IsabelleBonnard Descriptionofalgebraicallyconstructiblefunctions

(de Gruyter 2003

Description of algebraically constructible functions

Isabelle Bonnard

(Communicated by C. Scheiderer)

Abstract.The algebraically constructible functions on a real algebraic set are the sums of signs of polynomials on this set. We prove a formula giving the minimal number of polynomials needed to write generically a given algebraically constructible function as a sum of signs. We also prove a characterization of the polynomials appearing in a generic presentation of the function with the minimal number of polynomials. Both results are e¤ective.

Introduction

LetVHR^N be a real algebraic set. (All the algebraic sets we consider here are zero sets of polynomials in someR^N.) We will denote byPðVÞandRðVÞrespectively the ring of polynomials onV and the ring of regular functions onV. IfV is irreducible, letKðVÞbe the field of rational functions onV.Algebraically constructible functions on V have been defined by McCrory and Parusin´ski in [9], as linear combinations, with integer coe‰cients, of Euler characteristics of fibres of proper regular mor- phisms. These authors use them to study the topology of real algebraic sets: in [9]

they reformulate the Akbulut–King conditions of algebraicity in dimensionc3, and in [10] they give new necessary conditions for dimension 4.

If P is a polynomial function onV (or, more generally, a regular function or a Nash function), we define thesign of Pas the functionðsgnPÞ:V!Zsuch that for allxAV

ðsgnPÞðxÞ ¼

1 if PðxÞ>0;

1 if PðxÞ<0;

0 if PðxÞ ¼0:

8>

<

>:

Parusin´ski and Szafraniec on one hand ([12]), and Coste and Kurdyka on the other ([8]), have proved independently that the algebraically constructible functions on V are exactly the sums of signs of polynomials onV.

Let j:V !Zbe an algebraically constructible function. There are clearly many ways to write j as a sum of signs of polynomials on V. For instance, for any

PAPðVÞ, we have j¼jþsgnPþsgnðPÞ, so we can get a presentation as long as we want. We are interested here in a presentation as short as possible.

We will workgenerically, i.e. outside an algebraic subset ofVof dimension strictly smaller than the dimension ofV. We will write¼gen for an equality holding generi- cally onV.

We prove a formula giving the minimal number of polynomials (counted ‘‘with multiplicities’’) needed to write generically an algebraically constructible function as a sum of signs of polynomials. This formula allows us to calculate e¤ectively this minimal number, using an induction on the dimension of the space. The proof is a transposition to the geometric case of a result for quadratic forms over spaces of orderings: the isotropy theorem. There is a similar formula for Nash constructible functions.

Then, using the same type of proofs, we give results about the polynomials appear- ing in a generic presentation of a given algebraically constructible function with the minimal number of polynomials. Such polynomials are said to berepresentedby the function.

The paper is organized as follows: Section 1 is devoted to a short presentation of spaces of orderings, Section 2 contains the formula for the minimal number of poly- nomials, and Section 3 gives a characterization of the represented polynomials.

I wish to thank G. Stengle for rereading this paper, and C. Scheiderer for his helpful suggestions.

1 Spaces of orderings

1.1 Presentation.We present spaces of orderings in the context of the real spectrum of a field. For a complete definition we refer to [3] or [11]. HereZ=2Z¼ f1;1g, and we denote byFðX;Z=2ZÞthe set of functions from a setX toZ=2Z.

Let K be a real field. The set of the orderings of K (as a field) is called the real spectrum ofK and denoted Spec_rK. Ifa is a non-zero element of K, we define the functionðsgnaÞ: Spec_rK!Z=2Zwhich maps an orderingsto the sign of afors.

Denote

G¼ fsgnajaAK f0ggHFðSpec_rK;Z=2ZÞ:

ThenðSpec_rK;GÞis a space of orderings.

A subsetCof Spec_rK is said to beconstructibleif it is a finite union of sets of the form

fsASpec_rKj ðsgna₁ÞðsÞ ¼1;. . .;ðsgna_rÞðsÞ ¼1g

with a₁;. . .;a_rAK. We consider the constructible topology on Spec_rK, that is, the topology on Spec_rK for which the constructible subsets of Spec_rK form a basis.

Consider now a (non-empty) closed subset F of Spec_rK. The set F is a fan of K if

Es₁;s₂;s₃AF bsAF EgAG:gðsÞ ¼gðs1Þgðs2Þgðs3Þ

i.e. ‘‘the product of three elements of F is still in Spec_rK and belongs toF’’, where the product of orderings means the product of signs for these orderings.

Remark 1.1. Any subset of Spec_rK of one or two elements is a fan, and is called a trivial fan. The cardinality of a finite fan is always a power of two, since a fan has a structure of aðZ=2ZÞ-a‰ne space, with the product as inner operation, and the nat- ural scalar multiplication.

Let X be a (non-empty) closed subset of Spec_rK. Then the set X is a subspace of Spec_rK if there is no fan F of K such that XVF has exactly three elements. If H ¼ fðsgnaÞj_XjaAKnf0gg, then the coupleðX;HÞis a space of orderings.

Example 1.2.Let X be a set with a single element, and letH be the set of the two constant functions X ! f1g andX ! f1g. Then ðX;HÞ is a space of orderings, calledthe atomic spaceand denoted byE.

If sis an ordering of K andBis a valuation ring of K, we say thatsandB are compatible if for any a in K and anyb in the maximal ideal m of B, the relation 0<a<bforsimpliesaAm. Thensinduces an orderingson the residue fieldkof Bby

ðsgnaÞðsÞ ¼ ðsgnaÞðsÞ foraABnm

where a denotes the class of a in k. Conversely, if sASpec_rk, the orderings of K compatible withBand inducingsare calledpullbacks ofsvia B. IfXis a subspace of Spec_rk(respectively a fan ofk), then the set of the pullbacks of the elements ofXvia Bis a subspace of Spec_rK (respectively a fan ofK).

