The moment problem for non-compact semialgebraic sets

(de Gruyter 2001

The moment problem for non-compact semialgebraic sets

Victoria Powers and Claus Scheiderer

(Communicated by the Managing Editors)

1 Introduction and background Given a closed subsetKofRⁿ, and…m_a†_aAZn

‡a (multi-) sequence of real numbers, the K-moment problemasks whether this sequence can be realized as the moment sequence of some positive Borel measure onK. In other words, the question is whether there is a positive Borel measuremonKwhich satis®es

…

Kx^admˆm_a

for every aAZ_‡ⁿ. (By this notation, we imply tacitly that all moments ofm exist.) In slightly di¨erent terms, the question is to characterize those linear forms

L:R‰t1;. . .;tnŠ !R for which there exists a positive Borel measure mon Kwhose

moments exist and satisfy

…

Kf…x†dmˆL…f†

for everyf AR‰t1;. . .;tnŠ. In this case, we say that theK-moment problem is solvable for L. Obviously, it is necessary that L…f†X0 whenever fX0 on K. A classical theorem says that this condition is also su½cient:

1.1. Theorem(Haviland [12]). The K-moment problem is solvable for L if and only if L…f†X0for every polynomial f which is non-negative on K.

Moment problems were originally studied in the one-variable case. In 1894, Stieltjes [25] showed that forKˆ ‰0;y†, theK-moment problem is solvable forLif and only if L…f²‡tg²†X0 for allf;gAR‰tŠ. The most famous example is the case KˆR, which was solved by Hamburger in 1921 [9]. He showed that a necessary and su½- cient condition in this case is L…f²†X0 for all f AR‰tŠ. In 1923, Hausdor¨ [11]

studied the case K ˆ ‰0;1Š and showed that a necessary and su½cient condition is L…f²‡tg²‡ …1ÿt†h²†X0 for allf;g;hAR‰tŠ. These results can be viewed as par-

ticular cases of Haviland's theorem; in all three cases, every polynomial which is non- negative onKis a ®nite sum of polynomials which were used for testingL.

Extending the moment problem to more than one variable is a more recent idea. The multidimensional moment problem is mentioned brie¯y in the 1943 book of Shohat and Tamarkin [24]. Some partial results on the two-dimensional moment problem are given in a paper by Devinatz [8] in 1957. In 1979, SchmuÈdgen [21] and Berg, Chris- tensen, and Jensen [1] showed that the Hamburger result forKˆRdoes not extend toRⁿfornX2, using the fact that in more than one variable there exist polynomials which are globally non-negative but not sums of squares (see below). Later works studied the multidimensional moment problem for other (speci®c) setsK, see, e.g., [2], [6], [16].

In the following, we writeAˆR‰t1;. . .;tnŠ, and denote the set of sums of squares inAbySA².

1.2. De®nition. Given a closed subsetK ofRⁿ and a subset Pof A, we say thatP solves the moment problem for Kif

(1) Kˆ fxARⁿ:f…x†X0 for everyf APg;

(2) for every linear functionalLonAwith

L…a²f1. . .fr†X0

for everyaAA,rX0 andf1;. . .;f_rAP, there is a positive Borel measuremonK such thatLis integration with respect tom.

The classic examples above imply (fornˆ1) thatftgsolves the moment problem for‰0;y†, q solves the moment problem for R, and ft;1ÿtg solves the moment problem for‰0;1Š.

In 1991, K. SchmuÈdgen proved the following remarkable theorem:

1.3. Theorem (SchmuÈdgen [23]). Suppose f1;. . .;f_rAA are such that K :ˆ

ff₁X0;. . .;f_rX0gis compact.Thenff₁;. . .;f_rgsolves the moment problem for K.

Note that SchmuÈdgen's theorem holds regardless of the polynomials chosen to de®neKby inequalities. As a simple example, we obtain the following (non-obvious) variation of Hausdor¨ 's result: LetKˆ ‰0;1Š, then for any (®xed) odd integerskand m, theK-moment problem is solvable forLif and only if L…f²‡g²t^k…1ÿt†^m†X0 for allf;gAR‰tŠ.

It seems that SchmuÈdgen's result was the ®rst on the moment problem which cov- ers a truly general class of setsK, rather than just speci®c sets.

Our goal in this paper is to study the non-compact case. Recall that a subsetKof Rⁿ is called basic closed (semialgebraic) if it has the formKˆ ff₁X0;. . .;f_rX0g, where thefiare polynomials. Given suchfi, we ask, when doesff₁;. . .;f_rgsolve the moment problem forK? The result of Berg et al. and SchmuÈdgen mentioned above shows that SchmuÈdgen's theorem does not generalize to the non-compact case:

1.4. Example. Suppose AˆR‰t1;. . .;tnŠ where nX2. Then there exists a linear functionalLonAsuch thatL…f²†X0 for allf AAbut the Rⁿ-moment problem is not solvable forL. In particular,qdoes not solve the moment problem forRⁿ. Sketch of proof. Since nX2, there exists a polynomial p such that pX0 on all of Rⁿ, butpis not a sum of squares. This was proven by Hilbert in 1888 [13], although the ®rst explicit example of suchpto appear in the literature was given by Motzkin in 1965 [17]. (See [18] for more on this interesting subject.)

The coneSA²is closed in the ®nest locally convex vector space topology onA. By Hahn-Banach separation of convex sets, there is a linear functionalLonAsuch that L…p†<0 butL…f†X0 for everyf ASA². Trivially,L cannot come from a positive Borel measure onRⁿ.

Note that Haviland's theorem implies immediately that the setPconsisting of all everywhere non-negative polynomials inAsolves the moment problem forRⁿ. More generally, for any closedK, there exists a setPwhich solves the moment problem for K, namely the set of allfwhich are non-negative onK. However, even ifKis a basic closed semialgebraic set, it is not clear Ðand not trueÐin general that a ®nite setP of polynomials can be found which solves the moment problem forK. Therefore, we wish to study the following questions:

.

