Positivity, sums of squares and the multi-dimensional moment problem II ^{∗}

S. Kuhlmann^{†}, M. Marshall^{‡} and N. Schwartz^{§}

Revised 29. 02. 2004

Abstract

The paper is a continuation of work initiated by the first two authors in [K– M]. Section 1 is introductory. In Section 2 we prove a basic lemma, Lemma 2.1, and use it to give new proofs of key technical results of Scheiderer in

[S1] [S2] in the compact case; see Corollaries 2.3, 2.4 and 2.5. Lemma 2.1 is also used in Section 3 where we continue the examination of the case n = 1 initiated in [K–M], concentrating on the compact case. In Section 4 we prove certain uniform degree bounds for representations in the case n = 1, which we then use in Section 5 to prove that (‡) holds for basic closed semi-algebraic subsets of cylinders with compact cross-section, provided the generators satisfy certain conditions; see Theorem 5.3 and Corollary 5.5. Theorem 5.3 provides a partial answer to a question raised by Schmüdgen in [Sc2]. We also show that, for basic closed semi-algebraic subsets of cylinders with compact cross-section, the sufficient conditions for (SMP) given in [Sc2] are also necessary; see Corollary 5.2(b). In Section 6 we prove a module variant of the result in [Sc2], in the same spirit as Putinar’s variant [Pu] of the result in [Sc1] in the compact case; see Theorem 6.1. We apply this to basic closed semi-algebraic subsets of cylinders with compact cross-section; see Corollary 6.4. In Section 7 we apply the results from Section 5 to solve two of the open problems listed in [K–M]; see Corollary 7.1 and Corollary 7.4. In Section 8 we consider a number of examples in the plane. In Section 9 we list some open problems.

1 Introduction

Let ℝ[X] denote the polynomial ring in n variables X = (X_{1}, …, X_{n}), with real coefficients, and consider a finite set S = {g_{1}, ⋯, g_{s}} of polynomials in ℝ[X]. We denote by K_{S} the basic closed semi-algebraic set in ℝ^{n} defined by:

∗2000 Mathematics Subject Classification: Primary 14P10, Secondary 44A60.

†Partially supported by an NSERC research grant.

‡Partially supported by an NSERC research grant.

§Partially supported by the European RTN Network RAAG, Contract No. HPRN-CT-2001-00271

K_{S} = {x ∈ ℝ^{n} : g_{1}(x) ≥ 0, …, g_{s}(x) ≥ 0}.

We denote by T_{S} the preordering in ℝ[X] generated by S:

T_{S} = { ∑_{e∈{0,1}^{s}} σ_{e}g^{e} : σ_{e} ∈ ∑ ℝ[X]^{2}},

where g^{e} := g_{1}^{e_{1}} ⋯ g_{s}^{e_{s}}, if e = (e_{1}, ⋯, e_{s}), and ∑ ℝ[X]^{2} denotes the preordering of ℝ[X] consisting of sums of squares. Note: K_{∅} = ℝ^{n}, T_{∅} = ∑ ℝ[X]^{2}. By the Positivstellensatz [St], K_{S} = ∅ ⇔ −1 ∈ T_{S} ⇔ T_{S} = ℝ[X]. Set

T_{S}^{alg} = {f ∈ ℝ[X] : f ≥ 0 on K_{S}},
T_{S}^{∨} = {L : ℝ[X] → ℝ : L is linear (≠ 0) and L(T_{S}) ≥ 0},

and

T_{S}^{lin} = T_{S}^{∨∨} = {f ∈ ℝ[X] : L(f) ≥ 0 for all L ∈ T_{S}^{∨}}.

Note: We have the inclusions T_{S} ⊆ T_{S}^{lin} ⊆ T_{S}^{alg}. The sets T_{S}^{∨}, T_{S}^{lin} depend on T_{S}. The set T_{S}^{alg} depends only on K_{S}.

The set T_{S}^{alg} is a preordering, called the saturation of T_{S}. We say T_{S} is saturated if T_{S}^{alg} = T_{S}. The set T_{S}^{lin} is the closure of T_{S} in ℝ[X], giving ℝ[X] the unique finest locally convex topology [Po–S, p. 76]. The set T_{S}^{lin} is a preordering [Po–S, Lem. 1.2]. We say T_{S} is closed if T_{S}^{lin} = T_{S}. This holds in a variety of special cases, e.g., if K_{S} contains an n-dimensional cone [K–M, Th. 3.5] [Po–S, Prop. 3.7]. It also fails in many cases, e.g., if K_{S} is compact and dim(K_{S}) ≥ 3. This follows from [S1, Prop 6.1] using [Sc1, Cor. 2].

The general Moment Problem is the following: For a linear functional L on ℝ[X], when is there a positive Borel measure μ or ℝ^{n} such that ∀f ∈ ℝ[X] L(f) = ∫_{ℝ}^{n} fdμ? The following result is due to Haviland [H1] [H2]. For a proof based on the Riesz representation theorem, see [M2, Th. 3.1], for example.

Theorem 1.1 For a linear functional L on ℝ[X] and a closed set K of ℝ^{n}, the following are equivalent:

(i) ∃ a positive Borel measure μ on K such that ∀f ∈ ℝ[X], L(f) = ∫_{K} fdμ. (ii) ∀ f ∈ ℝ[X], f ≥ 0 on K ⇒ L(f) ≥ 0.