Example 1.3. We will use the following construction (see [4, Ex. 2.2]). Let A be a regular local ring of dimension d with quotient field K, and let ðx1;. . .;xdÞ be a regular system of parameters of A. Consider the valuation ring B of the place K ¼K0!K1U y! !KdU y, where Ki is the quotient field of the ring Ai¼A=ðx1;. . .;xiÞand the placeKi!Kiþ1U y corresponds to the valuation ring Aiðxiþ1Þ of Ki. The ringB is a discrete valuation ring of rankd. It dominates Aand has the same residue fieldk. Any orderingsofkhas exactly 2^d pullbackssinKvia B, and each of them is determined by the signs given tox1;. . .;xd.

We come now to the notion of form over a space of orderings.

Definition 1.4.LetðX;GÞbe a space of orderings. Aform of dimension roverX is a class ofr-tuples of elements ofGmodulo the relation

ðf₁;. . .;f_rÞ ðg1;. . .;g_rÞ i¤ EsAX: f₁ðsÞ þ þf_rðsÞ ¼g₁ðsÞ þ þg_rðsÞ:

We denote byhf₁;. . .;f_rithe class ofðf₁;. . .;f_rÞAG^r. IfX⁰is a subspace ofX, the restriction of the form r¼hf1;. . .;fri to X⁰ is the form rj_X⁰ ¼hf1j_X⁰;. . .;frj_X⁰i overX⁰.

Example 1.5.IfKis an ordered field, a quadratic form (in the usual sense) of dimen- sionroverK is a symmetric matrix of dimensionrwith entries inK. We can diago- nalize this matrix, and the diagonal matrix we get corresponds to the previous defi- nition for the real spectrum of K. The usual definition of signature of a quadratic form coincides with the following one.

Definition 1.6.Thesignatureof the formr¼hf₁;. . .;f_riover the space of orderings X is the functionrr^:X!Zdefined byrrðsÞ ¼^ f₁ðsÞ þ þf_rðsÞ.

A form over X isanisotropic if there is no form overX with the same signature and a strictly smaller dimension. A form which is not anisotropic is said to be iso- tropic.

Ifr¼hf1;. . .;friandr⁰¼hg1;. . .;gsiare two forms overX andhis an element ofG, we definerþr⁰¼hf1;. . .;fr;g1;. . .;gsiandhr¼hhf1;. . .;hfri. Then we have

d rþr⁰

rþr⁰¼rr^þrr^⁰andchrhr¼hrr.^

1.2 Structure. We present now two basic operations on spaces of orderings: addi- tion and extension.

LetðX1;G₁ÞandðX2;G₂Þbe two spaces of orderings. ThesumðY;HÞ ¼ ðX1;G₁Þ þ ðX2;G2Þis defined by Y¼X1tX2 (disjoint union) and ðg1;g2ÞAH ¼G1G2 act- ing as

ðg1;g2ÞðsÞ ¼ g₁ðsÞ if sAX₁; g₂ðsÞ if sAX₂:

The resulting space is a space of orderings.

If now ðY;HÞ is a space of orderings and H⁰ is a group of exponent two, the extension ðY;HÞ½H⁰ is the couple ðHHc⁰⁰Y;H⁰HÞ, where HHc⁰⁰ denotes the group of homomorphisms fromH⁰toZ=2Zand the functions are defined by

ðh⁰;hÞða;sÞ ¼aðh⁰ÞhðsÞ forða;sÞAHHc⁰⁰Y andðh⁰;hÞAH⁰H:

This defines a space of orderings.

Example 1.7. If B is a discrete valuation ring of rank d of a field K, and ifY is a subspace of the real spectrum of its residue field, then the set of pullbacks of the ele- ments ofY viaBis the extensionY½ðZ=2ZÞ^d.

These two operations are very important, as the following theorem shows ([3, IV.5.1], [11, 4.2.2]).

Theorem 1.8 (Structure theorem). Any finite space of orderings can be built from a finite number of atomic spaces,by a finite number of additions and extensions byZ=2Z.

This construction is unique up to isomorphism.

We now explain the behaviour of forms under addition and extension of spaces of orderings. LetX1;X2be spaces of orderings. It follows from the definition of the sum that a form rof dimensionroverX1þX2 can be seen as a coupleðr₁;r₂Þwhere r_l is a form of dimensionroverXl forl¼1;2. The formris anisotropic overX1þX2

if and only ifr₁ is anisotropic overX1orr₂is anisotropic overX2.

If now Y is a space of orderings andH⁰a group of exponent two, an anisotropic form r over Y½H⁰ can be written in a unique way as r¼P

h⁰AH⁰h⁰r_h0, where the r_h0’s are anisotropic forms overY, and only a finite number ofr_h0’s are di¤erent from the ‘‘empty’’ formh i.

1.3 Application to algebraically constructible functions.Let VHR^N be an irreduc- ible real algebraic set, and letjbe an algebraically constructible function onV. Thenj is in particular constructible, i.e. there exists a finite semi-algebraic partition of V such thatjis constant on each element of the partition.

Denote SV¼Spec_rKðVÞ. As in [7], we identify the algebraically constructible function j¼Pr

j¼1sgnPj considered generically on V, with the signature jj~of the formhf₁;. . .;f_rioverS_V, where f_j¼sgnP_jonS_V.