Given polynomialsf₁;. . .;f_r, can we give necessary or su½cient conditions under which they solve the moment problem forKˆ ff₁X0;. . .;f_rX0g?

.

Given a basic closed semialgebraic set K, when does there exist some ®nite set of polynomials which solves the moment problem forK?

If K is compact, the answer to the second question is ``always'', and no conditions are needed in the ®rst question, both by SchmuÈdgen's theorem. On the other hand, we will show that there are many cases of non-compact Kwhere the answer to the second question is negative.

If Pis any set of polynomials, it is easy to see thatPsolves the moment problem forKif and only if the closure ofPUSA² under addition and multiplication does. A subset of A closed under addition and multiplication and containing all squares is called a preorder. (See the next section for precise de®nitions.) It turns out to be ad- vantageous to replacePby the preorder it generates, and to study the moment problem for preorders. We will do so in the next section, after introducing the necessary technical language. Also, it will be important to allow arbitrary ®nitely generatedR- algebrasAin place of the polynomial ring. We equipAwith the ®nest locally convex vector space topology. The saturation of a preorderP is the set of all polynomials which are non-negative on the closed set associated toP. By Haviland's theorem,P solves the moment problem for its associated set if and only if the topological closure of Pis also closed under saturation (Cor. 3.1). For this reason we are interested in methods for deciding if a preorder is closed. A key notion in this regard is that of stable preorders. Our ®rst main result is Theorem 2.14, which provides large classes of examples of basic closed sets Kfor which every ®nitely generated preorder with associated set Kis topologically closed, but not saturated, and hence does not solve the moment problem forK.

Thus we have found many cases where the answer to our second question above is negative. Among them are even (many) cases where dim…K† ˆ1 (Thm. 3.9). We also have positive results for 1-dimensional setsK, which we obtain by applying theorems from [20] (Thm. 3.12). Combining 3.9 and 3.12, we have obtained a complete solu- tion for the moment problem on smooth algebraic curves. This is the second main result of our paper. Furthermore, in all cases, our results give more precise informa- tion than just whether the moment problem is solved or not.

When we were in the ®nal stages of writing this paper, we found out from Murray Marshall and Salma Kuhlmann that they had been studying similar questions. Their approach and techniques are somewhat di¨erent from ours, for example, they work only in the polynomial ring. Their preprint [15] contains a list of open questions, and at the end of our paper we settle some of these.

Acknowledgements. A talk of K. SchmuÈdgen at the 60th birthday celebration of M.

Marshall led the ®rst author to consider the questions studied in this paper. She thanks Prof. SchmuÈdgen for his inspiring talk, Prof. Marshall for having a birthday, and Salma and Franz-Viktor Kuhlmann for organizing the birthday conference and inviting her. The authors also thank M. Marshall and S. Kuhlmann for bringing their work to their attention.

2 Preorders

Preorders are important objects in real algebra, which in some sense play a role in semialgebraic geometry that is comparable to the role played by ideals in algebraic geometry. In this section we study preorders with a view towards answering the questions on the moment problem raised in the previous section. However, our results are also of independent interest.

We need to work with ®nitely generated (f. g.)R-algebras, rather than with poly- nomial rings only. In geometric terms, this means working with a½ne algebraicR- varieties rather than with n-dimensional a½ne space only. First recall the basic dictionary between such varieties and f. g. R-algebras.

To any f. g.R-algebra Athere corresponds an a½ne algebraic varietyVoverR.

(We use the word ``variety'' in a broad sense, it does not imply irreducible or reduced.) The correspondence between A and Vis expressed by writing V ˆSpecA or Aˆ R‰VŠ. The varietyVcomes together with its set V…R† of real points; by de®nition V…R† ˆHom_R…A;R†, the set of R-algebra homomorphisms from A to R. Given MAV…R†, the corresponding homomorphismA!R is written f 7!f…M† and is thought of as evaluation of the elements of Ain M. Note that ifAˆR‰t₁;. . .;t_nŠ, thenV ˆAⁿˆa½nen-space, andV…R†is simplyRⁿ.

The set V…R† comes with a natural topology, which has the sets fMAV…R†: f…M†>0g (f AA) as a subbasis of open sets. A subset of V…R† is called semi- algebraicif it is a ®nite boolean combination (unions, intersections, complements) of these sets.

Choosing a ®nite system x1;. . .;xn of generators of A gives an epimorphism

R‰t1;. . .;tnŠ !!A, ti7!xi, and correspondingly, an embedding of V…R†into Rⁿ as

a Zariski closed subset. Thus, one can think of V…R† as a (Zariski closed) alge-

braic subset of some Rⁿ, if one wishes, but it is often preferable not to ®x such an embedding.

Note that the full algebraAcannot be retrieved fromV…R†, ifV…R†is given as an algebraic subset ofRⁿ, say. For example,V…R†may be empty withoutAbeing trivial, e.g., for AˆR‰tŠ=…t²‡1†. All one gets back fromV…R†is the quotient ringA=N, whereNis the so-called real nilradical ofA(see [3, 4.1]). To get a complete dictionary between algebras and a½ne varieties, it would be necessary to employ the structure sheaves of the latter.

Therefore, the algebraAis our basic object of study. We ®xVˆSpecA, an a½ne R-variety. By this we mean that Ais a f. g.R-algebra and V ˆSpecAis the asso- ciated a½ne algebraicR-variety. A subsetPofAis called apreorder(in A) iff² AP for everyf AAandPis closed under addition and multiplication. Any intersection of preorders inAis again a preorder. Therefore, given a subsetFofA, there is a smallest preorder containing F, denoted PO…F†, or PO…f₁;. . .;f_r† if F ˆ ff₁;. . .;f_rg, and called the preorder generated by F. Explicitly, PO…F†is the set of all ®nite sums of elements of the forma²f1. . .fr, whereaAA,rX0 andf1;. . .;f_rAF.