Since T_{S}^{alg} is not finitely generated in general , one is interested in approximating it by T_{S}. Therefore one studies the following concrete Moment Problem: When is it true that every L ∈ T_{S}^{∨} comes from a positive Borel measure on K_{S}? By Theorem 1.1, this is equivalent to asking when the following condition holds:

(SMP)  T_{S}^{alg} = T_{S}^{lin}.

In the landmark paper [Sc1], Schmüdgen proves (SMP) when the basic closed semi-algebraic set K_{S} is compact. Moreover, for compact K_{S}, he gets the following substantial improvement of the Positivstellensatz:

(†)  ∀f ∈ ℝ[X], f ≥ 0 on K_{S} ⇒ ∀ real ϵ > 0, f + ϵ ∈ T_{S}.

See [B–W], [M1] or [P–D] for Wörmann’s proof of (†), which is based on the Kadison-Dubois representation theorem.

We turn now to the case where K_{S} is non-compact. Given a pair of sets S, S′ with K_{S} ⊆ K_{S′}, one can also ask: When is it true that every L ∈ T_{S}^{∨} comes from a positive Borel measure supported by K_{S′}? By Theorem 1.1 this is equivalent to asking if T_{S′}^{alg} ⊆ T_{S}^{lin}. We will consider this property in the special case S′ = ∅:

(MP)  T_{∅}^{alg} ⊆ T_{S}^{lin}.

Note: T_{S} is saturated ⇒ (†) holds ⇒ (SMP) holds ⇒ (MP) holds.

Proposition 1.2 If K_{S} contains a cone of dimension 2 then (MP) fails.

Note: The statement of Proposition 1.2 is stronger than [K–M, Cor. 3.10], but the proof is the same. Namely, one uses [K–M, Th. 3.5] along with [S1, Remark 6.7] to produce p ∈ T_{∅}^{alg}, p ∉ T_{S}^{lin}. See [Po–S, Cor. 3.10] for a more general criterion.

Note: For n ≥ 3 the condition that K_{S} contains a cone of dimension 2 does not imply that T_{S} is closed.

Example 1.3 Let n ≥ 1 and pick S_{1} in ℝ[X] such that K_{S_{1}} is compact and T_{S_{1}} is not closed. Define S in ℝ[X, Y, Z] by S = S_{1}. Then K_{S} = K_{S_{1}} × ℝ^{2} contains a 2-dimensional cone. Also T_{S_{1}}^{lin} ⊆ T_{S}^{lin} and T_{S_{1}} = T_{S} ∩ ℝ[X] so if p ∈ T_{S_{1}}^{lin}, p ∉ T_{S_{1}}, then p ∈ T_{S}^{lin}, p ∉ T_{S}. The possible choices for S_{1} are necessarily a bit artificial if n = 1, e.g., S_{1} = {X^{3}(1 − X)} will do. For n = 2 it seems that less contrived choices should exist. For n ≥ 3 any S_{1} with dim(K_{S_{1}}) ≥ 3 will do [S1, Prop. 6.1].

In [K–M], the authors consider the intermediate condition:

(‡) ∀ f ∈ ℝ[x], f ≥ 0 on K_{S} ⇒ ∃q ∈ ℝ[X] such that ∀ real ϵ > 0, f + ϵq ∈ T_{S}.

It is shown that (‡) is strictly weaker than (†), but implies (SMP). The condition (‡) holds in various non-compact cases, e.g., for basic closed semi-algebraic sets obtained by a certain natural dimension extension process [K–M, Cor. 4.5] and for cylinders with compact cross-section [K–M, Th. 5.1]. Define

T_{S}^{‡} = {f ∈ ℝ[X] : ∃q ∈ ℝ[X] such that ∀ real ϵ > 0 , f + ϵq ∈ T_{S}}.

Note: (i) T_{S} ⊆ T_{S}^{‡} ⊆ T_{S}^{lin}. (ii) (‡) holds iff T_{S}^{alg} = T_{S}^{‡}. (iii) It follows from the proof of [K–M, Cor. 4.2] that the element q appearing in the description of T_{S}^{‡} can be chosen from the set {p^{ℓ} : ℓ ≥ 0}, where p ∈ 1 + T_{S} is any polynomial which ‘grows sufficiently rapidly’ on K_{S}, e.g., p = 1 + ∥X∥^{2} always works. Here, ∥X∥^{2} := ∑_{i=1}^{n} X_{i}^{2}. If K_{S} is compact then p = 1 works.

Denote by P_{d} the (finite dimensional) vector space consisting of all polynomials in ℝ[X] of degree ≤ 2d, and by T_{d} = T_{S} ∩ P_{d}. The set T_{d} is obviously a cone in P_{d}, i.e., T_{d} + T_{d} ⊆ T_{d} and ℝ^{+}T_{d} ⊆ T_{d}. Denote by \overline{T}_{d} the closure of T_{d} in P_{d}.

Proposition 1.4 (i) T_{S}^{‡} = ∪_{d≥0}\overline{T}_{d}. (ii) T_{S}^{‡} is a preordering.

