = 4h if and only if there is a wall W of φ such that m(∂_{W} φ) + m(∂_{W}^{t} φ) = 4h.

Proof. This is immediate by the proof of Proposition 4.4.

Remark 4.9. If we do not assume that the walls have normal crossing intersections, then the result of Corollary 4.8 is not true. For instance, consider the function on M = ℝ^{2} defined on the left-hand side of the following figure:

Since it is constant mod 4, it is a sum of signs of analytic functions on ℝ^{2}. If π : M′ → ℝ^{2} is the blowing-up of the point of intersection of the walls, and if E is the exceptional divisor of π, then it can be checked that m(∂_{E}(φ∘π)) = m(∂_{E}^{t}(φ∘π)) = 4 (see the right-hand side of the figure). So we have n(φ∘π) = 8, and n(φ) ≥ n(φ∘π) (cf. proof of Proposition 4.4) so n(φ) = 8. But for each wall W of φ, we have m(∂_{W} φ) + m(∂_{W}^{t} φ) = 6.

If δ = 0 and k is odd (with the notations of Proposition 4.1), the situation is quite different. We have the following partial result.

Proposition 4.10. Let φ : M → [−2h + 1, 2h − 1] be a sum of signs of global analytic functions, with h ∈ ℕ. If we have n(φ) = 4h − 3, then the sets φ^{−1}(2h − 1) ∪ φ^{−1}(2h − 3) and φ^{−1}(−2h + 1) ∪ φ^{−1}(−2h + 3) are not generically separable.

Proof. Let φ̄ : M → {−1, 0, 1} be the function defined by

φ̄(x) = \{ \begin{matrix}1 & if φ(x) ≡ 2h − 1 mod 4, \\ −1 & if φ(x) ≡ −2h + 1 mod 4 \\ 0 & else \end{matrix}

Then φ−φ̄ is generically constant mod 4, so φ̄ is a sum of signs of analytic functions, and by Example 4.2 we have n(φ̄) = 1. In order to have n(φ) = 4h−3, we need that n(φ−φ̄) = 4h−4. As the values of φ−φ̄ are in [−2h+2, 2h−2], by Proposition 4.4 we have n(φ − φ̄) = 4h − 4 if and only if (φ − φ̄)^{−1}(2h − 2) and (φ − φ̄)^{−1}(−2h + 2) are not generically separable. As (φ − φ̄)^{−1}(2h − 2) = φ^{−1}(2h − 1) ∪ φ^{−1}(2h − 3) and (φ − φ̄)^{−1}(−2h + 2) = φ^{−1}(−2h + 1) ∪ φ^{−1}(−2h + 3), the proof is done.

Remark 4.11. The non separability of φ^{−1}(2h − 1) ∪ φ^{−1}(2h − 3) and φ^{−1}(−2h + 1) ∪ φ^{−1}(−2h + 3) does not guarantee that n(φ) = 4h − 3. Consider φ : ℝ^{2} → ℤ defined by φ = h − 1 + (h − 1)sign y − (h − 1)sign (xy) + (h − 2)sign x. Then n(φ) = 4h − 5.

Note also that we can have n(φ) = 4h − 3 even if φ^{−1}(2h − 1) = ∅ or φ^{−1}(−2h + 1) = ∅, as in the following example. For each wall of φ, we have m(∂_{W} φ)+m(∂_{W}^{t} φ) = 4h − 3.

4.3 Fans and obstructions

We recall the definition of a fan:

Definition 4.12. A (non empty) finite subset F of the real spectrum of a field K is a fan if for any triple σ_{1}, σ_{2}, σ_{3} ∈ F, the product σ_{1}σ_{2}σ_{3} is an ordering and belongs to F. Here the product of orderings means the product of signs for these orderings.

If the stability index of K is k ∈ ℕ, then the cardinality of a fan of K is 2^{r} with r ≤ k ([ABR]). If we work with K = ℳ(M), then we only have fans with one, two or four elements, since the stability index of ℳ(M) is two (cf. [DC-A]).

The Representation Theorem of Becker and Bröcker ([BB]) gives a characterization of signatures of quadratic forms in terms of fans:

Theorem 4.13. A constructible function φ̃ on the real spectrum of a field K is the signature of a quadratic form if and only if for each finite fan F of K, we have ∑_{σ∈F} φ̃(σ) ≡ 0 mod |F |.

If K = ℳ(M), it is enough to consider 2-element and 4-element fans when we use the Representation Theorem.

Given an irreducible curve W ⊂ M, denote by ℳ(W) = q.f.(𝒪(M)/ℐ(W)) the field of meromorphic functions on W. Since Artin-Lang property holds for ℳ(W) by [DC-A], there is a well-defined tilde map associating to a global semianalytic subset of W the constructible subset of Spec_{r}ℳ(W) with the same description. An ordering τ ∈ Spec_{r}ℳ(W) is a specialization of α ∈ Spec_{r}ℳ(M) if any f ∈ 𝒪(M) such that f_{|_{W}} is strictly positive for τ, is strictly positive for α. We write α → τ.

Similarly we write α → x for x ∈ M if for any f ∈ 𝒪(M) such that f(x) > 0, the function f is positive for α. We define in the same way τ → x for x ∈ W.