Given a preorderPinA, we write

S…P†:ˆ fMAV…R†:f…M†X0 for everyf APg:

This is a closed subset of V…R†. If PˆPO…f₁;. . .;f_r† is ®nitely generated, then S…P† ˆ fMAV…R†:f1…M†X0;. . .;f_r…M†X0g, and in particular, is a semi- algebraic set.

On the other hand, ifKis a closed subset ofV…R†, we write P…K†:ˆ ff AA:fX0 onKg:

This is a preorder inA, and clearlyS…P…K†† ˆK.

A preorder P will be called saturated if there exists a closed subset K of V…R†

with PˆP…K†. If so, then necessarily KˆS…P†. The saturation Sat…P† of a pre- order P is de®ned by Sat…P†:ˆP…S…P††; this is the smallest saturated preorder containingP. The correspondenceK7!P…K†is a bijection between closed subsets of V…R†and saturated preorders inA, the inverse map beingP7!S…P†.

2.1. Examples.The unique smallest preorder inAisSA², the set of sums of squares inA. Its saturation Sat…SA²†is the setA_‡of allpositive semide®nite(psd) functions in A, i.e., of all f AA with fX0 on V…R†. Consider in particular the case Aˆ

R‰t₁;. . .;t_nŠ. Ifnˆ1, thenSA²ˆA_‡. However, ifnX2, then as we saw in Example

1.4,A_‡is strictly larger thanSA².

As a matter of fact, the preorderA_‡of all psd polynomials is not ®nitely generated (as a preorder), if AˆR‰t₁;. . .;t_nŠ and nX2. An even stronger assertion is true:

A_‡is not the closure of any ®nitely generated preorder, with respect to the topology introduced below. This is a particular case of Thm. 3.7 from the next section.

Let A be any f. g.R-algebra. By a subspace ofA, we always mean anR-linear subspace of A. Recall that if W is any ®nite-dimensional (f.-d.) vector space over R, then asemialgebraic set inWis a ®nite boolean combination of sets of the form fxAW:f…x†>0g, wherefis a polynomial function onW. A subsetSofAwill be

called locally semialgebraic if SVU is a semialgebraic subset of U for every f.-d.

subspaceUofA. If, in addition,Sis contained in some f.-d. subspace ofA, thenSis calledsemialgebraic.

We will always equip A with the ®nest topology that makes A a locally convex topologicalR-vector space [4, II §4 no. 2]. Every subspace ofAis closed, and every linear mapA!Ris continuous (loc. cit., exercise 6). ByA⁴ we denote the dual of A, i.e., the space of all linear mapsA!R. A subset ofAis closed if and only if its intersection with every f.-d. subspace UofAis closed inU. (This uses thatAhas a countable linear basis, seeloc. cit., exercise 8.)

For preorders inA, we study the properties of being closed and of being saturated.

These will be key properties needed for our results on the moment problem. We begin with some simple observations.

2.2. Lemma.Any preorder in A is a convex cone in A.Any saturated preorder is closed in A.

Proof.The ®rst statement is obvious. For the second, note that ifPˆP…K†is satu- rated, thenPˆ7_MAKl^ÿ1_M‰0;y†, wherelM :A!Ris evaluation atM.

By the Hahn-Banach separation theorem for convex sets [4, II §5 no. 3], the closure Pof the preorderPis given by

Pˆ faAA:L…a†X0 for everyLAA⁴ withLX0 onPg:

However, in practice it is often not easy to give a more concrete description of P.

By Haviland's theorem, the problem of understanding P is in fact closely related to the moment problem, see 3.1 below. It is precisely this relation which leads us to introduce the topology onAand to study the closurePof a preorderP.

2.3. Lemma.If P is a preorder in A,then the closure P of P in A is again a preorder.

Moreover,S…P† ˆS…P†.

Proof. The multiplication map AA!A is continuous, sinceA has a countable linear basis [4, II exercise 9a]. Therefore, it is clear thatPis a preorder. Moreover, the saturated preorder Sat…P†is closed, and soPHPHSat…P†, which impliesS…P†I S…P†IS…SatP† ˆS…P†.

2.4. Proposition. Let K be a closed semialgebraic set in V…R†,and let PˆP…K†,the saturated preorder associated to K.Then P is a closed, convex,locally semialgebraic subset of A.

Proof. We already know that P is closed and convex. There is an epimorphism

p:R‰t₁;. . .;t_nŠ !!A, for somen, inducing an embeddingV…R†,!Rⁿ. Given a f.-d.

subspaceUofA, choose a f.-d. subspaceU⁰ofR‰t1;. . .;tnŠso thatp…U⁰† ˆU. Then U⁰Vp^ÿ1…P†consists of allf AU⁰which are non-negative on the closed semialgebraic subsetKofV…R†HRⁿ. SoU⁰Vp^ÿ1…P†is a semialgebraic subset ofU⁰, since it can be described by a formula in the ®rst order language of ordered ®elds.

2.5. Proposition.Let U be a f.-d. subspace of A.Then there is an integer pU such that every sum of squares of elements of U is a sum of pU squares of elements of U.

Proof.By [7, 4.2] this is true forAˆR‰t1;. . .;tnŠ. It is easy to see that this particular case implies the general one: Again use p:R‰t₁;. . .;t_nŠ !!A, and lift Uto a f.-d.

subspaceU⁰ ofR‰t₁;. . .;t_nŠ, as in the proof of Prop. 2.4. The result forU⁰ then im- plies the result forU.

Given a subspaceWofAandf₁;. . .;f_rAA, we denote byS…W;f₁;. . .;f_r†the set of all sums

X

iAf0;1g^r

sif₁ⁱ¹. . .f_r^ir

in which thesiare sums of squares of elements ofW. Recall that the ringAis said to bereducedifa²ˆ0 impliesaˆ0, for everyaAA.