Assume jtakes the valuekAZon a semi-algebraic subsetS of V. Consider the constructible subsetSS~ofS_V defined by the same boolean combination of equations and sign conditions asS. The setSS~is well-defined (see [5]). Then, the functionjj~takes the valuekonSS.~

The minimal number of polynomials needed to describejgenerically is the dimen- sion of the anisotropic form overSVwith signaturejj. In the same way, a polynomial~ Pappears in a generic presentation ofjwith the minimal number of polynomials if and only if the sign ofPis an entry of the anisotropic form overSVwith signaturejj.~ So instead of studying the geometric situation, we will study forms in the algebraic context of spaces of orderings.

2 Number of polynomials

Let VHR^N be a real algebraic set, and letj:V!Zbe an algebraically construc- tible function. We want to calculate the minimal numberNðjÞof polynomials needed to writejgenerically as a sum of signs of polynomials, i.e.

NðjÞ ¼min rANjbP₁;. . .;P_rAPðVÞ:j¼gen

X^r

j¼1

sgnP_jonV

( )

:

This means that if the same polynomial appears several times in the presentation, we will count it at each appearance. So we count the minimal number of polynomials

‘‘with multiplicities’’.

We will denote byMðjÞthe maximal generic value of the absolute value ofj. Since each polynomial in the presentation contributes the value 1 or1, we have

NðjÞdMðjÞ:

IfV1;. . .;Vrare the irreducible components ofV, then we have NðjÞ ¼maxfNðjj_V1Þ;. . .;Nðjj_VrÞg:

Example 2.1.If dimV ¼1, we haveNðjÞ ¼MðjÞfor any algebraically constructible functionj. Indeed, by the previous remark, we may assume thatV is irreducible. We can write genericallyjas the sum ofMðjÞconstructible functions, each of them tak- ing generically the values 1 and1. The spaceS_V is a so-called SAP-space, since its stability index issðSVÞ ¼dimV ¼1 (cf. [3, III.3.4], [11, 3.3]). Using [3, III.3.2] or [11, 3.3.1], we get that any constructible function onV with generic values in f1;1g is generically the sign of a polynomial.

If dimVd2, this equality no longer holds for a general algebraically constructible function. For instance, considerj:R²!Zdefined by

jðx;yÞ ¼ 2 if xd0 andyd0;

2 else.

Then MðjÞ ¼2 and NðjÞ is even. We have j¼gen sgnxþsgnyþsgnðxyÞ 1, so NðjÞc4. Ifjwas generically the sum of the signs of two polynomials, these poly- nomials should be generically positive on the first quadrant and generically negative outside. Such polynomials do not exist, soNðjÞ ¼4>MðjÞ.

To make the presentation clear we start by introducing the algebraic tools we use to estimate NðjÞ. Then we will present the formula, and prove it. Finally we will extend it to the Nash case.

2.1 Algebraic tools. From now on we denote Z=2Z¼ f1;ag and Z=2ZZ=2Zd ¼ f1;ag, that is,ais the identity.

Lemma 2.2. Let X be a space of orderings, and letj:X!Z be the signature of a form over X.We denote by NðX;jÞthe dimension of the anisotropic form over X with signaturej.

If X is a sum X ¼X1þX2,then

NðX;jÞ ¼maxfNðX1;jj_X1Þ;NðX2;jj_X2Þg:

If X is an extension X ¼Y½Z=2Z,we define two functionsc⁰;c⁰⁰on Y byc⁰ðsÞ ¼

1

2ðjð1;sÞ þjða;sÞÞand c⁰⁰ðsÞ ¼¹₂ðjð1;sÞ jða;sÞÞ. Then c⁰ and c⁰⁰ are signatures of forms over Y and

NðX;jÞ ¼NðY;c⁰Þ þNðY;c⁰⁰Þ:

Proof.Letrbe the anisotropic form overX such thatrr^¼j.

Assume firstX ¼X1þX2. Writer¼ ðr₁;r₂Þwherer_lis a form overXlforl¼1;2.

Then NðX;jÞ ¼dimr¼dimr₁¼dimr₂, and either r₁ or r₂ is anisotropic. As r_l representsjj_Xl, we haveNðXl;jj_XlÞcNðX;jÞ, with equality ifr_lis anisotropic. This proves the first point of the lemma.

Assume now that X is the extension X¼Y½Z=2Z. We can write r¼r₁þar_a where r₁ and r_a are anisotropic forms over Y. Then jð1;sÞ ¼rr^₁ðsÞ þrr^_aðsÞ and jða;sÞ ¼rr^₁ðsÞ rr^_aðsÞforsAY, soc⁰¼rr^₁ andc⁰⁰¼rr^_a. We get

NðX;jÞ ¼dimr¼dimr₁þdimr_a¼NðY;c⁰Þ þNðX;c⁰⁰Þ: r The formula of the next paragraph is a geometric version of the following result ([3, IV.6.4], [11, 4.3.1]):

Theorem 2.3 (Isotropy theorem).Let X be a space of orderings,and ran anisotropic form over X.Then there exists a finite subspace Y of X such thatrj_Yis still anisotropic.

2.2 The geometric result.LetVHR^N be a real algebraic set, and letj:V!Zbe an algebraically constructible function. We use the notion of walls of the functionj.

This notion has been used by Acquistapace, Andradas, Broglia and Ve´lez to study basicness ([2]) and separation ([1]) of semi-algebraic sets, and by the author to char- acterize algebraically constructible functions ([7]). We recall the definition.

IfSHVis a semi-algebraic set, we denote bySits regularized version S¼IntðAdhðIntðSÞVRegðVÞÞÞ:

A wall of j is an irreducible component, of codimension one in V, of the Zariski closure of the Euclidean boundary of aðj¹ðmÞÞ.

By the remark at the beginning of the section, we can work independently on each irreducible component ofV. So from now on, we assume thatV is irreducible.

If V is a one-point compactification of V, we can extend j to a function j on V, by giving any value at the additional point. Thenjis algebraically constructible on V, and we haveNðjÞ ¼NðjÞ. So from now on, we also assume that V is com- pact.

