Bounded Polynomials, Sums of Squares, and the Moment Problem

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Bounded Polynomials, Sums of Squares, and the Moment Problem

Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften am Fachbereich Mathematik und Statistik der Universität Konstanz

Daniel Plaumann

Tag der mündlichen Prüfung: 28. April 2008 Referenten: Prof. Dr. Claus Scheiderer

(Universität Konstanz) Prof. Dr. Murray Marshall (University of Saskatchewan)

Konstanzer Online-Publikations-System (KOPS)

URL: http://www.ub.uni-konstanz.de/kops/volltexte/2008/5579/

URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-55796

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Daniel Plaumann

Fachbereich Mathematik und Statistik Universität Konstanz

78457 Konstanz

Daniel.Plaumann@uni-konstanz.de

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Gewidmet meinen Eltern Marina und Michael Plaumann und meinem Onkel Peter Plaumann

in tiefer Dankbarkeit

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Acknowledgements

I would like to express my gratitude to my advisor Claus Scheiderer for the many ideas and insights into the topic that he shared with me, as well as for his constant encouragement. I am grateful for many interesting and helpful discussions with my colleagues in Konstanz and with people from the real algebraic community. Here, I want to name Sebastian Krug, Salma Kuhlmann, Murray Marshall, Tim Netzer, Alexander Prestel, Rainer Sinn, and Markus Schweighofer. I also wish to thank Wolf Barth, Fabrizio Catanese, and Michael Thaddeus, who have helped me understand some of the underlying algebraic geometry. From 2004 to 2006, I greatly profitted from the support of the RAAG research network, in particular during a wonderful six months stay in Paris and Rennes. For this I want to thank Michel Coste, Max Dickmann, Danielle Gondard, and Niels Schwartz.

Konstanz, January 2008 Daniel Plaumann

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INTRODUCTION

Leth_, . . . ,h_r be polynomials innvariablesx = (x_, . . . ,x_n)with real coefficients, and let

S = {a∈Rⁿ∣h(a) ⩾, . . . ,hr(a) ⩾}

be the basic closed semialgebraic subset determined by them. We study the cone

P (S) = {f ∈R[x] ∣ ∀a∈S∶f(a) ⩾}

of all polynomials that are non-negative onS, in particular how it relates to the cone (more accurately the preordering) generated byh, . . . ,hr:

T =PO(h_, . . . ,h_r) = { ∑

i∈{,}^r

s_ih_ⁱ^⋯hⁱ_r^r ∣s_i ∈ ∑R[x]^},

where∑R[x]^ = {∑^k_j=g^_j ∣g, . . . ,gk ∈ R[x],k ⩾ }is the set of all sums of squares inR[x]. The inclusionT ⊂ P (S)is obvious, and the natural question is whether equality holds or, more generally, to what extent something close to equality can be established. Questions of this kind have been extensively studied within the last 15 years. A classical result of Hilbert says that not every non-negative polynomial onRⁿ is a sum of squares inR[x]ifn⩾. More generally, Scheiderer has shown thatP (S)is not finitely generated, i.e. cannot coincide with a cone of the form PO(h_, . . . ,h_r), if S has dimension at least

. Equality betweenT andP (S)is a phenomenon of lower dimensional cases only. On the other hand, weaker statements can be proved in arbitrary dimensions: In 1991, Schmüdgen proved that if Sis compact, then every polynomial that is strictly positive on S is contained inT. More generally, he defined the moment property forT based on the moment problem of functional analysis, which he was studying. It requires thatTbe dense inP (S)in the sense thatT andP (S)cannot be separated by any linear functional. It can also be thought

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 INTRODUCTION

of as an approximation property for elements in P (S)by elements inT (see section 2.1 for precise statements and references for the above results).

Concerning the moment property, Schmüdgen went on to prove a much stronger version of his theorem, namely

Theorem (Schmüdgen 2003) — Let φ∶Rⁿ → R^m be a polynomial map such that the image φ(S)is bounded inR^m. Then T has the moment property if and only if the restriction of T to each fibre φ^−(a), a∈R^m, has the moment property.

(The restriction of a preordering is defined in section 2.1). In order to apply Schmüdgen’s theorem in a given situation, one must first know if any mapφas in the statement exists. We therefore consider the subring ofR[x]of polynomials that are bounded onS:

B(S) = {f ∈R[x] ∣ ∃λ∈R∀a∈S∶ ∣f(a)∣ ⩽λ}.

Clearly, B(S) = R[x] holds if and only if S is compact. More generally, the size of the ring B(S)can be regarded as a measure for the “compactness” of the setS. Schmüdgen’s theorem indicates that the bigger the ring B(S)is, the closer T should be toP (S). This point of view has been suggested to me for this thesis. Apart from Schmüdgen’s theorems, it is also motivated by results in the one-dimensional case, i.e. when S is contained in an algebraic curve.

Kuhlmann, Marshall, and Scheiderer have almost completely answered all of the above questions in this situation and found a dichotomy between the cases B(S) =RandB(S) ≠R.

We give a brief overview of the questions and results in this work:

(1) One task is to determineB(S), in particular, to decide whetherB(S) =R or B(S) ≠ R. This is achieved to some degree by interpretingB(S)as the ring of polynomial functions on a suitable algebraic compactification of Rⁿ (Prop. 1.12). This construction is carried out in chapter , but we can only prove the strongest result if S has dimension at most

 (Thm. 1.22). We also address the question whether B(S)is a finitely generatedR-algebra. We prove that this is true ifS is sufficiently regular and of dimension at most , while in higher dimensions there exist otherwise well-behaved counterexamples (section 1.6).

(2) If B(S) = R, the statement of Schmüdgen’s theorem becomes void, so some other means of deciding the moment problem for T have to be found. Here we use (and somewhat extend) a result due to Powers and

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INTRODUCTION 

Scheiderer showing that ifB(S) =R, thenTcan often be shown to be sta- ble, which roughly means the existence of degree bounds for represen- tations inT. This preventsT from having the moment property. These notions and results are explained in section 2.2.

(3) IfB(S) ≠R, then Schmüdgen’s theorem can be applied, and by restrict- ing to each fibre one can in principle hope to decide the moment problem by an inductive process. We try to make this as explicit as possible in the case whenS is of dimension . The fibres are then typically curves, and one can make use of Scheiderer’s extensive results for irreducible curves.

But even in good cases, the curves that come up as fibres need not be irreducible. Therefore, we try to extend Scheiderer’s results to the reducible case. This is completely achieved for the moment problem for sums of squares (section 3.2) while for general preorderings we only treat some special cases (section 3.3). We apply these results in chapter 4 to obtain new examples and criteria in dimension .

(4) Beyond the moment problem, we also prove a number of results about saturatedness of finitely generated preorderings, i.e. conditions that im- plyP (S) = PO(h_, . . . ,h_r)whenS is of dimension  or . Namely, for curves the above mentioned results in the reducible case often concern this stronger property. In dimension , we reprove a theorem by Rog- gero concerning the divisor class group of a real variety (Thm. 4.9) and combine this with Scheiderer’s results about saturated preorderings in the -dimensional compact case. This leads to a systematic way of pro- ducing new non-compact examples in which the correspoding finitely generated preordering is saturated (Thm. 4.12).