2.6. Proposition. Let f₁;. . .;f_rAA,and let KˆS…f₁;. . .;f_r†.Let W be a f.-d. sub- space of A.Then

(a) S…W;f₁;. . .;f_r†is a convex semialgebraic subset of A.

(b) If A is reduced and K is Zariski dense in V,thenS…W;f₁;. . .;f_r†is closed.

Proof.We abbreviateS…W;f₁;. . .;f_r†byS…W†. It is clear thatS…W†is contained in some f.-d. subspaceUofA, and that it is a convex set. LetNˆpW, see Proposition 2.5.ForaAA, let Ann…a† ˆ fbAA:abˆ0g, the annihilator ofa. This is an ideal of A, and the ring A=Ann…a† is reduced if A is reduced. For iAf0;1g^r, write fⁱ:ˆf₁ⁱ¹. . .f_r^ir. Consider the map

f: 0

iAf0;1g^r

…W=WVAnn…fⁱ††^N !A

de®ned by

wˆ …wij†_iAf0;1g^r_;jˆ1;...;N7!f…w† ˆ X

iAf0;1g^r

X

j

w_ij²

fⁱ: Here w_ijˆw_ij‡ …WVAnn…fⁱ††; note that the right hand side is well-de®ned.

The map fis a homogeneous quadratic polynomial map, and S…W†is its image set. In particular, it is clear that S…W†is a semialgebraic set. Now assume that the conditions of (b) hold. We ®rst show f^ÿ1…0† ˆ f0g. So letwˆ …w_ij†be a tuple with f…w† ˆ0. If M is any point in K, we have fⁱ…M†X0 for every i, and therefore …w_ij²fⁱ†…M† ˆ0 for every bi-index…i;j†. SinceKis Zariski dense andAis reduced, it

follows that w_ij²fⁱˆ0, and hence even wijfⁱˆ0, for every bi-index …i;j†. Hence wˆ0.

Now the next lemma shows thatfis a proper map. In particular, the image offis closed.

2.7. Lemma. Suppose fˆ …f₁;. . .;f_n†:R^m!Rⁿ is a homogeneous map of some

®xed degree dX1 (i.e., each componentf_i is homogeneous of degree d).Iff^ÿ1…0† ˆ f0g,thenfis a proper map.

Proof.Recall that a continuous map is proper i¨ it is closed and has compact ®bres [5, ch. I §10 no. 2]. If Kis a closed subset of R^m with 0BK, then 0Bf…K†, since 0Bf…K† and f is homogeneous. Therefore, it su½ces to show that the restriction f⁰:R^mnf0g !Rⁿnf0goffis proper. Consider the commutative square

R^mnf0g ƒƒƒ^f0! Rⁿnf0g

r

??

?y

??

?y^r S^mÿ1 ƒƒƒ^f! S^nÿ1

…†

in which the vertical arrows are the natural retractionsx7!x=jxjandfˆrfjS^mÿ1. One checks immediately that the square…†is cartesian. SinceS^mÿ1is compact,fis a proper map, and thereforef⁰is proper as well.

2.8. Remark. In general, the conditions in Prop. 2.6(b) cannot be dropped. For example, let AˆR‰tŠ, f ˆ ÿt², Kˆ f0g andW ˆR:1lR:t, the space of poly- nomials of degreeW1. Here K is not Zariski dense inV. We havet‡eAS…W;f† for everye>0, buttBS…W;f†, soS…W;f†is not closed. For essentially the same example, seen from a di¨erent viewpoint, take the ®nite (non-reduced) algebra Aˆ R‰tŠ=…t²†andrˆ0 (nof_i). Heret‡eis a sum of squares inAfor everye>0, but not foreˆ0. Again,SA²ˆS…A;q†is not closed.

Let again A be an arbitrary f. g. R-algebra, and assume now that we are given

®nitely many elements f₁;. . .;f_rAA. Suppose we know that S…W;f₁;. . .;f_r† is closed for every f.-d. subspaceWofA. Under suitable conditions, see 2.6(b), this will be the case. LetPˆPO…f₁;. . .;f_r†, the preorder generated by thefi. We would like to ®nd a condition under which we can conclude that P is itself closed. One such condition is the following:

…*† For every f.-d. subspaceUofA, there is a f.-d. subspaceWofAwithPVUH S…W;f₁;. . .;f_r†.

Indeed,PVUˆS…W;f₁;. . .;f_r†VU then, which under our assumption is a closed subset ofU.

We are therefore going to study condition…*†more closely.

2.9. Lemma. Let f1;. . .;f_rAA, and write PˆPO…f₁;. . .;f_r†. Given gAP (so also PˆPO…f₁;. . .;f_r;g†),condition…*†holds for f1;. . .;f_ri¨ it holds for f1;. . .;f_r;g.

Proof. The ``only if '' is obvious. Conversely, assume …*†holds for f₁;. . .;f_r;g. Let U and W be f.-d. subspaces ofA with UVPHS…W;f₁;. . .;f_r;g†. Choose a f.-d.

subspace W⁰ of A which contains 1 and the f_i and in addition satis®es gA

S…W⁰;f₁;. . .;f_r†. LetWW⁰ be the subspace spanned by the products ww⁰ …wAW;

w⁰AW⁰†. ThenUVPHS…WW⁰;f₁;. . .;f_r†.

2.10. De®nition. Let PˆPO…f₁;. . .;f_r† be a ®nitely generated preorder in A. We say thatPisstableif…*†holds. By Lemma 2.9, this is independent of the choice of generators ofP.

Note that if a f. g. preorder Pis stable, then it is a locally semialgebraic subset ofA.

2.11. Corollary. Let P be a ®nitely generated preorder in A which is stable. Assume that A is reduced and that KˆS…P†is Zariski dense in V.Then P is closed in A.