Proof: Assertion (i) is clear from the proof of [K–M, Prop. 1.3]. For polynomials f, g, the coefficients of f + g and fg are polynomial functions (in particular, continuous functions) of the coefficients of f and of g. Assertion (ii) is clear from this, using (i). □

Observe that (†), (‡), (SMP) and (MP) all depend on T_{S}. Consequently, if (†), resp., (‡), resp., (SMP), resp., (MP) holds then we say that (†), resp., (‡), resp., (SMP), resp., (MP) holds for T_{S} or that T_{S} satisfies (†), resp., (‡), resp., (SMP), resp., (MP).

In Section 2 we prove Lemma 2.1, which we need later, in Section 3, but which is also of independent interest in that it yields alternate proofs of key technical results in [S1] [S2]. In Sections 3 and 4 we continue the analysis of the 1-dimensional case initiated in [K–M]. The results in these two sections are of interest in their own right, but they are also essential in dealing with the fiber sets that arise in the study of subsets of cylinders with compact cross-section.

In [Sc2], Schmüdgen considers the following general set-up: Suppose h_{1}, …, h_{d} ∈ ℝ[X] are bounded on K_{S}. For λ = (λ_{1}, …, λ_{d}) ∈ ℝ^{d} define

S_{λ} = S ∪ {h_{1} − λ_{1}, −(h_{1} − λ_{1}), …, h_{d} − λ_{d}, −(h_{d} − λ_{d})}.

Thus K_{S_{λ}} = K_{S} ∩ C_{λ} where C_{λ} denotes the algebraic set in ℝ^{n} defined by the equations h_{i}(X) = λ_{i}, i = 1, …, d. By the assumption K_{S_{λ}} = ∅ (i.e., T_{S_{λ}} = ℝ[X]) for ∥λ∥ sufficiently large. Schmüdgen proves:

Theorem 1.5 Suppose h_{1}, …, h_{d} ∈ ℝ[X] are bounded on K_{S}. If (SMP) (resp., (MP)) holds for T_{S_{λ}} for each λ then (SMP) (resp. (MP)) holds for T_{S}.

Theorem 1.5 applies, in particular, to subsets of cylinders with compact crosssection. In this special case, we prove that the converse of Theorem 1.5 also holds; see Corollary 5.2.

The question of whether (SMP) is strictly weaker than (‡) is posed as an open problem in [K–M]. It seems likely that this is the case, but no examples are known. In fact, no examples are known where T_{S}^{‡} ≠ T_{S}^{lin}. In [Sc2], Schmüdgen asks the following related question: Is it true that Theorem 1.5 continues to hold with (SMP) replaced by (‡)?

We provide a partial answer to Schmüdgen’s question in the case of subsets of cylinders with compact cross-section; see Theorem 5.3. At the same time, we produce a variety of concrete examples where (SMP) holds but where we are unable to prove (‡). Our results are applied to settle two open problems in [K–M]; see Corollary 7.1 and Theorem 7.4.

As explained in [K–M], one can also consider the (generally smaller) quadratic module

M_{S} = {∑_{i=0}^{s} σ_{i}g_{i} : σ_{i} ∈ ∑ ℝ[X]^{2}},

generated by S (here g_{0} := 1) and the corresponding objects

M_{S}^{∨} = {L : ℝ[X] → ℝ | L is linear (≠ 0) and L(M_{S}) ≥ 0},
M_{S}^{lin} = M_{S}^{∨∨} = {f ∈ ℝ[X] : L(f) ≥ 0 for all L ∈ M_{S}^{∨}} and
M_{S}^{‡} = ∪_{d≥0}\overline{M}_{d} where M_{d} := M_{S} ∩ P_{d}.

M_{S}^{lin} is the closure of M_{S} in ℝ[X]. One can also consider the corresponding conditions

(SMP′)  T_{S}^{alg} = M_{S}^{lin},
(MP′)  T_{∅}^{alg} ⊆ M_{S}^{lin},
(†′)  ∀f ∈ ℝ[X], f ≥ 0 on K_{S} ⇒ ∀ real ϵ > 0, f + ϵ ∈ M_{S} and
(‡′) ∀ f ∈ ℝ[X], f ≥ 0 on K_{S} ⇒ ∃q ∈ ℝ[X] such that ∀ real ϵ > 0, f + ϵq ∈ M_{S}.

Again, (MP′) is strictly weaker than (SMP′) and (‡′) is strictly weaker than (†′), but implies (SMP′). If (†′), resp., (‡′), resp., (SMP′), resp., (MP′), holds then we say that (†), resp., (‡), resp., (SMP), resp., (MP) holds for M_{S} or that M_{S} satisfies (†), resp., (‡), resp., (SMP), resp., (MP).

According to Putinar [Pu], also see [J], if K_{S} is compact and there exists N such that N − ∥X∥^{2} ∈ M_{S}, then (†) holds for M_{S}. In certain cases the condition that there exists N such that N − ∥X∥^{2} ∈ M_{S} is automatically fulfilled; see [J–P]. In [M2] various results in [K–M], in particular, the results for dimension extension and for cylinders with compact cross-section, are extended under suitable assumptions to quadratic modules; see [M2, Cor. 4.3, Cor 5.3].

Insofar as it is possible, we develop quadratic module versions of the various results considered. In particular, we prove quadratic module versions of Theorem 1.5 and of Theorem 5.3; see Theorem 6.1 and Corollary 5.5.