(5) In some one-dimensional cases, the solvability of the moment problem forT =PO(h, . . . ,hr)is known to depend on the choice of generators h, . . . ,hr. Now if suitable generators can be found for the restriction of Tto the fibres ofφ, this poses the problem of lifting these generators to T. It comes down to the following general extension problem: Given a closed subvarietyZinRⁿ(or any ambient real variety) and a polynomial f ∈R[x]such that f is non-negative onS∩Z, does there existg∈R[x] such thatgis non-negative onS and such that f andgagree onZ? We extend a result of Scheiderer for the case whenZis a non-singular curve from global positivity to positivity on a semialgebraic set (Thm. 3.35).

Two technical notes are in place: In this introduction, we have put the ini- tial situation into affine space, i.e. S ⊂ Rⁿ, h_, . . . ,h_r ∈ R[x]. But it is much

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 INTRODUCTION

more natural and technically convenient to replace affine space by the Zariski closure V of S, an affineR-variety. The principal advantage is that S always has non-empty interior withinV(R), a hypothesis that is needed most of the time. The other remark concerns the ground field: Some of the above results use the topology and the archimedean property of the classical real numbers while others work over any real closed field. This distinction is important for connections to model theory, and while we do not make use of it at all, we will require the classical real numbers only when necessary.

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CHAPTER 

BOUNDED FUNCTIONS ON REAL VARIETIES

.. REAL VARIETIES

In this chapter,Rwill always denote a real closed field (e.g. the field of real numbers, which will be denoted by the boldface letterR). By anR-variety we mean a reduced, separated scheme of finite type over SpecR, not necessarily irreducible. The use of schemes is convenient in some places but not essential.

We now briefly establish notation and terminology for affineR-varieties in the simplest way and refer the reader to the appendix for a complete list of notations and some generalities on real varieties and semialgebraic sets.

Given polynomials f, . . . ,fr ∈R[x, . . . ,xn], letI=

√

(f, . . . , fr)be the rad- ical ideal inR[x_, . . . ,x_n]generated byf_, . . . ,f_r. PutV =Spec(R[x_, . . . ,x_n]/I) andR[V] = R[x_, . . . ,x_n]/I. ThenV is called an affineR-variety andR[V]its ring of regular functions. We writeV(R)for{a∈Rⁿ∣ f_(a) = ⋯ = f_r(a) =}, the set of real points ofV and considerR[V]as a ring of functions onV(R). We will usually identifyV(R)with the subset of points ofV that have residue fieldR. WriteK =R(

√

−), then the idealI(and hence the ringR[V]) can be recovered fromV(K) = {a∈ Kⁿ∣ f_(a) = ⋯ = f_r(a) = }, the set of complex points ofV, namelyI = I (V(K)) = {f ∈R[x, . . . ,xn] ∣ ∀a∈V(K)∶ f(a) =} by Hilbert’s Nullstellensatz. Thus we could work with justV(R)andV(K)and forget about the schemeV. It should be pointed out that our notion ofR-variety is different from the one often used in real geometry where only the set of real points is considered (for example in Bochnak, Coste, and Roy [6]). We say that anR-variety is real if it has a non-singularR-rational point (see also Prop. A.1).

IfV andW are two affineR-varieties andφ∶V →W is a morphism of varieties (i.e. a polynomial map), thenφinduces a mapV(R) →W(R)which we denote by φR. Furthermore,φ induces a homomorphismφ^#∶R[W] → R[V]

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 CHAPTER . BOUNDED FUNCTIONS ON REAL VARIETIES

ofR-algebras by the rule f ↦ f ○φ. The mapφ↦φ^#yields an equivalence of categories between finitely generated reducedR-algebras and affineR-varieties.

The setV(R)comes equipped with two topologies, theR-Zariski-topology, where the closed sets are given by the vanishing of polynomials with coefficients inR, and the strong topology induced by the ordering ofR(which is the euclidean topology forR=R). We will often consider the closure of subsets of V(R)with respect to either of these topologies. To avoid confusion, the closure ofS ⊂V(R)with respect to the Zariski topology will be denoted byS(also for subsets ofV) while the closure with respect to the strong topology will be denoted by clos_V_(R)(S)or just clos(S).

A subsetSofV(R)is called basic closed if there existh_, . . . ,h_r ∈R[V]such thatS= {a∈V(R) ∣h_(a) ⩾, . . . ,h_r(a) ⩾}. The setSis called semialgebraic if it can be written as a finite boolean combination of basic closed sets.

.. THE RING OF BOUNDED FUNCTIONS

LetVbe an affineR-variety andSa semialgebraic subset ofV(R). A function f ∈R[V]isbounded on S (over R)if there existsλ∈Rsuch that−λ⩽ f(a) ⩽λ holds in all pointsa∈S. The functions inR[V]that are bounded onSclearly form a subring ofR[V]. In this section, we will gather some fundamental properties of this ring.

Notation — We write

BV(S) = {f ∈R[V] ∣ ∃λ∈R∀a∈S∶ ∣f(a)∣ ⩽λ}

for thering of bounded functions on S (over R). In the caseS =V(R), we will also writeB(V)instead ofBV(V(R)).

Proposition 1.1 — Let V be an affine R-variety, Z a closed subvariety of V with vanishing ideal I_V(Z) ⊂ R[V], and let S, S^′ be two semialgebraic subsets of V(R).

(1) B_V(S∪S^′) =B_V(S) ∩B_V(S^′);

(2) B_V(S∩Z(R))/I_V(Z) =B_Z(S∩Z(R)); (3) B_V(S) =B_V(clos(S));

(4) B_V(S) =R[V]if and only ifclos(S)is semialgebraically compact.

Proof — (1) and (2) are immediate. (3) comes from the fact that we only consider functions that are regular on all ofV(R)and hence extend continuously to clos(S). (4) If clos(S)is semialgebraically compact, then its image under

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.. THE RING OF BOUNDED FUNCTIONS 

any continuous semialgebraic map is again semialgebraically compact, and in particular, any regular function onV is bounded onS. Conversely, ifB_V(S) = R[V], this implies thatx_^+ ⋯ +x_n^is bounded on S for any embedding ofV into affine space with coordinates x_, . . . ,x_n, thus clos(S)is semialgebraically

compact. ^◻

Proposition 1.2 — The ring B_V(S)is integrally closed in R[V].

Proof — The proof is well-known in the theory of real holomorphy rings. Sup- pose that f ∈ R[V] satisfies an equation fⁿ +b_n−f^n− + ⋯ + b_ =  with b_i ∈B_V(S),i=, . . . ,n−. Now we apply the following basic

Lemma — Let f = xⁿ +a_n−x^n−+ ⋯ +a_ ∈ R[x]. Every real root ξ ∈ R of f satisfies

∣ξ∣ ⩽max{,∣a_∣ + ⋯ + ∣a_n−∣}.