Proof.Choosef₁;. . .;f_rAAwithPˆPO…f₁;. . .;f_r†. ThenS…W;f₁;. . .;f_r†is closed for each f.-d. subspaceWofA(2.8). SincePis stable,Pis closed.

2.12. Remark.Prof. SchmuÈdgen informs us that a condition similar to…*†has been used in the context of general star algebras to prove that the positive cone is closed.

See [22], condition (III) on page 326.

2.13. Remark. This remark is for readers with a little background in real algebraic geometry. It won't be used in the sequel.

LetRbe any real closed ®eld, and letAbe a ®nitely generatedR-algebra. Consider the topology onAwhich is analogous to the one we use forR-algebras: A subset ofA is open i¨ its intersection with every f.-d.R-linear subspace ofAis open in this sub- space. Prop. 2.6 and Lemma 2.9 hold in this context as well,mutatis mutandis. Hence it is clear what we mean by saying that a ®nitely generated preorder inAis stable.

Now one can give the following di¨erent characterizations of stable preorders, which explain the reason for our choice of the word ``stable''. Given a real closed

®eldR, a ®nitely generatedR-algebraAand a ®nitely generated preorderPinA,Pis stable if and only if either one of the following two conditions is satis®ed:

(i) For every real closed ®eld extensionR⁰=R, the preorder inA⁰ˆAn_RR⁰generated byPis a locally semialgebraic subset ofA⁰;

(ii) Pis a locally semialgebraic subset ofA, and for every real closed extensionR⁰ofR, the preorder inA⁰generated byPis equal toP…R⁰†.

Here in (ii), we mean byP…R⁰†the subsetQofA⁰for whichQV…Un_RR⁰†is the base extension (toR⁰) of theR-semialgebraic setPVU, for every f.-d.R-linear subspaceU ofA.

We return to our usual setting, and assume that Ais a f. g.R-algebra. The next theorem, though it may appear somewhat technical, is the ®rst main result of our paper. In the next section we will apply it to concrete particular cases. It provides us with a large family of examples of ®nitely generated preorders which are stable and closed. On the other hand, if we add a suitable dimension hypothesis, they won't be saturated, by a result from [19].

Recall that VˆSpecAis called normalif Ais a direct product of ®nitely many integrally closed domains. This is a mildness condition on the nature of the singu- larities ofA.

2.14. Theorem. Suppose that the variety VˆSpecA is normal. Let P be a ®nitely generated preorder in A and set K ˆS…P†.Assume that V has an open embedding into a normal complete R-variety V such that the following is true: For any irreducible component Z of V ÿV, the subset KVZ…R†of Z…R†is Zariski dense in Z, where K denotes the closure of K in V…R†.Then the preorder P is stable and closed.

2.15. Example.To illustrate this, consider the polynomial ringA:ˆR‰t1;. . .;tnŠand the preorderPˆSA²of all sums of squares inA. The preorderPis stable (and there- fore closed). Indeed, if a polynomial fis a sum of squares, say f ˆP

if_i², and iff has (total) degreeWd, then it is obvious that the f_i must have (total) degreesW^d₂, since leading terms cannot cancel. The theorem, and its proof, are a generalization of this simple remark.

For the proof of the theorem, we need to use some (easy) ideas from algebraic ge- ometry and from real algebraic geometry. For the ®rst we refer to Hartshorne's book [10], for the second to [3] or [14]. In particular, we need the notion of the real spec- trum SperAofA, and how it relates to the semialgebraic subsets ofV…R†.

Proof. We can assume that V is irreducible. Fix an irreducible component Z of VÿV, thenZhas codimension one inV, and hence de®nes a discrete valuationvZ

ofR…V†, the function ®eld ofV. Namely,vZ…f†is the (vanishing, resp. pole) order of falongZ. The residue ®eld ofvZisR…Z†, the function ®eld ofZ.

The condition in the theorem implies that Z…R† is Zariski dense in Z. It is well known that this is equivalent to the condition that the function ®eld R…Z†of Z is (formally) real. Therefore, the valuationv_Zhas a real residue ®eld.

The subset Z…R†VK of Z…R† is semialgebraic. Therefore, the hypothesis that Z…R†VK is Zariski dense in Z means that the associated constructible subset …Z…R†VK†^@ in the real spectrum of Z contains an element whose support is the generic point of Z. This, in turn, means that the constructible set K~ of SperR‰VŠ

contains an elementawith support (0) which is compatible with the valuationv_Z. Let now Z1;. . .;Zr be the irreducible components of VÿV, and let vZi be the discrete valuation ofR…V†associated toZi, as above. FornX0, let theR-subspace UnofAbe de®ned by

Un ˆ ff AA:vZi…f†Xÿnforiˆ1;. . .;rg:

Then dimR…Un†<yfor eachn. This is a particular case of [10, Thm. II.5.19], noticing that UnˆG…V;O_V…nD††, where Dis the Weil divisor DˆP

iZi on V. Moreover, U0HU1H H 6_nUnˆA.

Now assume thata₁;. . .;a_mAPare such thata₁‡ ‡a_mAU_n, for somenX0.

Eacha_iis positive ina. Sinceais compatible withv_Zi, for eachi, it follows from [19, Lemma 0.2] that v_Zi…P

ja_j† ˆmin_jv_Zi…a_j†, for each i. In particular, each summand a_jlies itself inU_n.

From this observation it is easy to see that the preorderPis stable. Indeed, ifPˆ PO…f₁;. . .;f_r†, and if aAAand 1Wj₁< <j_sWr are such that a²f_j1. . .f_js AU_n, then v_Zi…a²†XÿnÿP_s

kˆ1v_Zi…f_jk† for each i, and so it is clear that PVU_nH

S…U_N;f₁;. . .;f_r† for su½ciently large N. (Explicitly, it su½ces to take 2NXn‡

maxiP_r

jˆ1mij, wheremijˆmaxf0;vZi…f_j†g.)