2 Basic Lemma

The following general result is useful:

Lemma 2.1 Let C be a compact Hausdorff space, A a commutative ring with 1 with \frac{1}{n} ∈ A for some integer n ≥ 2 and ϕ : A → Cont(C, ℝ) a (unitary) ring homomorphism. Suppose f, g ∈ A are such that ϕ(f) ≥ 0, ϕ(g) ≥ 0 and sf +tg = 1 for some s, t ∈ A. Then there exist σ, τ ∈ A such that σf + τg = 1 and ϕ(σ), ϕ(τ) are strictly positive.

Proof: We suppress ϕ from the notation. Let s, t ∈ A be such that 1 = sf + tg. On the compact set

L_{1} := {p ∈ C | s(p) ≤ 0},

tg = 1 − sf ≥ 1. Thus g > 0 on L_{1} so, for N sufficiently large, s + Ng > 0 on L_{1}. On C\L_{1} this is obviously also true. Define s_{1} = s + Ng, t_{1} = t − Nf. Thus 1 = s_{1}f + t_{1}g in A and s_{1} > 0 on C. Choose a positive rational δ ∈ A so small that δfg < 1 on C. Choose a positive rational ϵ ∈ A so small that, on the compact set

L_{2} = {p ∈ C : g(p) ≤ ϵ},

f > 0, 1 > ϵδf and ϵt_{1} + s_{1}f > 0. Choose k so large that, on the set L_{2}, ϵt_{1} + s_{1}f > s_{1}f(1 − ϵδf)^{k} and, on the set

L_{3} = {p ∈ C : g(p) ≥ ϵ},

s_{1}f(1−ϵδf)^{k} < 1. Choose r = s_{1}δf ∑_{i=0}^{k−1}(1−δfg)^{i}. Choose σ = s_{1}−rg, τ = t_{1}+rf. Thus 1 = σf + τg in A. It remains to verify that σ, τ > 0 on C. Using the identity (1 − z) ∑_{i=0}^{k−1} z^{i} = 1 − z^{k}, we see that, on C,

\begin{aligned}σ & = & s_{1} − rg = s_{1} − s_{1}δfg ∑_{i=0}^{k−1}(1 − δfg)^{i} \\ & = & s_{1} − s_{1}(1 − (1 − δfg)^{k}) = s_{1}(1 − δfg)^{k} > 0. \end{aligned}

On L_{2},

\begin{aligned}τ & = & t_{1} + rf = t_{1} + s_{1}δf^{2} ∑_{i=0}^{k−1}(1 − δfg)^{i} \\ & ≥ & t_{1} + s_{1}δf^{2} ∑_{i=0}^{k−1}(1 − ϵδf)^{i} = t_{1} + (s_{1}f/ϵ)(1 − (1 − ϵδf)^{k}) > 0. \end{aligned}

On L_{3},

\begin{aligned}τ & = & t_{1} + rf = t_{1} + (f/g)rg = t_{1} + (f/g)s_{1}(1 − (1 − δfg)^{k}) \\ & = & t_{1} + s_{1}f/g − (s_{1}f/g)(1 − δfg)^{k} = 1/g − (s_{1}f/g)(1 − δfg)^{k} > 0. \end{aligned}

This completes the proof. □

We apply Lemma 2.1 to C = K_{S} and A = ℝ[X]/I, where I is some ideal of polynomials vanishing on K_{S}. In the next section we will be interested in the case where n = 1 and I = {0}. Note: For K_{S} compact, σ, τ > 0 on K_{S} implies, by [Sc1, Cor. 3], that σ, τ ∈ T_{S}.

Corollary 2.2 Let K_{S} be compact, f, g ≥ 0 on K_{S}. Assume that f and g are relatively prime modulo T_{S} ∩ −T_{S} and that fg ∈ T_{S}. Then f ∈ T_{S} and g ∈ T_{S}.

Proof: Apply Lemma 2.1 with C = K_{S}, A = ℝ[X]/(T_{S} ∩ −T_{S}), together with [Sc1, Cor. 3], to obtain σ, τ ∈ T_{S}, υ ∈ T_{S} ∩ −T_{S} such that 1 = σf + τg + υ. Then f = σf^{2} + τfg + fυ ∈ T_{S} and g = σfg + τg^{2} + gυ ∈ T_{S}. □

Corollary 2.3 Assume f, g ∈ ℝ[X] are relatively prime. Assume further that K_{{fg}} is compact and f, g ≥ 0 on K_{{fg}}. Then

(i) There exist σ, τ ∈ ∑ ℝ[X]^{2} such that 1 = σf + τg. (ii) M_{{fg}} = M_{{f,g}} = T_{{fg}} = T_{{f,g}}.

See [S1, Prop. 4.8] for another proof of Corollary 2.3 (i) in the case n = 1.

Proof: Apply Lemma 2.1 with C = K_{{fg}} and A = ℝ[X] and [Sc1, Cor. 3] to obtain 1 = σf + τg, σ, τ ∈ T_{{fg}}. Thus there exist σ_{i}, τ_{i} ∈ ∑ ℝ[X]^{2}, i = 0, 1 such that σ = σ_{0} + σ_{1}fg, τ = τ_{0} + τ_{1}fg. Then

1 = σf + τg = (σ_{0} + σ_{1}fg)f + (τ_{0} + τ_{1}fg)g = (σ_{0} + τ_{1}g^{2})f + (τ_{0} + σ_{1}f^{2})g.