Proof — Ifξ≠, thenξ= −(a_n−ξ^−+⋯+a_ξ^−n). Apply the triangle inequality

to complete the proof. ^◻

In our situation, this gives

∣f(a)∣ ⩽max{,∣b_(a)∣ + ⋯ + ∣b_n−(a)∣}

for alla∈V(R). Since∣λ∣ ⩽+λ^for anyλ∈R, it follows that

∣f(a)∣ ⩽ (n++b_^(a) + ⋯ +b^_n−(a))

for alla∈V(R), so∣f∣is bounded onSby an element ofRsince the right hand

side is. ^◻

Definition — LetVbe anR-variety andSa semialgebraic subset ofV(R). We call the morphismV →Spec(BV(S))induced by the inclusionBV(S) ⊂R[V] thecanonical bounded morphism over S.

Letφ∶V → Spec(BV(S))denote the canonical bounded morphism overS.

Any morphismψ∶V → Aⁿ_R such that closψR(S)is semialgebraically compact factors uniquely throughφ:

V

ψJJJJJJJ$$

JJ JJ

J ^φ // Spec(B_V(S))

∃!

Aⁿ_R.

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 CHAPTER . BOUNDED FUNCTIONS ON REAL VARIETIES

For if closψ_R(S)is semialgebraically compact, then the image of the homo- morphismψ^#∶R[x_, . . . ,x_n] → R[V]is contained inB_V(S), so the decomposition ψ^#∶R[x_, . . . ,x_n] → B_V(S) → R[V] yields the desired decomposition ψ∶V →Spec(B_V(S)) →Aⁿ_R.

It is important to point out that the ring BV(S)is not in general a finitely generated R-algebra, hence Spec(BV(S))is a not a variety but only an affine scheme. This is a major obstacle in the study of rings of bounded functions. It can be overcome by putting everything in the context of schemes (or real spec- tra) and working with the more general notion of rings of bounded elements or real holomorphy rings (see section 2.3). The question of when the ring of bounded functions is in fact finitely generated will be addressed in section 1.6.

.. FIBRES OF BOUNDED MORPHISMS

LetV be an irreducible affineR-variety,S a semialgebraic subset ofV(R), andW = Spec(B_V(S)). Assume that B_V(S)is a finitely generated R-algebra (c.f. section 1.6), so that W is a variety. The goal of this section is to study the fibres of the canonical bounded morphism V → W which will always be denoted byφ. This will gain us some understanding of the fibres of any regular map that is bounded onSsince the fibres of such a map are unions of fibres of φ.

We writeφ_Rfor the induced mapV(R) →W(R)onR-rational points. Then φ^−_R(w) = φ^−(w) ∩V(R)is the real fibre for everyw ∈ W(R). We begin by showing that a function that is bounded outside a Zariski-closed subset con- sisting of fibres ofφ_Rcan be expressed as a quotient of bounded functions: For

f ∈R[V], let

Ω(f) = {w∈φR(S) ∣ f ∉BV(φ^−_R(w) ∩S)}.

Note that Ω(f)is a semialgebraic subset ofW(R)(since it spells out to Ω(f) = {w∈W(R) ∣ ∀λ∈R∃x∈S∶φR(x) =w∧ ∣f(x)∣ >λ}).

Proposition 1.3 — Assume that BV(S)is a finitely generated R-algebra. For f ∈ R[V], the following statements are equivalent:

(1) f ∈qf(BV(S));

(2) φ^−_R(Ω(f))is not Zariski-dense in V.

(3) Ω(f)is not Zariski-dense in W.

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.. FIBRES OF BOUNDED MORPHISMS 

Proof — WriteA=R[V]andB=B_V(S).

(1)⇒(2): Leth ∈ B,h ≠  andh f ∈ B. Thenh is constant along all fibres φ^−(w),w ∈ W(R). Now ifw ∈ Ω(f), i.e. if f is unbounded on φ^−_R(w) ∩S, then we must haveh= onφ^−(w), sinceh f is bounded onφ^−_R(w) ∩S. Hence φ^−(Ω(f)) ⊂ V (h), so (2) holds.

(2)⇒(3): Note first that ifW is a real variety, then so isV. For ifV is non- real, then there exists an ideal ⊊I⊊R[V]such that all elements ofIvanish on V(R). Clearly,I ⊂ B_V(S)which implies qf(B_V(S)) = R(V) = R(W). Hence W cannot be real either (by Prop. A.1).

Now suppose that Ω(f)is Zariski-dense in W, so that W andV are real.

Since Ω(f)is semialgebraic and Zariski-dense inWandφis a dominant morphism, Ω(f) ∩φ_R(V_reg(R))contains a non-empty open subsetU. This implies thatφ^−_R(U)contains a non-empty open subset of real regular points ofV and is therefore Zariski-dense inV, hence so isφ^−_R(Ω(f)).

(3)⇒(1): To show thatf ∈qf(B), we define a map ̃f∶clos(φ_R(S)) →R∪{∞}

by

̃f(w) = 

sup{∣f(v)∣ ∣v ∈φ^−_R(w) ∩S}

for w ∈ φR(S) (where /∞ ∶=  and / ∶= ∞) and put ̃f(w) =  ifw ∈ clos(φ_R(S)) ∖φ_R(S). The map ̃f is semialgebraic. LetN be the set of points where ̃f is not continuous. ThenNis not Zariski-dense inW since ̃f is semialgebraic. Furthermore, Ω(f) = {w∈φR(S) ∣ ̃f(w) = }is not Zariski-dense, by hypothesis. By the subsequent lemma, there exists a continuous semialgebraic function ̃h∶clos(φ_R(S)) → R such that∣̃h∣ ⩽ ̃f and such that the set Z = {w ∈ clos(φR(S)) ∣̃h(w) = } is contained in clos(N) ∪ Ω(f)and is therefore not Zariski-dense.

Let h ∈ B, h ≠ , be such that Z ⊂ V_R(h). Since clos(φR(S))is compact, there exist c > and N ∈Nsuch that∣h^N∣ ⩽ c̃hholds on clos(φ_R(S)), by the Łojasiewicz inequality (see e.g. Bochnak, Coste, and Roy [6], Cor. 2.6.7). Now letv∈S andw=φ(v). We have

∣(c^−h^Nf)(v)∣ =c^−∣h^N(w)f(v)∣ ⩽ ∣̃h(w)f(v)∣ ⩽ ̃f(w)∣f(v)∣ ⩽.

It follows thatc^−h^Nf ∈B, hence f ∈qf(B). ^◻ Lemma 1.4 — Let M be a compact semialgebraic subset of Rⁿ, and let f∶M → R∪{∞}be a semialgebraic map with f ⩾on M. Let N be the set of points where f is not continuous. Then there exists a continuous semialgebraic map g∶M→R

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 CHAPTER . BOUNDED FUNCTIONS ON REAL VARIETIES

such that∣g∣ ⩽ f on M and such that

g(x) = Ô⇒ (f(x) = ∨ x ∈clos(N)) holds for all x∈M.

Proof — Upon replacing f by min{f, }, we may assume that f is bounded on M. The function d_N∶M → R, x ↦ inf{∥x − y∥ ∣ y ∈ N}is continuous, semialgebraic and vanishes precisely on clos(N). Letmbe its maximum onM.