From the hypotheses, it follows thatKis Zariski dense inV. ThereforePis closed by Corollary 2.11.

2.16. Remark.Observe that the hypotheses in the theorem depend only onVandK, but not on the particular choice of the preorder P. In the next section we will apply the theorem to exhibit classes of basic closed sets which admit no ®nite presentation by non-strict inequalities which would solve the moment problem.

In the next section, we will apply the following result proved in [19, Prop. 6.1]:

2.17. Proposition. Let A be a f. g.R-algebra, let V ˆSpecA,and let P be a ®nitely generated preorder in A. Assume that KˆS…P† has (topological) dimensionX3.

Then there exists f AR‰VŠ with fX0 on V…R† but f BP. In particular, P is not saturated.

2.18. Example. Let P be a f. g. preorder in A for which KˆS…P† is compact. If K has dimensionX3, then, by 2.17, Pis not saturated, and hence is not closed by SchmuÈdgen's theorem (c.f. 3.2 below). In particular, ifAis reduced andKis Zariski dense inV,Pcannot be stable, by 2.11.

If dim…K†W2, however, the question of stability forPis less clear, sincePmay be saturated. We mention a non-trivial example: LetCbe a smooth a½ne curve overR for whichC…R†is compact. LetPˆSR‰CŠ², the preorder of all sums of squares. It can be shown thatPis saturated [20]. In any case,Pis stable if, and only if, for each integerdX0, there is an integerN…d†X0, such that every sum of squaresf AR‰CŠ

whose pole orders (in the points at in®nity) are Wd is a sum of squares of regular functions whose pole orders are WN…d†. We do not know whether this condition always holds. One can show that it does indeed hold ifC has genus one, using the approach from [19, §4].

3 Applications to the moment problem

We use the previous results on preorders to study the questions raised in §1. First, we can rephrase Haviland's theorem (1.1) in our terminology as follows:

3.1. Corollary.Suppose P is a preorder inR‰t1;. . .;tnŠand KˆS…P†.Then P solves the moment problem for K if and only if the closure P of P is a saturated preorder.

Proof. IfPis not saturated, then there is a polynomialfwithfX0 onKbutf BP.

The argument of 1.4 now generalizes: SincePis a convex cone, there exists, by Hahn- Banach separation, a linear formL:R‰t₁;. . .;t_nŠ !RwithLX0 onP, butL…f†<

0. Obviously,Lcannot come from a positive Borel measure onK. Conversely, every linear mapLwithLX0 onPsatis®esLX0 onP. So ifPis saturated,Psolves the moment problem forKby Haviland's theorem.

3.2. Example.Let Abe a f. g. R-algebra andPa f. g. preorder inAfor whichKˆ S…P†is compact. Then the saturation Sat…P†ofPis equal to the closurePofPinA.

Indeed, Sat…P†is closed by 2.4. Conversely, for everyf ASat…P†ande>0 one has f ‡eAP, by SchmuÈdgen's theorem (1.3), and sof AP.

3.3. Corollary. Suppose P is a preorder inR‰t1;. . .;tnŠand K ˆS…P†.If P is closed but not saturated,then P does not solve the moment problem for K.

3.4. Remark.For the question of whether a given preorder solves the moment prob- lem, we can often pass from a½nen-space to a smaller algebraic variety; and, as we will see, it is in fact useful to do so. The reason is the following.

Let P be a preorder in AˆR‰t₁;. . .;t_nŠ, and let KˆS…P†. The supportof P is the ideal supp…P†:ˆPV…ÿP†ofA. WriteB:ˆA=supp…P†, and letp:A!Bbe the natural epimorphism. ThenVˆSpec…B†is a closed subvariety ofAⁿ, andKis con- tained inV…R†. IfPis ®nitely generated, thenKis Zariski dense inV. This is seen as follows: Let f AA with f10 on K. By the real Nullstellensatz (e. g. [14, p. 143]), there is an identityf^2m‡aˆ0 inA, withmX1 andaAP. Thereforef^2mAsupp…P†, and so the elementp…f†is nilpotent inB. Note however that the ringBneed not be reduced in general.

Let Qˆp…P†, the preorder inB induced byP. The linear maps L:A!Rwith LX0 on P are precisely those of the formLˆLp, where L:B!R is a linear map withLX0 onQ. Therefore, the closures ofP(inA) and ofQ(inB) are related byPˆp^ÿ1…Q†. On the other hand, the respective saturations are obviously related by SatA…P† ˆp^ÿ1…SatB…Q††. Therefore,Psolves the moment problem forK,Pˆ Sat_A…P† ,p^ÿ1…Q† ˆp^ÿ1…Sat_B…Q†† ,QˆSat_B…Q† , the closure of Q in B is saturated.

Conversely, if Iis an ideal in AˆR‰t₁;. . .;t_nŠ, if BˆA=I andp:A!Bis the canonical map, and ifQis a preorder inB, then starting from a system of generators forQit is easy to obtain a system of generators for the preorderP:ˆp^ÿ1…Q†ofA.

Indeed, ifQis generated by elementsp…f_l†,lAL, withfl AA, thenPis generated by the fl together with Ga1;. . .;Gam, where a1;. . .;am is a system of generators for the ideal I. In particular,Q is ®nitely generated i¨P is ®nitely generated. Summa- rizing some of the above discussion, we have:

3.5. Lemma.Let P be a preorder in AˆR‰t1;. . .;tnŠ,let KˆS…P†,and let Q be the preorder induced by P in BˆA=supp…P†.Write V ˆSpec…B†.

(a) P solves the moment problem for K(inRⁿ)if and only if the closure of Q in B is saturated.

(b) P is ®nitely generated if and only if Q is ®nitely generated.In either case,the subset K of V…R†is Zariski dense in V.