An ordering σ ∈ Spec_{r}ℳ(M) is called unbounded if it does not specialize to any point in M, see for instance [CA], [DC-A]. Here we prove a slight improvement of the Representation Theorem for the special case K = ℳ(M), namely, that unbounded orderings have no role. This result, with similar ones found in [BP], shows that the unbounded orderings, because of which Artin-Lang property fails in Spec_{r}𝒪(M), are not relevant as far as it concerns many geometric problems.

Denote by Σ_{W} [ℤ_{2}] the set of the orderings in Spec_{r}ℳ(M) specializing to some ordering in Spec_{r}ℳ(W) (see [1]). We recall a well known way to build fans in Σ_{W} [ℤ_{2}]. Let τ_{1}, τ_{2} be distinct orderings of ℳ(W). The ring 𝒪(M)_{ℐ(W)} is a discrete valuation ring of rank 1, and we can consider t ∈ 𝒪(M)_{ℐ(W)} an uniformizer of this ring. Then, each f ∈ 𝒪(M)_{ℐ(W)} can be written as f = t^{k}u where k is the valuation of f and u is a unit in 𝒪(M)_{ℐ(W)}. Denote by \overline{u} the class of u in ℳ(W). We set:

α_{1}(f) = τ_{1}(\overline{u}) α_{3}(f) = (−1)^{k}τ_{1}(\overline{u})
α_{2}(f) = τ_{2}(\overline{u}) α_{4}(f) = (−1)^{k}τ_{2}(\overline{u})

This defines four orderings α_{1}, α_{2}, α_{3}, α_{4} of ℳ(M), and these orderings form a 4element fan contained in Σ_{W} [ℤ_{2}].

We come back to our signature problem. Let φ : M → ℤ be a semianalytically constructible function.

Lemma 4.14. Assume that there exists a wall W of φ such that ∂_{W} φ is not generically constant mod 2. Then there exists a four-element fan F = {α_{1}, α_{2}, α_{3}, α_{4}} ⊂ Σ_{W} [ℤ_{2}], made of bounded orderings, such that ∑_{i} φ̃(α_{i}) ≢ 0 mod 4.

Proof. There exist Ω_{1} and Ω_{2} ⊂ W ∩ Reg ( ⋃_{n} ∂_{Z}(φ^{−1}(n))) non empty open semianalytic sets such that (∂_{W} φ)_{|Ω_{1}} is even and (∂_{W} φ)_{|Ω_{2}} is odd. Consider two points, x_{1} ∈ Ω_{1} and x_{2} ∈ Ω_{2}. Let τ_{1} and τ_{2} be orderings in Spec_{r}ℳ(W) such that τ_{1} → x_{1} and τ_{2} → x_{2} in Spec_{r}𝒪(W). Then we have τ_{1} ∈ Ω̃_{1} and τ_{2} ∈ Ω̃_{2}. Let F be the fan defined by τ_{1} and τ_{2} as above. Then α_{1}, α_{3} → x_{1} and α_{2}, α_{4} → x_{2}, so the α_{i}’s are bounded. Moreover,

{ \atop ∂_{W} φ}(τ_{1}) + { \atop ∂_{W} φ}(τ_{2}) = \frac{φ̃(α_{1}) + φ̃(α_{3})}{2} + \frac{φ̃(α_{2}) + φ̃(α_{4})}{2}

is odd, and this implies that ∑_{i} φ̃(α_{i}) ≢ 0 mod 4.

Similarly we have:

Lemma 4.15. Assume that there exists a wall W of φ such that ∂_{W} φ is not generically integer-valued. Then there exists a two-element fan F = {α_{1}, α_{2}} ⊂ Σ_{W} [ℤ_{2}], made of bounded orderings, such that ∑_{i} φ̃(α_{i}) ≢ 0 mod 2.

Using Theorem 2.5 and these two lemmas, we conclude:

Theorem 4.16. Let φ : M → ℤ be a semianalytically constructible function. It is generically a sum of signs of global analytic functions on M if and only if for each fan F ⊂ Spec_{r}ℳ(M) made only of bounded orderings, we have ∑_{σ∈F} φ̃(σ) ≡ 0 mod |F |.

This is a generic result. To state a complete result we consider, if W ⊂ M is an irreducible curve, the function φ̃_{|_{W}} : Spec_{r}ℳ(W) → ℤ induced by φ_{|_{W}}. The function φ̃_{|_{W}} is defined as φ̃ above, using that Artin-Lang property holds for ℳ(W).

Proposition 4.17. Let φ : M → ℤ be a semianalytically constructible function. It is a sum of signs of global analytic functions on M if and only if for each fan F ⊂ Spec_{r}ℳ(M) made only of bounded orderings, ∑_{σ∈F} φ̃(σ) ≡ 0 mod |F |, and for each irreducible curve W of M contained in ⋃_{n∈ℤ} ∂(φ^{−1}(n)), for each fan F ⊂ Spec_{r}ℳ(W) made only of bounded orderings, ∑_{σ∈F} φ̃_{|_{W}}(σ) ≡ 0 mod |F |.

This is a consequence of Proposition 1.9.

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