Theng(x) = f(x) ⋅d_N(x) ⋅m^−,x ∈M, has the desired properties. ^◻ Example — LetV =A^_Rwith coordinatesx,y, letS = S (x,y, −x, −x y). We haveB = R[x,x y],φR∶R^ → R^,(x,y) ↦ (x,x y); hence y ∉ B, but y = ^xy_x ∈ qf(B). This is because the functionxvanishes onφ^−_R(Ω(y)) = {(,a) ∣a∈R}. Now let Ω= ⋃_f_∈R[V]Ω(f), then

Ω= {w∈φR(S) ∣clos(φ^−_R (w) ∩S)is not semialgebraically compact}. by Prop. 1.1 (4). Ifx_, . . . ,x_n are generators for theR-algebra R[V], then Ω = Ω(x_) ∪ ⋯ ∪Ω(x_n). Thus Ω is again a semialgebraic subset ofW(R).

Corollary 1.5 — Assume that BV(S)is a finitely generated R-algebra, and let W, φ∶V →W be as before. The following are equivalent:

(1) φ is birational;

(2) dimW =dimV;

(3) φ^−_R(Ω)is not Zariski-dense in V.

(4) Ωis not Zariski-dense in W;

Proof — (1)⇒(2): is clear, since birationally equivalent varieties have the same dimension.

(2)⇒(3): If dimW =dimV, then there is a Zariski-open subsetUofWsuch thatφ^−(w)is a finite set for allw ∈ U (see Hartshorne [9], chap. 2, ex. 3.22).

Clearly,U(R) ∩Ω= ∅, so Ω is not Zariski-dense inW.

(3)⇒(4): with the same reasoning as in the proof of the proposition.

(4)⇒(1): For every f ∈R[V], Ω(f) ⊂Ω is not Zariski-dense, so f ∈qf(B) by the proposition. Therefore, qf(B) =qf(R[V]) = R(V), so φis birational.

◻

Corollary 1.6 — If all fibres φ^−_R (w) ∩S, w∈W(R), are semialgebraically compact, then φ is birational.

Proof — In this case, we have Ω= ∅. ^◻

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.. COMPLETIONS OF QUASI-PROJECTIVE VARIETIES 

Remark 1.7 — The implication (2)⇒(1) in Cor. 1.5 says that if the canonical bounded morphism has generically finite fibres, then it must be birational, i.e.

generically one-to-one. In algebraic terms this is equivalent to saying that if the quotient fields of R[V]and B_V(S)have the same transcendence degree over R, then they conincide. IfV is normal, thenR[V]is integrally closed inR(V) and this claim can be directly deduced from the fact that B_V(S)is integrally closed in R[V](Prop. 1.2). Namely, it amounts to the following, purely algebraic, statement:

Lemma — Let B be a domain, K=qf(B)and L/K an algebraic field extension.

If B is integrally closed in L, then L=K.

Proof — Takea∈L, and let f ∈K[x]be the minimal polynomial ofaoverK.

After clearing denominators in f(a) =, we findb∈Bsuch thatabis integral

overB; henceab∈Band thusa∈K. ^◻

.. COMPLETIONS OF QUASI-PROJECTIVE VARIETIES

In this section, we will show how the ring of bounded functions can often be described in terms of a prescribed distribution of poles on a suitable compactification. Let X be an irreducible R-variety, and let Z be an irreducible subvariety of codimension  in X. IfX is normal alongZ, then the local ring O_X,Z ofZinXis a discrete valuation ring with valuationv_Z (see section A.3).

For any rational function f ∈ R(X), the value−v_Z(f)is called thepole order of f along Z; we say that f has a pole along Z if the pole order of f along Zis strictly positive.

Lemma 1.8 — Let X be an irreducible R-variety, and let Z be an irreducible subvariety of codimensionin X such that X is normal along Z. Let V =X∖Z, and let S be a semialgebraic subset of V(R). Assume thatclosX(R)(S) ∩Z(R)is Zariski-dense in Z. If a rational function f ∈R(X)has a pole along Z, then f is unbounded on S∩dom(f).

Proof — PutS^′ =clos_X(R)(S), and lett ∈ O_X,Z be a regular parameter, i.e. an element satisfyingvz(t) = . LetU be an open affine subset of X with U ∩ Z ≠ ∅and t ∈ O_X(U). Since Z is a pole of f, we can write f = g/tⁱ where i ⩾  and g ∈ O^×_X,Z does not vanish identically onZ(R). For any λ ∈ R, set U_λ = {x ∈ U(R) ∩dom(g) ∣ ∣g(x)∣ > λ}. Then ⋃_λ∈RU_λ is dense in Z(R), so there exists µ ∈ R such that S^′ ∩U_µ ∩Z(R) is non-empty. By the curve

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 CHAPTER . BOUNDED FUNCTIONS ON REAL VARIETIES

selection lemma (see Thm. 2.5.5. in Bochnak, Coste, and Roy [6]), there exists a continuous semialgebraic mapφ∶ [, ] → X(R)such thatφ((, ]) ⊂S∩U_µ andφ() ∈S^′∩U_µ∩Z(R). Sincet(φ()) = andg(φ(a)) >µfor alla∈ (, ], we see that f =g/tⁱ is unbounded onφ((, ]) ⊂S. ^◻ Examples 1.9 — (1) LetV be the affine planeA^_Rwith coordinatesu,v. A non- constant polynomial f ∈R[u,v]is never bounded onR^. Of course, this is easy to see directly, but it can also be deduced from the lemma: Choose homogeneous coordinates x,y,z onP^_R and putu = ^x_z, v = ^y_z, so thatV is identified withP^∖ {z =}. Then closX(R)(V(R)) =P^(R)so that any function with a pole along{z =}must be unbounded onV(R). We haveR(V) ≅R(u,v) = {_G^F ∣F,G ∈ R[x,y,z]homogeneous and degF =degG}. If f has total degree n, we write f = ∑ai juⁱv^j = ^∑^a^{i j}^x

iy^jzⁿ⁻ⁱ⁻^j

zⁿ , then the enumerator is a polynomial not divisible byz, so thatnis the pole order off along the line{z=}. We find

BV(R^) = {f ∈R[u,v] ∣ f has no pole along{z=}}

= {f ∈R[u,v] ∣deg(f) =} =R.

(2) WithV as before, letS = {(u,v) ∈ R^∣ −⩽ u ⩽}. We use the lemma to show that B_V(S) = R[u]. Let f ∈ R[V] = R[u,v], degf = n, and write

f = ∑ai juⁱv^j = ^∑^a^{i j}^x

iy^jzⁿ⁻ⁱ⁻^j

zⁿ . The set clos_P^_(R)(S)meets the line L = {z =} in the single point P = ( ∶  ∶ ). Therefore, the lemma does not apply, and that f has a pole alongL, i.e.n>, does not imply that f is unbounded onS.

However, there is a simple remedy: Letφ∶X →P^_R be the blowing-up ofP^_Rin (∶  ∶)with exceptional divisorE, and let̂Lbe the strict transform ofLin P^_R. In affine coordinates this means that the local chart(^x

y,^z_y)centered around (∶∶)onP^_R is replaced by a new chartU with coordinates(r,s) = (^x_y,^z_x). LetS^′=closX(S), then the situation looks as follows:

L S

̂L S^′ E

X P^_R

!!