Now, consider p:V⁰!V, a sequence of blowings-up with smooth centers, such that V⁰ is non-singular, and that the walls in V⁰ of the algebraically constructible functionjpare non-singular with normal crossings intersections. (By this we mean that there is a family of polynomials P1;. . .;PsAPðV⁰Þ describing ððjpÞ¹ðkÞÞ, forkAZ, such that all thePj’s are at normal crossings inV⁰.) We haveNðjpÞ ¼ NðjÞ. So we consider from now on this non-singular situation.

Remark 2.4. Since V is non-singular, j is generically constant on each of the con- nected components of the complement of the union of the walls.

Indeed, letCbe such a connected component. Denote byYthe union of the Eucli- dean boundaries of the ðj¹ðmÞÞ for mAZ, and let j¼P

mAZm1_ðj¹_ðmÞÞ. The

function j is constant on each connected component of VnY. If xAC, the codi- mension of the germ Yx inVxis at least two, soðVnYÞ_xis connected andjis con- stant onðVnYÞ_x. Sincejandjare generically equal onV, the functionjis generi- cally constant on a neighbourhood ofx.

Fix nowxAC. We define a function f :C! f0;1gbyfðyÞ ¼1 if the generic value of jnear y is the same as nearx, and fðyÞ ¼0 else. Then f is continuous, so it is constant and equal to 0, andjis generically constant onC.

LetW be a wall ofj. We consider the algebraically constructible functionqWjon W defined in [7]. We only recall here the generic definition in our non-singular case.

Letxbe a point ofW, such thatxbelongs to no other wall ofj. The functionjis generically constant nearxon each of the two sides ofW. Then,q_WjðxÞis the average of the generic values ofjon each of these two sides.

We define another function on W. Let tbe a polynomial on V which is an uni- formizer of the regular local ringRðVÞ_IðWÞ. Then we set

q_W^t j¼q_WðjsgntÞ:

The functionq_W^t jis algebraically constructible onW, and is generically equal to the half of the di¤erence of the generic values ofjon the two sides ofW. The definition ofq_Wjandq_W^t jcan be compared to the definition of the shadow and countershadow of a semi-algebraic set on a wall in [1].

Remark 2.5. If t⁰APðVÞ is another uniformizer of RðVÞ_IðWÞ, then the functions q_W^t j and q_W^t0j are a priori di¤erent, but Nðq_W^t jÞ ¼Nðq_W^t0jÞ. Indeed, if q_W^t j¼gen

Pr

j¼1sgnP_j on W, then q_W^t0j¼gen Pr

j¼1sgnðtt⁰P_jÞ on W. This allows us to talk aboutNðq_W^t jÞwithout making precise the chosen uniformizert.

Theorem 2.6. Let VHR^N be an irreducible real algebraic set which is compact and non-singular.Letj:V!Zbe an algebraically constructible function whose walls are non-singular with normal crossings intersections.Then

NðjÞ ¼maxn

MðjÞ; max

Wwall ofjðNðqWjÞ þNðq_W^t jÞÞo :

This theorem reduces the problem of calculatingNðjÞto a finite number of similar problems in lower dimension. By induction on the dimension, it is su‰cient to cal- culate this number of polynomials in dimension one. By Example 2.1, in dimension one we have NðjÞ ¼MðjÞ, and we can calculate NðjÞ in an e¤ective way for any dimension ofV.

Remark 2.7.If the generic values ofjare contained in an interval½dk;dþkwith dAZandkAN, then the generic values ofq_Wjare in½dk;dþk, and the generic values ofq_W^t jare in½k;k. AsMðjÞckþ jdj, we get by induction on the dimension that

NðjÞc2^dimV1kþ jdj:

This bound was already in [7], where its sharpness is proved if kis even. (For odd values ofk, this bound can be improved a bit to get a sharp one, see [7].)

2.3 Proof of Theorem 2.6.We have seen that NðjÞdMðjÞ. LetW be a wall ofj, and let tbe an uniformizer ofRðVÞ_IðWÞ which is a polynomial onV. Consider the space of orderingsX ¼SW½Z=2Z. The residue field ofRðVÞ_IðWÞ isKðWÞ, and we embedX inSV via this ringRðVÞ_IðWÞ, using t, as in Example 1.7. We have clearly NðjÞdNðX;jjj~_XÞwith the notations of Lemma 2.2, and by the second point of this lemma we get

NðX;jjj~_XÞ ¼NðSW;qqgWWjjÞ þNðSW;qqg_WW^tt jjÞ ¼NðqWjÞ þNðq_W^t jÞ:

This proves the inequalityd.

To prove the other inequality, we consider the anisotropic form r over SV such that rr^¼jj. We want to calculate~ NðjÞ ¼dimr. By the isotropy Theorem 2.3, there exists a finite subspace X⁰ ofSV such thatrj_X0 is still anisotropic. We chooseX⁰of minimal cardinality for this property. ThenX⁰cannot be a sumX₁⁰þX₂⁰, since then either rj_X⁰

1 or rj_X⁰

2 would be anisotropic, which would contradict the minimality of the cardinality ofX⁰. Thus, it follows from the structure Theorem 1.8 thatX⁰is the atomic space or an extension.

IfX⁰is the atomic space,X⁰¼ fsg, thenNðjÞ ¼ jjjðsÞj~ cMðjÞ.

If X⁰ is an extension, we write X⁰¼Y⁰½ðZ=2ZÞ^r, where Y⁰ is not an extension.

There is a valuation ringBofKðVÞsuch thatY⁰is a subspace of the real spectrum of the residue fieldkofB, and such thatX⁰is a subspace of the pullback ofY⁰inS_V viaB. AsV is compact, we haveRðVÞHB. Letpbe the intersection withRðVÞof the maximal ideal ofB. It is a prime ideal of RðVÞ. Denote byZ the zero set ofp inV: this is an irreducible algebraic set, and by constructionKðZÞis a subfield ofk.