This proves (i): 1 = σf + τg, σ, τ ∈ ∑ ℝ[X]^{2}. Thus fg = σf^{2}g + τg^{2}f ∈ M_{{f,g}}. Also, f = σf^{2} + τfg ∈ T_{{fg}} = M_{{fg}} and, similarly, g ∈ T_{{fg}} = M_{{fg}}. This proves (ii). □

Lemma 2.1 is also closely related to results in [S2]. We illustrate by giving a proof of two key results in [S2]:

Corollary 2.4 Let K_{S} be compact, f ≥ 0 on K_{S}, and suppose

f ∈ T_{S} + f\sqrt{(f) + (T_{S} ∩ −T_{S})}.

Then f ∈ T_{S}.

Proof: f = σ + fg, σ ∈ T_{S}, g^{k} = af + τ, k ≥ 1, a ∈ ℝ[X], τ ∈ T_{S} ∩ −T_{S}. af + (1 − g^{k}) = 1 − τ so f, 1 − g are relatively prime modulo T_{S} ∩ −T_{S}. Also f(1 − g) = σ ∈ T_{S}. Using this, and the definition of g, one checks that 1 − g ≥ 0 on K_{S}. Corollary 2.2 yields f ∈ T_{S}. □

Corollary 2.5 Let K_{S} be compact, f ≥ 0 on K_{S} and suppose f = σ + τb, where σ, τ ∈ T_{S} and b is such that f = 0 ⇒ b > 0 on K_{S}. Then f ∈ T_{S}.

Proof: By [Sc1, Cor. 3] applied to the set S ∪ {−f^{2}}, b = α − βf^{2}, α, β ∈ T_{S}. Then f = σ + τb = σ + τ(α − βf^{2}) = (σ + τα) − τβf^{2}. The conclusion follows by Corollary 2.4. □

3 Compact Subsets of Lines

In this section, we continue the analysis begun in [K–M, Sect. 2], focusing on the compact case. For related results see [Po–R] and [S2].

If K ⊆ ℝ is a non-empty closed semi-algebraic set then K = K_{N}, for N the set of polynomials defined as follows:

— If a ∈ K and (−∞, a) ∩ K = ∅, then X − a ∈ N.

— If a ∈ K and (a, ∞) ∩ K = ∅, then a − X ∈ N.

— If a, b ∈ K, a < b, (a, b) ∩ K = ∅, then (X − a)(X − b) ∈ N.

— N has no other elements except these. We call N the natural set of generators for K. We adopt the convention that the natural set of generators of the empty set is {−1}. The following can be easily deduced from [K–M] and [Sc1]:

Theorem 3.1 Suppose n = 1.

(a) If K_{S} is not compact, then T_{S} is closed, for any finite set of generators S. Moreover the following are equivalent:

(i) T_{S} is saturated. (ii) T_{S} contains the natural set of generators for K_{S}. (iii) S contains the natural set of generators of K_{S} (up to scalings by positive reals).

(b) If K_{S} is compact, then (†) holds for T_{S}, for any finite set of generators S. Moreover the following are equivalent:

(i) T_{S} is saturated. (ii) T_{S} is closed. (iii) T_{S} contains the natural set of generators for K_{S}.

Note: Unlike the non-compact case (a), T_{S} may be saturated in the compact case (b), even when S does not contain a scaling of the natural set of generators (cf. [K-M, Notes 2.3(4)]). Theorem 3.1 (b) is unsatisfactory unless we have a practical criterion on S for determining when T_{S} is saturated. The following is such a criterion. Here, g′ denotes the derivative of g.

Theorem 3.2 Let K_{S} = ∪_{j=0}^{k}[a_{j} , b_{j}], b_{j−1} < a_{j}, j = 1, …, k, S = {g_{1}, ⋯, g_{s}}. Then T_{S} is saturated if and only if the following two conditions hold:

(a) for each endpoint a_{j} ∃ i ∈ {1, ⋯, s} such that g_{i}(a_{j}) = 0 and g′_{i}(a_{j}) > 0,

(b) for each endpoint b_{j} ∃ i ∈ {1, ⋯, s} such that g_{i}(b_{j}) = 0 and g′_{i}(b_{j}) < 0.

In fact, Theorem 3.2 is just a special case of a general criterion for curves proved in [S2, Th. 5.17]. For completeness we include our own proof.