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.. COMPLETIONS OF QUASI-PROJECTIVE VARIETIES 

Indeed, we haveE∩U = {r=}and̂L∩U= {s=}. Furthermore,S^′∩U = {(r,s) ∣ (s⩽ −) ∨ (s⩾)}. This shows thatS^′∩ ̂L(R) = ∅whileS^′∩E(R)is an interval and hence Zariski-dense inE(R). Since f = ∑ai juⁱv^j = ∑ai jr^−js^−i−j, we see that f has a pole alongEif and only if j>, hence

BV(S) = {f ∈R[u,v] ∣ f does not have a pole alongE} =R[u].

We will now carry out this procedure in greater generality. The example we have just seen suggests that we should pass from the natural compactification A^_R ↪P^_R to other compactifications, obtained by blowing up. But it is best to first deal with this from a more general point of view. We will be looking for compactifications with the following properties:

Definition 1.10 — LetV be a quasi-projectiveR-variety.

(1) An open dense embeddingV ↪ X ofV into a projective variety X is called acompletion of V.We will say that a completionV ↪Xisgoodif

– X∖V has pure codimension  inX;

– Xis normal alongX∖V (c.f. section A.3).

(2) LetSbe a semialgebraic subset ofV(R). We say that a completionV ↪ Xhasdense boundaries for Sor thatS has dense boundaries in Xif every irreducible componentZofX∖V satisfies the following condition:

– closX(R)(S) ∩Z(R)is either empty or Zariski-dense inZ.

Remarks — (1) Note that if clos_X(R)(S) ∩Z(R)is Zariski-dense in Z for some irreducible componentZofX∖Y, then, in particular,Z(R)itself must be Zariski-dense inZso thatZis a real variety (see Prop. A.1).

(2) Since clos_X(R)(S) =clos_X(R)(clos_V(R)(S)), we see thatShas dense boundaries in a completionXif and only if clos_V_(R)(S)has dense boundaries inX.

Examples 1.11 — (1) In Examples 1.9 (1) and (2),P^_Rresp.Xwere good completions ofV =A^_R with dense boundaries forR^resp.S.

(2) LetV =Aⁿ_Rbe affine space and consider the standard embeddingAⁿ_R ↪ Pⁿ_R into projective space, which identifiesAⁿ_Rwith the complement of a hyperplane inPⁿ_R. A semialgebraic subsetS ⊂Rⁿ has dense boundaries inPⁿ_R if and only if it contains an open convex cone (i.e. if there exist x ∈ Rⁿ and an open subsetU of Rⁿ such thatx +λU ⊂ S holds for all λ> inR).

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 CHAPTER . BOUNDED FUNCTIONS ON REAL VARIETIES

We will prove the existence of good completions with dense boundaries for several particular cases in the next section. Let us first prove that such a completion yields a representation of the ring of bounded functions:

Proposition 1.12 — Let V be an affine R-variety and S a semialgebraic subset of V(R). Assume that V possesses a good completion V ↪X with dense boundaries for S. Let Z be the union of those irreducible components of X∖V that are disjoint fromclosX(R)(S). Then

BV(S) = O_X(X∖Z).

Thus a function is bounded on S if and only if it has poles only in components of Z.

Proof — If V, . . . ,Vr are the irreducible components ofV, let Xi denote the Zariski-closure of Vi in X, and put Si = S ∩Vi(R), and Zi the union of all irreducible components of Xi ∖Vi that are disjoint from clos_X_i_(R)(Si). Then Vi ↪ Xi is a good completion with dense boundaries forSi, and for every f ∈ R[V], we have f ∈ BV(S)(resp. f ∈ O_X(X∖Z)) if and only if f∣_V_i ∈ BVi(Si) (resp. f∣_V_i ∈ O_X_i(Xi ∖Zi)) for eachi ∈ {, . . . ,r}. We may therefore assume thatV is irreducible.

PutW = X∖Zand S^′ = clos_X(R)(S). Since X is projective,S^′is semialgebraically compact (see Prop. A.4) andS^′⊂W(R), so every element ofO_X(W) is bounded on S^′, henceO_X(W) ⊂ BV(S). For the converse, let f ∈ R[V] = O_X(V)and assume f ∉ O_X(W). LetZ_, . . . ,Z_r be those irreducible components ofX∖Z not contained inZ, then, by hypothesis, S^′∩Z_i(R)is Zariski- dense inZ_i fori=, . . . ,r. We have

O_X(W) = O_X(V) ∩ O_X,Z_∩ ⋯ ∩ O_X,Z_r

by Prop. A.5, becauseXis normal alongX∖V. Therefore, f ∈ O_X(V) ∖ O_X(W) implies f ∉ O_X,Z_i for somei, in other words f has a pole alongZi. Therefore, f is unbounded onSby Lemma 1.8 which means f ∉BV(S). ^◻

.. EXISTENCE OF GOOD COMPLETIONS WITH DENSE BOUNDARIES

If V is a quasi-projectiveR-variety, then for any embedding V ↪ Pⁿ_R the Zariski-closureV ofV inPⁿ_R is a completion ofV. In constructing good completions and completions with dense boundaries, we always start from such an embedding which is then gradually improved. The case of curves is quite simple:

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.. EXISTENCE OF GOOD COMPLETIONS WITH DENSE BOUNDARIES 

Proposition 1.13 — Let C be a curve over R.

(1) C has a good completion, unique up to isomorphism.

(2) If C is non-singular, then so is the good completion of C.

(3) Any completion of C has dense boundaries for all semialgebraic subsets of C(R).

Proof — LetC ↪ X_be any completion. SinceC is dense in X_o, the comple- mentX_∖Cconsists of finitely many points. That any semialgebraic subset of C(R)has dense boundaries in X(R)is therefore obvious. We can remove any singularities inX_∖Cby successive blow-ups or by taking the normalization of X∖Csingand then glueing withC(see for example Serre [35], ch. IV, §1). Thus we obtain a completionC ↪ Xsuch that all points ofX∖Care non-singular.

Clearly, this completion is good, and ifCis non-singular, then so isX. IfC ↪X andC ↪X^′are two good completions, the birational mapX⇢X^′given by the identityC →Cextends to an isomorphism, sinceX∖C andX^′∖Cconsist of non-singular points (see Hartshorne [9], ch. I, Prop. 6.8.). ^◻ Definition — LetCbe a curve overR, and letC↪Xbe a good completion of C. The finitely many points inX∖Care called thepoints at infinity of C.

We have just seen that the points at infinity of C are uniquely determined up to isomorphism as a zero-dimensional variety, in other words their number and their residue fields are uniquely determined.

Corollary 1.14 — Let C be a curve over R, with good completion X, and let S be a semialgebraic subset of C(R). Then BC(S) = R if and only if all points at infinity of C are real and contained inclos_X(R)(S).