LetYbe the subspace ofSZgenerated by the restrictions toKðZÞof the elements of Y⁰. Note that ifsAX⁰is a pullback of the elementgAY⁰, andtAY is the restriction of g, then tis a specialization ofs in Spec_rRðVÞ. Indeed, if f ARðVÞis such that (sgnfÞðtÞ ¼1, thenðsgnfÞðgÞ ¼1, and soðsgnfÞðsÞ ¼1.

We claim that at least one wall ofjcontainsZ. (Note that we do not claim that two orderings in X⁰ have a common specialization on a wall.) For, otherwise, each elementtinY would be in the constructible subsetCC~ of Spec_rPðVÞfor some con- nected componentC of the complement of the union of the walls. But then, all the elements ofX⁰specializing totwould be inCC~too. Sincejis generically constant on C by Remark 2.4, the value ofjj~would be the same on the 2^r elements ofX⁰ which are pullbacks of the same element of Y⁰, and the restriction of r to the subspace X⁰⁰¼ fð1;. . .;1Þg Y⁰ofX⁰¼ðZ=2ZÞðZ=2ZÞd ^rY⁰would be anisotropic. Indeed, the res- idue form ðrj_X0Þ_ð1;...;1Þ would be anisotropic of dimension dimr over Y⁰, and the image of this form via the isomorphism between Y⁰ and X⁰⁰ is rj_X⁰⁰. This way we would get again a contradiction with the minimality of the cardinality ofX⁰.

Denote byW₁;. . .;W_d⁰ the walls containingZ, and byd the codimension ofZin V. As the walls have normal crossings intersections, we haveddd⁰. LetP1;. . .;Pr

be polynomials on V describing the ðj¹ðmÞÞ formAZ. Since by assumption all the Pj’s are at normal crossings, there is a regular system of parametersðx1;. . .;xdÞ of RðVÞ_p such that each Pj is a monomial in RðVÞ_p for this system, i.e.

P_j¼u_jx₁^m1;j. . .x_d^md;j with u_j a unit of RðVÞ_p, and m_1;j;. . .;m_d;j some integers. For i¼1;. . .;d⁰, at least one of theP_j’s vanishes onW_i. So we may assume thatfxi¼0g corresponds to the wallW_ifori¼1;. . .;d⁰.

Consider now the discrete valuation ringCdominatingRðVÞ_pwith the same resi- due field, as explained in Example 1.3, using the parameters ðx1;. . .;xdÞ. LetX be the pullback ofY viaC. In particular,X is the pullback of a subspaceX1ofSW1via RðVÞ_IðW1Þ. We will prove thatrj_X is anisotropic.

We consider the mappingy:X⁰!X, which maps an elements⁰ ofX⁰, pullback oft⁰AY⁰, to the elementsofX, which is a pullback of the restriction oft⁰inY and satisfiesðsgnxiÞðsÞ ¼ ðsgnxiÞðs⁰Þfori¼1;. . .;d. We want to prove thatyis a mor- phism of spaces of orderings (cf. [11, 2.1]).

Let f :X! f1;1gbe the restriction toXof the sign of an element ofKðVÞ. We have to prove thatg¼ f y:X⁰! f1;1gis the restriction toX⁰of the sign of an element ofKðVÞ. If this were not the case, by [3, IV.7.2.a)], there would be a four- element fanF⁰ofX⁰such thatgis positive on exactly an odd number of elements of F⁰. DenoteF⁰¼ fs₁⁰;s₂⁰;s₃⁰;s₄⁰gwith, say,g positive onfs₁⁰;s₂⁰;s₃⁰gand negative on fs₄⁰g. Lets_l¼yðs_l⁰Þforl¼1;2;3;4 andF¼ fs1;s₂;s₃;s₄g. Then f would be posi- tive onfs1;s₂;s₃gand negative onfs4g.

We prove thatFis a four-element fan ofX. Denote byt_l⁰the element ofY⁰induced by s_l⁰ and bytl the restriction of t_l⁰ in Y, for l¼1;2;3;4. As F⁰ is a fan, we have t₄⁰ ¼t₁⁰t₂⁰t₃⁰ hencet4¼t1t2t3, and ifiAf1;. . .;dgthen

ðsgnxiÞðs4Þ ¼ ðsgnxiÞðs₄⁰Þ ¼Y³

l¼1

ðsgnxiÞðs_l⁰Þ ¼Y³

l¼1

ðsgnxiÞðslÞ:

This proves thats4¼s1s2s3, andF is a fan. The possible cardinalities forF are 1;2 and 4. If the cardinality of F is not four, the value of f impliess1 ¼s2¼s30s4. But on the other hand s4¼s³₁ ¼s1, a contradiction. Therefore F is a four-element fan and we get that f is positive on exactly three elements ofF, which is not possible by [3, III.3.8]. We conclude thatgis the restriction toX⁰of the sign of an element of KðVÞ, and thaty:X⁰!X is a morphism of spaces of orderings.

Remark that the value ofjat an element ofSVinducing an ordering onKðZÞ, is determined by this induced ordering, and by the signs given tox1;. . .;xd. So for any s⁰AX⁰we havejjðyðs~ ⁰ÞÞ ¼jjðs~ ⁰Þ.

Let hf₁;. . .;f_si be a form over X, of signature jjj~_X. Then jjj~_X0¼ ðjj~yÞj_X0 ¼ Ps

j¼1 fjy, where fjy is the restriction toX⁰ of the sign of an element ofKðVÞ since y is a morphism of spaces of orderings. We conclude thatsdNðjÞ, and that rj_X is anisotropic.