3.6. Corollary.Let K be a basic closed semialgebraic subset ofRⁿ.Let I be the ideal of all polynomials in AˆR‰t1;. . .;tnŠwhich vanish identically on K,and put BˆA=I.

The following two conditions are equivalent:

(i) There exists a ®nite family of polynomials which solves the moment problem for K;

(ii) the saturated preorder P…K† in B contains a dense preorder which is ®nitely generated.

In view of the lemma and its corollary, Thm. 2.14 allows us to obtain many exam- ples of basic closed semialgebraic setsKfor which the moment problem is not solved by any ®nite family of polynomials.

3.7. Proposition. Let f1;. . .;f_rAR‰t1;. . .;tnŠ be such that K ˆ ff₁X0;. . .;f_rX0g contains an open cone(i.e.,there are xARⁿand a non-empty open subset U ofRⁿwith x‡luAK for everylX0and uAU).Then the preorder PˆPO…f₁;. . .;f_r†is stable and closed.If nX2,P is not saturated,and hence f₁;. . .;f_r do not solve the moment problem for K.

Proof. By Thm. 2.14, P is stable and closed, and by Prop. 2.17,P is not saturated if nX3. Consider the case nˆ2 (c.f. also [19, Rem. 6.7]). We can ®nd a smooth irreducible curveCin the a½ne plane, of genusX1, which has exactly one point at in®nity and for which the setKVC…R†is unbounded. Indeed, after an a½ne change of coordinates, Kcontains the positive quadrant, and we can take C to be the curve t₂²ˆt₁…t₁²‡1†, for example.

Let p:R‰t₁;t₂Š !!R‰CŠbe the natural (restriction) homomorphism. By [19, Cor.

3.9], there exists a psd functiongAR‰CŠwhich is not contained inp…P†, the preorder induced by P in R‰CŠ. By loc. cit., Thm. 5.6, g can be lifted to a psd polynomial f AR‰t₁;t₂Š, i.e.,p…f† ˆg. Clearlyf BP, and in particular,Pis not saturated.

3.8. Remarks.1. The fact thatPis closed is also proven by Kuhlmann and Marshall in [15, Thm. 3.5]. Applying results from [19], they deduce from this that thefido not solve the moment problem fornX2, in the same way that we do.

2. In contrast to 3.7, it is not enough to assume that Kcontains a cylinder. For example, in [15, 5.1], it is shown ifKis a cylinder inR² with compact cross section, then there is a ®nite set of polynomials which solves the moment problem forK.

We recall some terminology: IfCis a smooth irreducible a½ne curve overR, there exists (up to isomorphism) a unique smooth irreducible projective curve C overR which containsCas a Zariski open subset, i.e., for which there is a ®nite subsetTof Csuch thatCGCnT. The points inTare called thepoints at in®nityofC. They are calledrealornonrealaccording to whether they areR-rational or not.

We can now give a complete solution to the moment problem for one-dimensional closed semialgebraic sets K contained in a smooth curveC. There are two possible situations, depending on properties ofCandK. We start with the ®rst case (for the remaining case, see 3.12 below):

3.9. Theorem. Let C be a smooth a½ne curve overR,of genus gX1,and let P be a f. g. preorder inR‰CŠ.Put K ˆS…P†.Assume that every point of C at in®nity is real, and is contained in the closure of K(inside C…R†).Then P is closed and stable,but not saturated.

Proof. P is stable and closed by Thm. 2.14. On the other hand, by [19, Thm. 3.5], there exists a psd functionf AR‰CŠwhich is not contained in P. In particular,P is not saturated.

In particular, ifCis a smooth a½ne curve overR, of genusgX1, whose points at in®nity are all real, and ifKHC…R†is a closed semialgebraic set whose closure in C…R†contains all points at in®nity, then the moment problem forKis not solvable by ®nitely many polynomial functions.

We can even go further and, using a restriction-extension argument, generalize this last fact considerably. The following corollary was inspired by, and generalizes, [15, Cor. 3.10]:

3.10. Corollary.Let K be a basic closed semialgebraic subset ofRⁿ.Assume that there exists a smooth curve C inAⁿ,of genusX1,all of whose points at in®nity are real and are contained in the closure of KVC…R†(insidePⁿ…R†).Then the moment problem for K is not solvable by ®nitely many polynomials.

Proof. Let Pbe a f. g. preorder inAˆR‰t1;. . .;tnŠ withS…P† ˆK. We show that the closurePis not saturated. Letp:A!R‰CŠbe the natural epimorphism, and let Qˆp…P†, the preorder induced by P on C. Then Q is ®nitely generated and has S…Q† ˆKVC…R†. By [19, Thm. 3.5], there existsgAR‰CŠwithgX0 onC…R†but gBQ. The latter is witnessed by a linear map L⁰:R‰CŠ !R with L⁰…g†<0 but L⁰X0 onQ. The linear mapL:ˆL⁰p:A!Rsatis®esLX0 onP. Byloc. cit., Thm. 5.6, we can ®nd an everywhere non-negative polynomialf AAwithp…f† ˆg.

In particular,f ASat…P†, butf BPsinceL…f†<0.

On the other hand, there are the following positive results for the moment problem on curves. We state them without proofs here, referring instead to [20].

3.11. Theorem ([20]).Let C be an irreducible smooth a½ne curve over R,let f₁;. . .; f_rAR‰CŠ, and put K ˆ ff₁X0;. . .;f_rX0gHC…R†. Suppose that the following three conditions are satis®ed:

(1) Either C has a nonreal point at in®nity,or it has a real point at in®nity which does not lie in the closure of K(inside C…R†);

(2) f1. . .frhas vanishing orderW2in each point of K;

(3) if M is an isolated point of K,thenordM…f_i†W1for every i.

Then the preorderPO…f₁;. . .;f_r†inR‰CŠis saturated.