Proof — LetT = {P ∈ X∖C ∣P ∉ X(R)orP ∈ X(R) ∖closX(R)(S)}. Then BC(S) = O_X(X∖T)by Prop. 1.12. We haveO_X(X∖T) =Rif and only ifT= ∅,

as a consequence of the Riemann-Roch theorem. ^◻

Remark 1.15 — On a side note, the statement of the corollary does not gener- alise to higher dimensions in any reasonable way because if Xhas dimension at least , it may happen thatO_X(X∖Z) = Rfor some non-empty divisor Z on X. For concrete examples where this occurs in the construction of a good completion, see Examples 4.5.

In higher dimensions, it is more difficult to prove the existence of good completions with dense boundaries. It is not hard to see that every normal variety

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 CHAPTER . BOUNDED FUNCTIONS ON REAL VARIETIES

has a good completion: IfV is normal andV ↪Xis any completion, then the normalizationX^′ →Xis an isomorphism overV, and should any component ofX^′∖V have codimension greater than , this can be resolved by blowing up.

It is the condition of dense boundaries that is critical. We begin with the case S =V(R):

Proposition 1.16 — Let V be a quasi-projective variety over R, and assume that V has a good completion V ↪X such that every irreducible component of X∖V is non-singular. Then V(R)has dense boundaries in X.

Proof — Letd =dimV. The local dimension ofX(R)in any regular point isd while dim(X(R)∖V(R)) ⩽d−. Therefore, the complement of closX(R)(V(R)) inX(R)must be contained in the singular locus ofXand is therefore of dimension at mostd−, sinceXis normal alongX∖V. Now ifZis any irreducible component of X∖V, then eitherZ(R)is empty or dimZ(R) =d−, sinceZ is non-singular, so clos_X(R)(V(R)) ∩Z(R)is Zariski-dense inZ. ^◻ Proposition 1.17 — Let V be a non-singular, quasi-projective R-variety. There is a good completion V ↪X such that X is non-singular and all irreducible components of X∖V are also non-singular.

Proof — Starting from any open embedding ofV into a projective varietyX, we may first apply resolution of singularities (as stated in Thm. A.7) toXand obtain an open embeddingV ↪ X such that Xis again non-singular. Then apply embedded resolution of singularities (Thm. A.8) toX∖V to get a non- singular completionV ↪ X ofV where all irreducible components of X∖V

are non-singular and of codimension  inX. ^◻

Corollary 1.18 — Let V be a non-singular, quasi-projective variety over R. Then V possesses a good completion with dense boundaries for V(R). ^◻ Remark 1.19 — Good completions in dimensions⩾  are not unique, but if V ↪ X_ andV ↪ X_are two good completions, one may ask whether there exists a common refinement, i.e. a good completion V ↪ X^′ together with birational morphisms X^′ → X_ and X^′ → X_that are isomorphisms over V. This is always true in the case of non-singular surfaces, since any birational mapφ∶X_⇢X_of non-singular surfaces can be factored into a series of blow- downs X_ ← X^′ followed by a series of blow-ups X^′ → X_, both with centers away from dom(φ) (see for example Beauville [3], Cor. II.12; the analogous statement in dimensions⩾ goes by the name “strong factorization conjecture”,

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.. EXISTENCE OF GOOD COMPLETIONS WITH DENSE BOUNDARIES 

and no proof is known so far; see Abramovich, Karu, Matsuki, and Włodarczyk [1]). Applying this to the mapφinduced by the identity onV gives the desired refinement X^′. It is not hard to see that if some semialgebraic subset ofV(R) has dense boundaries in X_(R) or X_(R), then this can be arranged also for X^′(R).

As an example in dimension , considerP^_RandP^_R×P^_Ras completions ofA^_R in the natural way. The two are not directly comparable, since the first has only one component at infinity while the second has two, and there is no birational morphism (and hence no non-constant morphism that preserves the embedding ofA^_R) from one to the other. But the birational mapφ∶P^_R ×P^_R ⇢ P^_R, obtained by embeddingP^_R×P^_RintoP^_Ras a non-singular quadric and project- ing from a point, fits into a diagram

Q̂

||yyyyyyyyy

>

>>

>

P^_R×P^_R ^_ ^_ ^_^φ^_ ^_ ^_ ^_^// P^_R

whereQ̂ →P^_R×P^_R is a blow-up in one point andQ̂ → P^_R a blow-up in two points.

We now turn to the case of general semialgebraic subsets ofV(R).

Definition 1.20 — LetVbe anR-variety andSa semialgebraic subset ofV(R). (1) S is called regular if there exists an open subset U of V(R)such that

clos(S) =clos(U).

(2) S is calledregular at infinity if clos(S)is the union of a regular and a semialgebraically compact subset ofV(R).

Examples 1.21 — (1) Put V = A^_R. The set S = {(x,y) ∈ R^∣ (x^+ y^ ⩽

) ∨ (x =∧ −⩽y⩽)}is not regular, but it is regular at infinity. The setS = {(x,y) ∈R^∣ (x^+y^⩽) ∨ (x =)}is not even regular at infinity.

(2) For every semialgebraic subsetS ofV(R), both int(S)and clos(int(S)) are always regular. IfV is non-singular and of dimensiond, thenS ≠ ∅ is regular if and only ifS is of pure dimension d, i.e. dim_x(S) = d for allx ∈ S (where dim_x denotes the local semialgebraic dimension inx).

Furthermore, if V is regular at infinity and V ↪ X is any completion of V, then dim_x(clos_X(R)(S)) = d holds for every x ∈ clos_X(R)(S) ∩

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 CHAPTER . BOUNDED FUNCTIONS ON REAL VARIETIES

(X∖Z)(R). This can be seen by taking a suitable semialgebraic cellular decomposition of clos_X(R)(S).

(3) LetV = A^_R,S = {(x,y) ∈ R^∣ − ⩽ x ⩽}, S = {(x, ) ∣x ∈ R}, and S=S∪S. ThenVhas a good completionV ↪Xwith dense boundaries forS, namely the blow-up ofP^_Rin a point;X∖V consists of two linesL andE, and closX(R)(S) ∩E(R)is an interval while closX(R)(S) ∩L(R) =

∅ (see Example 1.9 (2)). But S does not have dense boundaries in X because closX(R)(S) ∩L(R)is a point and therefore not Zariski-dense in L. In fact, there is no way to fix this:V does not have a good completion with dense boundaries forS.

Theorem 1.22 — Let V be a quasi-projective surface over R with at most isolated singularities. Let S be a semialgebraic subset of V(R)that is regular at infinity.

Then V possesses a good completion with dense boundaries for S. If, in addition, V is non-singular (resp. normal), there exists such a completion that is again non- singular (resp. normal).

Proof — Assume first thatVis real and non-singular and thatSis basic closed, sayS = S (h, . . . ,hr). By Prop. 1.17, there exists a non-singular good comple- tionXofV such that all irreducible components ofE =X∖V are also non- singular. LetC = ⋃_iV_V(hi)(whereV_V(hi)is the Zariski-closure ofV_V(hi)in X). Now apply embedded resolution of curves in surfaces as stated in Thm. A.9 to the surfaceX, the curveC∪Eand the finite set of intersection points ofC∪E.

We obtain a birational morphismπ∶X→Xwith the following properties:

(1) Xis non-singular and projective;

(2) π∣_π⁻(V)∶π^−(V) →V is an isomorphism;

(3) all irreducible components ofX∖V =π^−(E)are non-singular andX∖V has only normal crossings;

(4) letĈdenote the strict transform ofCinX. Then all points inĈ∩π^−(E) are also normal crossings.