We consider now the space X₁. The functions q_W1j and q_W^t ₁j correspond to the functionsc⁰andc⁰⁰of Lemma 2.2, and we have

NðjÞ ¼NðX;jjj~_XÞ ¼NðX1;qqgWW11jjj_X1Þ þNðX1;qqg_WW^tt ₁₁jjj_X1ÞcNðqW1jÞ þNðq_W^t ₁jÞ:

The proof is complete.

2.4 The Nash case.Recall that a Nash function onR^N is a function which is both analytic and semi-algebraic, and a Nash subset ofR^Nis a zero set of Nash functions.

Let VHR^N be a Nash set. In [9] McCrory and Parusin´ski introducedNash con- structible functionson V: their definition is similar to that of algebraically construc- tible functions, but now the fibres are restricted to connected components of algebraic sets. More precisely,j:V!Zis Nash constructible if fori¼1;. . .;r, there are an integer mi, a regular proper morphism fi from an algebraic setZi toV, and a con- nected componentTiofZi such that

jðxÞ ¼X^r

i¼1

miwðf_i¹ðxÞVTiÞ forxAV:

(Here w denotes the Euler characteristic.) In particular, algebraically constructible functions are Nash constructible.

If j:V !Z is a constructible function, we define aNash wall of j as a Nash- irreducible component, of codimension one inV, of the Nash closure of the Euclidian boundary of aðj¹ðkÞÞ. In [6] we proved that ifV is compact and non-singular, and if the Nash walls of jare non-singular with normal crossings intersections, thenjis generically Nash constructible on V if and only if jis generically a sum of signs of Nash functions onV. IfV is compact, but these regularity assumptions do not hold, this is not true: in this case, Nash constructible functions coincide with sums of signs of semi-algebraic arc-analytic functions.

Assume thatV is compact and non-singular, and that the Nash walls ofjare non- singular with normal crossings intersections. In this case, for a Nash wall W, the functions q_Wj and q_W^t j also are generically sums of signs of Nash functions. We transpose Theorem 2.6 to this case:

Proposition 2.8. Let VHR^N be a compact Nash set which is Nash-irreducible and non-singular. Let j:V!Zbe a Nash constructible function whose Nash walls are non-singular with normal crossings intersections.Denote by N_NðjÞthe minimal number of Nash functions(counted with multiplicities)needed to write genericallyjas a sum of signs of Nash functions.Then

N_NðjÞ ¼maxn

MðjÞ; max

WNash wall ofjðNNðqWjÞ þN_Nðq_W^t jÞÞo :

Proof. We repeat the proof of Theorem 2.6, working with the ring NðVÞof Nash functions onV instead of the ring of rational functions.

The ring NðVÞ is noetherian by [5, Theorem 8.7.18]. Ifm is a maximal ideal of NðVÞ, thenm is the ideal of Nash functions vanishing at a point inM by [5, Cor- ollary 8.6.3] andNðVÞ_mis a regular local ring by [5, Proposition 8.7.15]. SoNðVÞ_p is a regular local ring for any prime idealpofNðVÞ, i.e. the ringNðVÞis regular.

This allows us to build the discrete valuation ring denoted byCin the proof of The-

orem 2.6. r

As in the algebraic case, we can determineNNðjÞby induction on the dimension, since in dimension one we haveNNðjÞ ¼MðjÞ.

Remark 2.9.The formulas given in Theorem 2.6 and Proposition 2.8 are very similar.

However NðjÞ andN_NðjÞ are not the same in general, even if VHR^N is a com- pact real algebraic set which is Nash-irreducible and non-singular, and if j is an algebraically constructible function on V with non-singular and normal crossings walls.

For instance, letV ¼RP²with the coordinatesðx0:x₁:x₂Þ, and letCbe the cubic ofV with the equationx0x₂²¼x1ðx₁²x₀²Þ. We define an algebraically constructible functionjonV in the following way:

There is only one (algebraic) wall: the cubicC, and we haveNðqWjÞ ¼Nðq_W^t jÞ ¼2, soNðjÞ ¼4. There is also only one Nash wall: the connected componentC₁ of C, and we haveN_NðqWjÞ ¼0 andN_Nðq_W^t jÞ ¼2. SoN_NðjÞ ¼20NðjÞ.

3 Represented polynomials

3.1 Algebraic tools.

Definition 3.1 ([3], III.1.18). Let ðX;GÞ be a space of orderings and r a form of dimensiond overX. We define

D_XðrÞ ¼ fgAGjbg₂;. . .;g_d AG:r¼hg;g₂;. . .;g_dioverXg:

An element ofD_XðrÞis said to berepresented byrover X.

Example 3.2. If r is isotropic, then it is clear that D_XðrÞ ¼G. Actually, these two conditions are equivalent ([11, 2.2.6(3)]).

The following lemma explains the behaviour of DXðrÞ under additions and extensions.

Lemma 3.3.Let X be a space of orderings andra form over X.

If X ¼X₁þX₂ andr¼ ðr₁;r₂Þwherer_j is a form over X_j for j¼1;2,then DXðrÞ ¼DX1ðr₁Þ DX2ðr₂Þ:

If X ¼Y½Handris anisotropic,writer¼P

hAHhr_h.Then D_XðrÞ ¼ G

hAH

hD_Yðr_hÞ:

Proof.The first point is clear, the second one is given by [3, IV.2.12.b)]. r Remark 3.4.LetX be a space of orderings, and letr₁;. . .;r_n be forms overX.

Assume that X ¼X1þX2 and denote r_i¼ ðr_i;1;r_i;2Þ. Then 7ⁿ

i¼1DXðr_iÞ ¼qif and only if7ⁿ

i¼1DX1ðr_i;1Þ ¼qor7ⁿ

i¼1DX2ðr_i;2Þ ¼q.