Proof: We prove the necessity of condition (a). The necessity of condition (b) is proved similarly. Let a = a_{j}. There exists f ∈ ℝ[X] (of degree two), f ≥ 0 on K_{S}, f(a) = 0, f′(a) > 0. Since T_{S} is saturated, f has a presentation f = ∑ σ_{e}g^{e}, σ_{e} ∈ ∑ ℝ[X]^{2}. Since f(a) = 0, each term σ_{e}g^{e} is divisible by X − a. Since f′(a) > 0, there exists e with (σ_{e}g^{e})′(a) > 0. If (X − a)^{2}|σ_{e}g^{e} then (σ_{e}g^{e})′(a) = 0, a contradiction. Thus σ_{e}(a) ≠ 0 and there is some unique i such that g_{i} appears in g^{e} and g_{i}(a) = 0. Then σ_{e}g^{e}/g_{i} is strictly positive at a so (σ_{e}g^{e})′(a) has the same sign as g′_{i}(a). We prove the sufficiency of conditions (a) and (b). Assume (a) and (b) hold. According to Theorem 3.1 (b), it suffices to show that T_{S} contains the natural generators. We begin with the linear generators. Let a = a_{0}. By (a) there exists i such that g_{i} = (X − a)h_{i}, h_{i}(a) = g′_{i}(a) > 0. Then h_{i} and X − a are each ≥ 0 on K_{S} and gcd(h_{i}, X − a) = 1 so, by Corollary 2.2, X − a ∈ T_{S}. A similar argument shows that b − X ∈ T_{S}, where b = b_{k}. Now consider a generator of the form (X − a)(X − b), a = a_{ℓ}, b = b_{ℓ−1}, ℓ ∈ {1, …, k}. By (a) and (b) there exists i, j such that g_{i}(a) = 0, g′_{i}(a) > 0, g_{j}(b) = 0, g′_{j}(b) < 0. Factor g_{i}, g_{j} as g_{i} = (X − a)h_{i}, g_{j} = (X − b)h_{j}. Then

(X − a)(X − b)((X − b)h_{i} + (X − a)h_{j}) = (X − b)^{2}g_{i} + (X − a)^{2}g_{j} ∈ T_{S},

and (X −a)(X −b) and (X −b)h_{i} +(X −a)h_{j} are each ≥ 0 on K_{S} and are relatively prime, so by Corollary 2.2, (X − a)(X − b) ∈ T_{S}. □

Corollary 3.3 Let K_{S} be compact, S = {g_{1}, ⋯, g_{s}}. Assume that K_{S} has no isolated points. Then T_{S} is saturated if and only if, for each endpoint a ∈ K_{S}, there exists i ∈ {1, ⋯, s} such that x − a divides g_{i} but (x − a)^{2} does not.

Corollary 3.4 Let K be a compact semi-algebraic set in ℝ without isolated points, N its set of natural generators, and π the product of the elements of N. Then

M_{{π}} = M_{N} = T_{{π}} = T_{N} .

Proof: Applying Corollary 3.3 to S = {π} we see that T_{{π}} is saturated. Since K_{{π}} = K = K_{N}, and T_{N} is saturated, this implies T_{{π}} = T_{N}. It remains to show that π ∈ M_{N}. Decompose π as

π = fq_{1} … q_{t},

where f = (X − a)(b − X) is the product of the linear natural generators and q_{1}, …, q_{t} are the quadratic natural generators, t ≥ 0. Suppose t = 0. Then π = (X − a)(b − X) ∈ M_{{X−a,b−X}} by Corollary 2.3. Suppose t > 0. Then π = π′q_{t}, where π′ = fq_{1} … q_{t−1}. By Corollary 2.3, π ∈ M_{{π′,q_{t}}}. By induction on t, π′ ∈ M_{N′}, where N′ = {X − a, b − X, q_{1}, …, q_{t−1}}. Thus π ∈ M_{N′∪{q_{t}}} = M_{N}. □

We need “no isolated points” in Corollaries 3.3 and 3.4: Take K = {0}, so N = {X, −X} and π = (X)(−X) = −X^{2}. Also M_{{−X^{2}}} ≠ M_{{X,−X}}. On the other hand, writing −X = a^{2} − b^{2}, a, b ∈ ℝ[X], we see that

−X^{2} = (a^{2} − b^{2})X = a^{2}X + b^{2}(−X) ∈ M_{{X,−X}},

so M_{{X,−X}} = T_{{X,−X}}. This suggests that M_{N} = T_{N} may hold even when K has isolated points. In fact this is the case.

Theorem 3.5 Let K be a compact semi-algebraic set in ℝ, N its set of natural generators. Then M_{N} = T_{N}.

Note: If K is not compact, then M_{N} = T_{N} holds only in a few cases: if |N | = 1 or if |N | = 2 and K has an isolated point; see [K–M, Th. 2.5].

Proof: Let N = {g_{1}, …, g_{s}}. It suffices to show that M_{N} is closed under multiplication, equivalently, that g_{i}g_{j} ∈ M_{N} for all i ≠ j.

Case 1: g_{i}, g_{j} are both linear. g_{i} = X −a, g_{j} = b−X, a ≤ b. If a = b then g_{j} = −g_{i}. Write g_{j} = f^{2} − g^{2}, f, g ∈ ℝ[X]. Then

g_{i}g_{j} = f^{2}g_{i} − g^{2}g_{i} = f^{2}g_{i} + g^{2}g_{j} ∈ M_{{g_{i},g_{j}}}.

If a < b then {g_{i}, g_{j}} are the natural generators of the compact set [a, b], so g_{i}g_{j} ∈ M_{{g_{i},g_{j}}} by Corollary 3.4.

Case 2: g_{i} is linear, g_{j} is quadratic. Replacing X by −X if necessary, we can assume g_{i} = X − a. g_{j} = (X − c)(X − d), a ≤ c < d. If a = c then g_{i}g_{j} ∈ M_{{g_{i},g_{j}}} by [K–M, Th. 2.5]. Suppose a < c. Fix β ∈ ℝ so large that β − X > 0 on K. β − X ∈ M_{N} by [J–P, Remark 4.7]. Applying Corollary 3.4 to [a, c] ∪ [d, β], we see that

g_{i}g_{j} = (X − a)(X − c)(X − d) ∈ M_{{X−a,(X−c)(X−d),β−X}} ⊆ M_{N} .