Let us now convince ourselves thatV ↪ Xhas the desired properties: It is clearly a good completion ofV, becauseXis non-singular andX∖V =π^−(E) is a curve. To see that S has dense boundaries in X, let Z be an irreducible component of X∖V and assume that clos_X(R)(S) ∩Z(R)is non-empty. We have to show that it is Zariski-dense inZ. SinceZis non-singular,Z(R)must be Zariski-dense in Z. Hence it suffices to show that clos_X(R)(S) ∩ Z(R) is Zariski-dense inZ(R).

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.. EXISTENCE OF GOOD COMPLETIONS WITH DENSE BOUNDARIES 

Letx ∈clos_X(R)(S) ∩Z(R), and letu∈ O_X,xbe a local equation forZ. There can be at most one other irreducible component Z^′ ≠ Z ofĈ∪ (X∖V)with x ∈ Z^′(R) because x must be a normal crossing point. If such Z^′ exists, let v ∈ O_X,xbe a local equation forZ^′. Thenu,vis a regular system of parameters forO_X,x. If no suchZ^′exist, letv∈ O_X,xbe some element ofO_X,xsuch thatu,v is a regular system of parameters forO_X,x.

Now choose an open neighbourhoodUofxinX(R)such thatU∩Z^′′(R) =

∅for all irreducible componentsZ^′′ofĈ∪ (X∖V)other thanZand possibly Z^′. Then the sign ofhk,k =, . . . ,r, onV(R) ∩U depends only on the sign of uandvsince all poles and zeros of thehi are contained inĈ∪ (X∖V).

After intersectingU with a suitable Zariski-open subset of X, we can also assume thatuandvare regular onU. SinceSis regular at infinity, clos_X(R)(S) ∩ Uhas dimension  (see Example 1.21 (2)). From all this we see that clos_X(R)(S)∩

U must contain a quadrant, i.e. a set of the form {x ∈U∣εu(x) ⩾∧δv(x) ⩾}

whereε,δ ∈ {±}. Sinceu,vis a regular system of parameters andZ(R) ∩U = {u =} ∩U, bothZ(R) ∩ {v ⩾ }and Z(R) ∩ {v ⩽ }are Zariski-dense in Z(R). This shows that clos_X(R)(S) ∩U∩Z(R)is Zariski-dense inZ(R).

This completes the proof ifVis real and non-singular andSis basic closed. If Sis not basic closed, we may first replaceS by clos_V_(R)(S)(see remark 1.4 (2)), so we can assume thatSis closed. By the finiteness theorem (see e.g. Bochnak, Coste, and Roy [6], Thm. 2.7.2), S can then be written as a union of finitely many basic closed sets. The construction used above can now be applied con- secutively to each of these.

IfV has isolated singularities, letV ↪Xbe a completion, and letφ∶ ̃X →X be a resolution of singularities for X (Thm. A.7). Put Ṽ = φ^−(V)and S̃ = φ^−_R(S). Consider the commutative square

̃S⊂ ̃V(R) ⊂ ̃V ^//

X̃

S⊂V(R) ⊂V ^// X.

By what we have already proved, we can construct a good completion ofṼwith dense boundaries for̃Sby blowing up in points ofX̃∖ ̃V. This completion will fit into the same commutative square in place of X, so we may assume without̃ loss of generality thatX̃is a good completion ofṼwith dense boundaries for

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 CHAPTER . BOUNDED FUNCTIONS ON REAL VARIETIES

S̃. SinceV_singis finite (and in particularV_sing∩ (X∖V) = ∅), the divisorφ^−(y) ofX̃is contractible for everyy∈Vsing, so there existsψ∶ ̃X→X^′,X^′projective, such thatψ( ̃V) ≅V andψ( ̃X∖φ^−(Vsing))Ð→ (^∼ X∖Vsing)so thatX^′is a good completion ofV with dense boundaries forS.

Finally, ifV is not real, thenV(R)and henceSare contained inVsing, soSis finite. Therefore, any good completion ofV (which exists by the construction in the preceding paragraph) has dense boundaries forS. The additional claim about non-singularity resp. normality ofXis clear from the construction. ^◻ We have shown that under certain conditions a ring of bounded functions can be represented as the ring of regular functions of a quasi-projective variety.

The converse also holds. To show this, we need the following

Lemma 1.23 — Let V be a non-singular, irreducible, affine R-variety, V ↪X a good completion of V, and Z, . . . ,Zr the irreducible components of X ∖V. Let s⩾, and assume that Z, . . . ,Zsare real. Then there exists a regular, basic closed subset S of V(R)such thatclosX(R)(S) ∩Zi(R)is Zariski-dense in Zi for all i⩽s andclosX(R)(S) ∩Zi(R) = ∅for all i>s.

Proof — Let U be an open affine subvariety of X with U(R) = X(R) (see Lemma A.2), and fix an embedding of U into affine space with coordinates x, . . . ,xn. If s = , there is nothing to show. Otherwise, let  ⩽ i ⩽ s and write Ei = ⋃_j≠iZj. We will first show that there exists hi ∈ R[V]such that clos_X(R)(S_V(hi)) ∩Zi(R)is Zariski-dense and clos_X(R)(S_V(hi)) ∩Ei(R) = ∅. Let(a, . . . ,an) ∈ Zi(R) ∖Ei(R)be a non-singular point ofZi, and for every ε >  inR, let gi,εbe the class of (x−a)^+ ⋯ + (xn −an)^−εin R[U], so thatS (gi,ε)is the intersection ofU(R)with a closed ball of radius εaround (a, . . . ,an). Chooseε >  small enough, such that S_U(gi) ∩Ei(R) = ∅for gi = gi,ε. On the other hand, S_U(gi) ∩ Zi(R)is Zariski-dense in Zi. Since U(R) =X(R), there existsti ∈R[V]such thatt^_igi ∈R[V]andti has no zeros onV(R). Puthi =t^_igi, thenS_V(hi)behaves as desired.

Now choosehi as above for each  ⩽ i ⩽ s. We may assume that theS (hi) are pairwise disjoint, after choosing balls of smaller radius if necessary. This implies⋃_iS (hi) = S ((−)^r+h⋯hr), soS = ⋃_iS (hi)is basic closed and has

the desired properties. ^◻

Theorem 1.24 — Let W be a non-singular, irreducible, real, quasi-projective R- variety. Then there exists an affine open subvariety V of W and a regular, basic

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.. FINITE GENERATION OF THE RING OF BOUNDED FUNCTIONS 

closed subset S of V(R)such that

B_V(S) ≅ O_W(W).