Assume that X¼Y½Hand that the r_i’s are anisotropic. Write r_i¼P

hAHhr_i;h. Then7ⁿ

i¼1DXðrÞ ¼qif and only if for anyhAH, we have7ⁿ

i¼1DYðr_i;hÞ ¼q. These two conditions follow easily from the previous lemma.

The aim of this section is to derive a geometric version of the following result ([3, IV.6.1.b)], [11, 4.3.2]) in the frame of algebraically constructible functions:

Theorem 3.5 (Local-global principle).Let r₁;. . .;r_n be forms over a space of order- ings X. If 7ⁿ

i¼1DXðrÞ ¼q, then there exists a finite subspace Y of X such that 7ⁿ

i¼1DYðrj_YÞ ¼q.

3.2 The geometric result.LetVHR^N be an irreducible real algebraic set. Ifjis an algebraically constructible function onV, we copy the definition given in Part 3.1 for forms. We say that a polynomial PAPðVÞ is represented by j on V if there exist P2;. . .;PNðjÞAPðVÞ such thatj is generically equal to sgnPþPNðjÞ

j¼2 sgnPj on V.

In this case, we will say also that sgnPis represented byj. We denote byDVðjÞthe set of the polynomials represented byjonV.

So, ifr is the anisotropic form overS_V representingjj~andPa polynomial onV, thenPbelongs toD_VðjÞif and only if the sign ofP(onS_V) belongs toD_SVðrÞ.

Theorem 3.6. Let VHR^N be an irreducible real algebraic set which is compact and non-singular.Letj₁;. . .;j_n be algebraically constructible functions on V such that all

the walls of the j_i’s are non-singular with normal crossings intersections.We assume that none of thej_i’s is generically equal to zero.

Then7ⁿ

i¼1D_Vðj_iÞ ¼qif and only if

there exist i;i⁰Af1;. . .;ng,and SHV semi-algebraic of dimensiondimV,such that Mðj_iÞ ¼Nðj_iÞ,Mðj_i⁰Þ ¼Nðj_i⁰Þ,andj_ij_S¼ Mðj_iÞ,j_i0j_S ¼Mðj_i⁰Þ,

or

there is a wall W of one of thej_i’s such that,if we fix a polynomial uniformizer t of RðVÞ_IðWÞ and if we denote JW ¼ fiAf1;. . .;ng jNðj_iÞ ¼NðqWj_iÞ þNðq_W^t j_iÞg, then7

iAJWD_WðqWj_iÞ ¼qand7

iAJWD_Wðq_W^t j_iÞ ¼q. Remark 3.7. The condition 7

iAJW DWðq_W^t j_iÞ ¼q is independent of the chosen t.

Indeed, let t⁰ be another polynomial uniformizer of RðVÞ_IðWÞ. If a polynomial P belongs toDWðq_W^t j_iÞ, thentt⁰Pbelongs toDWðq_W^t0j_iÞ.

Remark 3.8.If there isiAf1;. . .;ng such thatj_i¼gen0 on V, then D_Vðj_iÞ ¼q, so 7ⁿ

i¼1D_Vðj_iÞ ¼q.

As Theorem 2.6, Theorem 3.6 reduces a problem in dimension dimV to a finite number of similar problems in lower dimension. By induction on the dimension, we have to solve similar problems in dimension 1. In this case, only the first condition of Theorem 3.6 remains, and for any function j_i we have Nðj_iÞ ¼Mðj_iÞ, so we can check easily if7ⁿ

i¼1DVðj_iÞ ¼q.

Remark 3.9. Again, we can transpose Theorem 3.6 from the algebraic case to the Nash case, by working with the ring of Nash functions instead of the ring of poly- nomials. We get the same results with Nash walls instead of walls.

3.3 Proof of Theorem 3.6.We denoteG¼ fðsgnPÞ:SV!Z=2ZjPAPðVÞnf0gg.

Fori¼1;. . .;n, letr_i be the anisotropic form overSV representingj_i. We assume first that there is a polynomial PA7ⁿ

i¼1D_Vðj_iÞ. Then, if i satisfies Mðj_iÞ ¼Nðj_iÞ, the sign ofPand the sign ofj_imust be generically equal on the semi- algebraic setfjj_ij ¼Mðj_iÞg. So the first point of the theorem is not possible.

Consider a wallW of one of thej_i’s, and a uniformizertofRðVÞ_IðWÞ. We embed the space X ¼SW½Z=2Z into SV via this ring, so that for any sASW we have ðsgntÞð1;sÞ ¼1. If iAJW, then r_ij_X is anisotropic. For such an i, write r_ij_X ¼ r_i⁰þar_i⁰⁰, wherer_i⁰andr_i⁰⁰are anisotropic forms overSW representingqWjandq_W^t j respectively. AsðsgnPÞj_X belongs to7

iAJWDXðr_iÞ, we get that7

iAJWDSWðr_i⁰Þ0 q or7

iAJWDSWðr_i⁰⁰Þ0 qby Remark 3.4. In terms of algebraically constructible func- tions, this means that 7

iAJWD_WðqWj_iÞ0 q or 7

iAJWD_Wðq_W^t j_iÞ0 q. The first implication is proved.

Conversely, assume that 7ⁿ

i¼1D_Vðj_iÞ ¼q, so 7ⁿ

i¼1D_SVðr_iÞ ¼q. By Theorem 3.5, there is a finite subspace X⁰ of S_V such that7ⁿ

i¼1D_X⁰ðr_ij_X⁰Þ ¼q. We choose

X⁰ of minimal cardinality for this property. Then, by Remark 3.4, the space X⁰ is not a sum. According to the structure theorem, X⁰ is the atomic space or an extension.