3.12. Theorem([20]).Let C be an irreducible smooth a½ne curve overR.Let K be a closed semi-algebraic subset of C…R†, and let PˆP…K†, the saturated preorder in R‰CŠassociated with K. Assume that at least one of the following three conditions is satis®ed:

(i) C is rational;

(ii) C has at least one nonreal point at in®nity;

(iii) C has a real point at in®nity which does not lie in the closure of K(in C…R†).

Then the following are true:

(a) P is ®nitely generated. In particular,the moment problem for K can be solved by

®nitely many functions.

(b) If either(ii)or(iii)holds,then P is in fact generated by two elements,and even by one element if K has no isolated points.

Sketch of proof.In case (i), one can reduce toCˆA¹, i.e. to the polynomial ringR‰tŠ.

In this case the proof becomes elementary, see [15, Thm. 2.2], for example. For the remaining cases one has, by Thm. 3.11, to show that one can ®nd polynomial func- tions f₁;. . .;f_rAR‰CŠ such thatKˆ fMAC…R†:f₁…M†X0;. . .;f_r…M†X0g, and such that thef_i satisfy conditions (2) and (3) from 3.11; moreover, that this is even possible withrˆ2 orrˆ1, respectively. This can be achieved using methods similar to those used in [19, §2]. For details, we refer to [20].

3.13. Remark.Observe that the condition that at least one of (i)±(iii) holds in 3.12 is the precise complement of the condition in 3.9. Therefore, taking together 3.9 and 3.12, we have obtained a complete answer to the question whether the moment prob- lem forKis solvable by ®nitely many polynomials, in the case whenKis contained in a smooth curve.

3.14. Example.We illustrate our results on curves by a series of examples.

1. Let Cbe an irreducible smooth a½ne curve over R, embedded into Aⁿ as a Zariski closed subset, for some n. Let Iˆ …g₁;. . .;g_m† be the vanishing ideal ofC insideR‰t₁;. . .;t_nŠ, and letKbe a closed semialgebraic subset ofC…R†. Assume that CandKsatisfy (ii) or (iii) from 3.12. Then the moment problem forK(considered as a closed subset ofRⁿ) can be solved by 2m‡2 polynomials, and even by 2m‡1 if K does not contain an isolated point. Namely, it su½ces to take two polynomials whose restrictions to C generate PˆP…K† in R‰CŠ, together with Gg1;. . .;Ggm. If KˆC…R† (soC has a nonreal point at in®nity), the moment problem for K is solved byGg1;. . .;Ggm alone. Both assertions follow from 3.12.

2. For an explicit example, let C be the plane a½ne curve with equation f…x;y† ˆx³‡y³‡1ˆ0. Then C is smooth, of genus one, and has 3 geometric points at in®nity, of which one is real and the other two are complex conjugate.

According to the remark before, the moment problem can be solved by W4 poly- nomials for any closed semialgebraic subset K of C…R†HR². Using the criterion from 3.11, it is easy to verify, for example, that the moment problem for K₁ˆ C…R†VfxX0gis solved byx,f,ÿf, that the moment problem forK₁Uf…ÿ1;0†gis solved byx‡y‡1,ÿy,Gf, and that the moment problem forKˆC…R†is solved byGf.

3. LetCbe the plane hyperbola with equationy²ˆx²‡1. The moment problem for the set C…R†HR² is solved by G…y²ÿx²ÿ1†. The moment problem for C…R†VfxX0;yX0gis solved byx‡yÿ1,G…y²ÿx²ÿ1†; and so on.

4. Letq…x†AR‰xŠbe a square-free polynomial, and letKbe a closed semialgebraic subset of C…R†:ˆ f…x;y†AR²:y²ˆq…x†g. If deg…q†W2, then the moment prob- lem for Kcan be solved by ®nitely many polynomials, since the curve y²ˆq…x† is rational. If deg…q†X3 andKis not compact, the moment problem forKcannot be solved by ®nitely many polynomials (3.9). In the caseKˆC…R†, this is also shown directly in [15, 3.7].

5. Generalizing part of the last example, let C be an irreducible smooth plane curve with equation xⁿ‡f…x;y† ˆ0, where every monomial in f…x;y† has total degreeWnÿ1. IfnX3, then the moment problem cannot be solved by ®nitely many polynomials for any non-compact closed semialgebraic subset ofC…R†(in particular, for C…R† itself, since C…R† is not compact). Indeed, C has precisely one point at in®nity, and this point is real; moreover, Cis not rational. So the assertion follows from 3.9.

6. We remark that our questions from the introduction are completely settled for (closed semialgebraic) subsetsKofR, in [15, §2].

Finally, we can settle some of the open problems raised in [15]. Open Problem 4 asks for an irreducible smooth a½ne curve overRfor which the sums of squares are not closed. Such a curve cannot exist:

3.15. Corollary. If C is an irreducible smooth a½ne curve over R, then the preorder SR‰CŠ²of all sums of squares is closed inR‰CŠ.

Proof.IfChas a nonreal point at in®nity, thenSR‰CŠ²is even saturated ([20]; ifCis not rational, this is also contained in Thm. 3.11 as a special case; ifCis rational, it is clear anyway). If all points ofCat in®nity are real, 3.9 applies.

We can also answer Open Problems 6 and 7 from [15]. Namely, the preorders P1:ˆPO…x;1ÿx†andP2:ˆPO…1‡x;1ÿx;1‡y;1ÿy†inR‰x;yŠare both satu- rated, and in particular, are closed. This is proved in [20]. Problem 7 asked whether P2 is closed. Problem 6 asked whether for everyf ASat…P1†it is true thatf ‡eAP1

for alle>0.

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Received 19 October, 2000.

V. Powers, Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA

E-mail: [email protected]

C. Scheiderer, FakultaÈt fuÈr Mathematik, UniversitaÈt Duisburg, 47048 Duisburg, Germany E-mail: [email protected]