Proof — LetW ↪ X be a good completion of W such that X is again non- singular (see Prop. 1.17). SinceWis real, there exists a real divisorZof Xsuch thatV =W∖ (Z∩W)is affine. LetZ_, . . . ,Z_sbe the irreducible components of ZandZs+, . . . ,Zr those ofX∖W. By the preceding lemma, there exists a basic closed subsetSofV(R)such that closX(R)(S) ∩Zi is Zariski-dense inZi

for  ⩽ i ⩽ s and empty otherwise. Therefore, we haveBV(S) = O_W(W)by

Prop. 1.12. ^◻

.. FINITE GENERATION OF THE RING OF BOUNDED FUNCTIONS

In this section, we ask when the ring of bounded functions is a finitely gen- eratedR-algebra. The case of curves is very simple:

Proposition 1.25 — Let C be an irreducible curve over R and S a semialgebraic subset of C(R). Then BC(S)is a finitely generated R-algebra.

Proof — By Prop. 1.13, there exists a quasi-projective curveC^′such thatBC(S) = O_C′(C^′). But every quasi-projective curve is either affine or projective (an easy consequence of the Riemann-Roch theorem; see Hartshorne [9], ch. IV, ex. 1.1), soO_C′(C^′)is a finitely generatedR-algebra. ^◻ But in higher dimensions, the situation is again much more complicated:

Example 1.26 — LetV be the affine plane overRandSa semialgebraic subset of R^. IfS is not regular at infinity,BV(S)need not be a finitely generatedR- algebra. For example, letS = S (x^(−x^−y^). Clearly,BV(S)consists of all polynomials that are bounded and hence constant on the y-axis. Therefore, it is the subalgebra ofR[x,y]generated by all monomialsx yⁱ,i⩾. This cannot be finitely generated overR, for the corresponding lattice{(i,j) ∣i ⩾,j⩾} inN^_is not finitely generated overN.

However, this cannot happen ifVis a normal surface andSis regular at infinity:

Theorem 1.27 (Zariski) — Let W be a normal, quasi-projective surface over a field k of characteristic zero. Then the ring of regular functions O_W(W) is a finitely generated k-algebra.

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 CHAPTER . BOUNDED FUNCTIONS ON REAL VARIETIES

Proof — See Zariski [40], remarks on Thm. 1 at the end of the article. See Krug [13] for an alternative proof in modern language, based on methods by

Nagata. ^◻

Corollary 1.28 — Let V be a normal, quasi-projective surface over R, and let S a semialgebraic subset of V(R)that is regular at infinity. Then the ring of bounded functions BV(S)is a finitely generated R-algebra.

Proof — Indeed, under the hypotheses,Vpossesses a normal good completion with dense boundaries for S by Thm. 1.22, henceB_V(S)is the ring of regular functions of a normal quasi-projective surface by Prop. 1.12 which is finitely

generated overRby the above theorem. ^◻

Zariski’s theorem and our corollary do not hold in higher dimensions: For example, there exist non-singular quasi-projective threefolds overR(or indeed over any field that is not algebraic over a finite field) whose ring of regular functions is not a finitely generatedR-algebra. By Thm. 1.24, such a ring is also a ring of bounded functions of a suitable regular, basic closed subset of some non-singular affineR-variety. Examples of such quasi-projective varieties can be constructed in various ways but are not entirely elementary; see for example Krug [13]; in an unpublished note [37], Vakil has given an example that is the total space of a vector bundle of rank  over an elliptic curve.

If the semialgebraic setS is not regular at infinity, then at least the interior int(S)ofS is regular, and one can hope to understand the ringBV(S)by first consideringBV(int(S))and then proceed inductively. We need the following lemma (see also [4], Prop. 2.3):

Lemma 1.29 — Let A be a domain, and B a subring of A that is integrally closed in A. If I is a non-trivial ideal of A contained in B, thenqf(A) =qf(B)and either B=A or B is not noetherian.

Proof — Let c ∈ I,c ≠ . Then for any a/b ∈ qf(A), we have a/b = ca/cb ∈ qf(B); hence qf(A) = qf(B). Now suppose that B is noetherian. Since A is a domain and I ≠ (), we have an embedding λ∶A ↪ EndB(I) given by multiplication from the left. Let a ∈ A. Since B is noetherian, I is a finitely generated B-module. By the Cayley-Hamilton theorem, there exists a monic polynomial f with coefficients inB such that f(λ(a)) =  in EndB(I). Since f(λ(a)) = λ(f(a)), we have f(a) =  in A. Therefore, a is integral overB,

hencea∈B. ^◻

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.. FINITE GENERATION OF THE RING OF BOUNDED FUNCTIONS 

Proposition 1.30 — Let V be an irreducible affine R-variety, S a semialgebraic subset of V(R)and S_=int(S). Assume thatdim(S∖S_) <trdegB_V(S_). Then B_V(S) =B_V(S_)or B_V(S)is not noetherian.

Proof — LetS^′ = S ∖S, and let I = {f ∈ BV(S) ∣ ∀x ∈ S^′ ∶ f(x) = }. If S^′ = ∅, the claim is trivial. So we may assume thatS^′ ≠ ∅, henceI ≠ BV(S). We claim that I ≠ (). Indeed, let W = Spec(BV(S)), and let φ∶V → W be the canonical bounded morphism. Then Iis the vanishing ideal ofφR(S^′). But dim(S^′) < trdegBV(S)by hypothesis, hence φR(S^′)cannot be Zariski- dense in W which shows I ≠ (). Now the preceding lemma applies (with A = BV(S),B = BV(S)), since obviouslyI ⊂ BV(S)andBV(S)is integrally

closed inR[V], hence also inBV(S). ^◻

Corollary 1.31 — Let V be an irreducible affine R-variety, S a semialgebraic subset of V(R). IfdimS <dimV, then B_V(S) =R[V]or B_V(S)is not noetherian.

Proof — EitherV is real, then dimV(R) =dimV. In this case, int(S) = ∅, so dimS <dimV =trdegR[V] =trdegBV(int(S)), and the proposition applies.

OrVis non-real, then there is a non-zero idealI⊂R[V]such thatV(R) ⊂ V (I) by Prop. A.1 which implies I ⊂ BV(S). Thus BV(S) = R[V]orBV(S)is not

noetherian by the lemma above. ^◻

Corollary 1.32 — Let V be an irreducible, normal, real, affine surface over R, S a semialgebraic subset of V(R)and S_=int(S). Then B_V(S)is a finitely generated R-algebra if and only iftrdeg(B_V(S_)) ⩽or B_V(S) =B_V(S_).

Proof — We do sufficiency first: By Cor. 1.28 above,BV(S)is finitely generated. If trdeg(BV(S)) = , thenBV(S) = BV(S) = R, so there is nothing to show. If trdeg(BV(S)) = , thenW = Spec(BV(S)) is a curve over Rand BV(S) = BW(φR(S ∖S)), where φ∶V → W is the canonical bounded morphism. Therefore, BV(S) is finitely generated by 1.25. For necessity, assume that trdeg(BV(S)) = andBV(S)is finitely generated. ThenBV(S) =BV(S)

by the proposition. ^◻

Remark 1.33 — If S is regular at infinity, then clearly BV(S) = BV(S). The converse, however, is not true, even if trdeg(BV(S)) =. For example, letV = A^_R with coordinatesu,v, and putS = S (u, −u,u^(uv−),u^(−u^v)), then BV(S) =BV(S) =R[u,uv]butSis not regular at infinity.

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Bounded Polynomials, Sums of Squares, and the Moment Problem