First case. X⁰¼E. Denote by s the element of X⁰. There is iAf1;. . .;ng such that r_ij_X0 is anisotropic (else 7ⁿ

i¼1DX⁰ðr_ij_X0Þ would be fX⁰! f1g;X⁰! f1gg).

For such an i, we have j^rr_iðsÞj ¼dimr_i¼Nðj_iÞ ¼Mðj_iÞ, and DX⁰ðr_ij_X0Þ only contains the function s7!sgnðjj~_iðsÞÞ. We deduce from this the first point of the theorem.

Second case. X⁰¼Y⁰½ðZ=2ZÞ^rwhereY⁰is not an extension. We copy the construc- tion of the proof of Theorem 2.6: letBbe a valuation ring ofKðVÞsuch thatY⁰is a subspace of the real spectrum of the residue fieldkofB, andX⁰is a subspace of the pullback of Y⁰ viaB. As before we denote bypthe restriction of the maximal ideal ofBtoRðVÞ, byZthe zero set ofpinV, and byY the subspace ofSZgenerated by the restrictions toKðZÞof the elements ofY⁰.

We claim that the set Z is contained in at least one wall of one of the j_i’s.

Otherwise, as in the proof of Theorem 2.6, the value of j_i would be the same on the 2^r pullbacks in X⁰ of the same element of Y⁰, for i¼1;. . .;n. Consider X⁰⁰¼ fð1;. . .;1Þg Y⁰, and denote J⁰¼ fiAf1;. . .;ng jr_ij_X⁰ anisotropicg. Then, foriAJ⁰, we would haveD_X⁰ðr_ij_X⁰Þ ¼D_Y⁰ððr_ij_X⁰Þ_ð1;...;1ÞÞ. As7

iAJ⁰D_X⁰ðr_ij_X⁰Þ ¼q, we would get 7

iAJ⁰DY⁰ððr_ij_X0Þ_ð1;...;1ÞÞ ¼q, and by the isomorphism between X⁰⁰ andY⁰we would have7

iAJ⁰DX⁰⁰ðr_ij_X⁰⁰Þ ¼q. This would contradict the minimality of the cardinality ofX⁰.

Let W1;. . .;Wd⁰ be the walls of thej_i’s containingZ. We repeat the construction of the proof of Theorem 2.6. We get a space of orderings X and a morphism of spaces of orderingsy:X⁰!X. We prove that7ⁿ

i¼1DXðr_ij_XÞ ¼q.

Else, there would be an element f AGsuch that for everyiAf1;. . .;ng, there exist gi;2;. . .;gi;ri AG withri¼dimr_i andr_ijX ¼hf;gi;2;. . .;gi;rii_jX. For everys⁰AX⁰, we would havejj~_iðs⁰Þ ¼ ðjj~_iyÞðs⁰Þ ¼ ðf yÞðs⁰Þ þPri

j¼2ðgi;jyÞðs⁰Þ. Asyis a mor- phism of spaces of orderings, f yand all thegi;jy would be restrictions toX⁰of elements ofG, and f ywould be in7ⁿ

i¼1DX⁰ðr_ij_X0Þ. We would get a contradiction.

So we have7ⁿ

i¼1DXðr_ij_XÞ ¼q.

We can write X ¼X1½Z=2Z where X1 is a subspace of SW1. Denote J ¼ fiAf1;. . .;ng jr_ij_X anisotropicg. We have7

iAJDXðr_ij_XÞ ¼q. IfiAJ, the restric- tion r_ij_SW

1½Z=2Z is a fortiori anisotropic, so iAJW1 and 7

iAJW1DXðr_ij_XÞ ¼q. By Remark 3.4, we get7

iAJW1DX1ððr_ij_XÞ₁Þ ¼qand7

iAJW1DX1ððr_ij_XÞ_aÞ ¼q. In terms of algebraically constructible functions, this means that7

iAJW1DW1ðqW1j_iÞ ¼qand 7iAJW1

D_W1ðq_W^t ₁j_iÞ ¼q. The theorem is proved.

3.4 Recognizing represented polynomials.Ifjis an algebraically constructible func- tion on an irreducible real algebraic setVHR^N, and ifPis a polynomial onV, we can ask if P is represented byj. We give the following answer, using Theorem 2.6 and the fact thatPis represented byjif and only ifNðjsgnPÞ ¼NðjÞ 1. Note

that, if W is an hypersurface, then q_WsgnP and q_W^t sgnP are generically signs of polynomials.

Corollary 3.10. Let VHR^N be an irreducible real algebraic set which is compact and non-singular. Let j:V !Z be an algebraically constructible function, and let PAPðVÞnf0g.We assume that the walls ofjandsgnP are non-singular with normal crossings intersections,and thatjis not generically equal to zero.

Then P is represented byjif an only if

if MðjÞ ¼NðjÞ,then P has generically the same sign asjon the semi-algebraic set wherejjj ¼MðjÞ,

and

for any wall W ofjsuch that NðjÞ ¼NðqWjÞ þNðq_W^t jÞ, – if W is a wall ofsgnP,thenq_W^t sgnP is represented byq_W^t j, – if W is not a wall ofsgnP,thenqWsgnP is represented byqWj, and

for any wall ofsgnP which is not a wall ofj,we have NðjÞ>NðqWjÞ.

Remark 3.11.In dimension 1, only the first condition remains:Pis represented byj if and only if sgnP¼sgnjonfjjj ¼MðjÞgexcept at a finite number of points. As before, using induction on the dimension, we can reduce to this case.

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Received 18 October, 2001; revised 24 March, 2002 and 30 July, 2002

I. Bonnard, Institut Mathe´matique de Jussieu, Equipe d’Analyse Alge´brique, 175 rue du Chevaleret, 75013 Paris, France

Email: [email protected]