Case 3: g_{i} = (X − c)(X − d), g_{j} = (X − c′)(X − d′), c < d ≤ c′ < d′. If c′ = d then g_{i}g_{j} ∈ M_{{g_{i},g_{j}}} by [K–M, Th. 2.5]. Suppose c′ > d. Use [J–P, Remark 4.7] to choose α < c and β > d′ so that X − α, β − X ∈ M_{N}. Applying Corollary 3.4 to [α, c] ∪ [d, c′] ∪ [d′, β] yields

g_{i}g_{j} ∈ M_{{X−α,(X−c)(X−d),(X−c′)(X−d′),β−X}} ⊆ M_{N} .
□

Corollary 3.6 For n = 1, K_{S} compact, the following are equivalent:

(i) M_{S} is saturated, i.e., M_{S} = T_{S}^{alg} (ii) M_{S} contains the natural set of generators of K_{S} (iii) M_{S} is closed.

Proof: By [J–P, Remark 4.7] (†) holds for M_{S} so, in particular, (SMP) holds for M_{S}. (i) ⇔ (iii) follows from this fact. (i) ⇒ (ii) is clear. (ii) ⇒ (i) follows from Theorem 3.5 and Theorem 3.1(b). □

Corollary 3.6 can be viewed as the module analog of Theorem 3.1 (b). One would also like to have a module analog of Theorem 3.2.

4 Uniform Degree Bounds

We continue to assume n = 1. Theorems 4.1 and 4.5 and Corollary 4.3, in this section, are essential in the next section, in the proof of Theorem 5.3 and Corollary 5.5. These results, suitably formulated, hold over an arbitrary real closed field.

Fix a non-empty basic closed semi-algebraic set K in ℝ, with N = {s_{1}, …, s_{t}} its set of natural generators. T_{N} is saturated, by Theorem 3.1, so each P ∈ ℝ[X] non-negative on K is expressible as

(1)  P = ∑_{e∈{0,1}^{t}} σ_{e}s_{1}^{e_{1}} … s_{t}^{e_{t}}

with σ_{e} ∈ ∑ ℝ[X]^{2}.

Theorem 4.1 Each P nonnegative on K has a presentation (1), with the degree of each term σ_{e}s_{1}^{e_{1}} … s_{t}^{e_{t}} bounded by the degree of P.

Proof: This is just a matter of keeping track of degrees in the proof of [K–M, Th. 2.2]. □

Corollary 4.2 Suppose K = K_{{g_{1},...,g_{k}}} and T_{{g_{1},...,g_{k}}} is saturated. Then each P nonnegative on K has a presentation

(2)  P = ∑_{i∈{0,1}^{k}} τ_{i}g_{1}^{i_{1}} … g_{k}^{i_{k}},

τ_{i} ∈ ∑ ℝ[X]^{2}, where the degrees of the terms are bounded by some number of the form deg(P) + N, with N depending only on g_{1}, …, g_{k}.

Proof: There exist presentations

s_{j} = ∑_{p∈{0,1}^{k}} ρ_{jp}g_{1}^{p_{1}} … g_{k}^{p_{k}},

with ρ_{jp} ∈ ∑ ℝ[X]^{2}, for each j. Substituting these into (1) and rearranging terms, one obtains (2), with the degree of each term at most

deg(P) + t max{deg(ρ_{jp}g_{1}^{p_{1}} … g_{k}^{p_{k}}) : j, p}.
□

It does not seem possible to choose N in Corollary 4.2 so as to depend only on the degrees of g_{1}, …, g_{k}. Unfortunately, this limits what we are able to prove later, in Section 5, in Theorem 5.3.

We consider module analogs of Theorem 4.1. We continue to denote by s_{1}, …, s_{t} the natural generators of K. Is it possible to find presentations

P = σ_{0} + σ_{1}s_{1} + … + σ_{t}s_{t},

with σ_{i} ∈ ∑ ℝ[X]^{2}, with the terms of bounded degree depending only on the degree of P? For t ≤ 1 it is trivially true, by Theorem 4.1, (with deg(P) serving as the bound). If K is not compact and t ≥ 2, then such a presentation may not exist. (M_{N} is saturated iff either t ≤ 1 or if t = 2 and K has an isolated point; see [K–M, Th. 2.5].) If such a presentation does exist, then deg(P) serves as a degree bound. If K is compact and t = 2, then one checks that deg(P) + 1 serves as a degree bound. To summarize:

Corollary 4.3 If either t ≤ 1 or t = 2 and K either has an isolated point or is compact, then every polynomial P non-negative on K has a presentation

P = σ_{0} + σ_{1}s_{1} + … + σ_{t}s_{t},

with the degree of each term bounded by deg(P) + 1.

What if K is compact and t ≥ 3? In view of Theorem 4.1, this reduces to the question of whether a degree bound exists for representations of the products s_{i}s_{j}, i < j. Such a bound exists if s_{i}, s_{j} are both linear, or, if at least one of s_{i}, s_{j} is quadratic and K_{{s_{i},s_{j}}} has an isolated point. In remaining cases (i.e, the cases where at least one of s_{i}, s_{j} is quadratic and K_{{s_{i},s_{j}}} has no isolated point), there is no degree bound in general.

Example 4.4 (a) Suppose s_{1} = X − a, s_{2} = (X − b)(X − c), where a < b < c are fixed reals. Then, for any real β > c, s_{1}s_{2} has a presentation

(3),  s_{1}s_{2} = σ_{0} + σ_{1}s_{1} + σ_{2}s_{2} + σ_{3}(β − X)

with σ_{i} ∈ ∑ ℝ[X]^{2} depending on β. We claim that, for any such presentation, as β → ∞, the maximum of the degrees of the σ_{i} necessarily tends to infinity. Otherwise, by the Transfer Principle, for any real closed field R ⊇ ℝ and any β ∈ R, β > c, we would have a presentation (3) with σ_{i} ∈ ∑ R[X]^{2}. Choose any such (nonarchimedean) R, e.g., take R to be the field of formal Puiseux series with real coefficients. Denote by v the unique finest valuation on R compatible with the ordering, and choose β to be positive and infinitely large relative to ℝ, i.e., v(β) < 0. Write σ_{i} = ∑ f_{ij}^{2}, f_{ij} ∈ R[X]. Choose d ∈ R, d > 0, with v(d) equal to the minimum of the values of the coefficients of the f_{ij}, i ∈ \{0, 1, 2\} and the \sqrt{β}f_{3j}.

Case 1: v(d) ≥ 0. Then (3) pushes down to the residue field giving an equation of the form s_{1}s_{2} = α_{0} + α_{1}s_{1} + α_{2}s_{2}, α_{i} ∈ ∑ ℝ[X]^{2}. This contradicts [K-M, Th. 2.5].

Case 2: v(d) < 0. Then dividing (3) by d^{2} and pushing down to the residue field yields an equation of the form 0 = α_{0} + α_{1}s_{1} + α_{2}s_{2}, α_{i} ∈ ∑ ℝ[X]^{2}, α_{i} ≠ 0 for some i. Since s_{1} and s_{2} are both strictly positive on the infinite set (a, b), this is not possible.

(b) A similar remark applies to presentations of the form

s_{1}s_{2} = σ_{0} + σ_{1}s_{1} + σ_{2}s_{2} + σ_{3}(β + X) + σ_{4}(β − X), σ_{i} ∈ ℝ[X]^{2},

where s_{1} = (X − a)(X − b), s_{2} = (X − c)(X − d), a < b < c < d are fixed reals, β > max{−a, d}.

We turn now to the case where K is the empty set. We use the next result in the proof of both Theorem 5.3 and Corollary 5.5. Actually, for the proof of Theorem 5.3, we need only a weaker preordering version of the result. The weaker preordering version holds even in the multivariable situation, and is a consequence of the Positivstellensatz.

Theorem 4.5 Given a positive integer k and non-negative integers d_{1}, …, d_{k}, there exists a positive integer N such that for any real closed field R and any set of polynomials S = {g_{1}, …, g_{k}} in one variable X with coefficients in R with deg(g_{j}) ≤ d_{j}, j = 1, …, k, if K_{S} = ∅, then there exist σ_{i} ∈ ∑ R[X]^{2} of degree ≤ N such that −1 = ∑_{i=0}^{k} σ_{i}g_{i}. (Convention: g_{0} = 1.)

Proof: If −1 ∉ M_{S} then, by [Br, Satz 1.8], there exists a real prime ideal p in R[X] such that the quadratic form ϕ^{∗}, which is obtained from ϕ = ⟨1, g_{1}, …, g_{k}⟩ by deleting the entries belonging to p, is strongly anisotropic over the residue field, call it k(p), of R[X] at p. (This works even in the multivariable situation, in fact it works for any commutative ring.) The point is, since we are in the 1-variable situation, k(p) has transcendence degree ≤ 1 over R, so there exists an ordering of k(p) making ϕ^{∗} positive definite [P, Th. 9.4]. By the Transfer Principle, this implies K_{S} ≠ ∅. The degree bounds follow by a standard ultrapower argument. □

Note: Using the identity P = (\frac{P +1)}{2})^{2} − (\frac{P −1}{2})^{2}, we also get degree bounds for presentations P = ∑_{i=0}^{k} σ_{i}g_{i} depending only on k and the degree of P and the degrees of g_{1}, …, g_{k}, for each P ∈ R[X].

5 Subsets of Cylinders

We fix the terminology. We consider the polynomial ring ℝ[X, Y] in n+1 variables X_{1}, …, X_{n}, Y with coefficients in ℝ. We suppose S is a finite subset of ℝ[X, Y]. When is it true that (SMP) (or (‡) or (†)) holds for T_{S}?

Our first result gives a general necessary condition. Let L ⊆ ℝ^{n+1} be any line,

λ : ℝ →^{≅} L ⊆ ℝ^{n+1},

an affine-linear isomorphism onto this line, and λ^{∗} : ℝ[X, Y] → ℝ[Z] the corresponding map between the polynomial rings. Thus λ^{−1}(K_{S}) is the basic closed semi-algebraic subset of ℝ defined by the set λ^{∗}(S).

Theorem 5.1 Suppose that (SMP) holds for T_{S} and λ^{−1}(K_{S}) is not compact. Then λ^{∗}(S) contains the natural generators of λ^{−1}(K_{S}) (up to scaling by positive reals).

Note: If λ^{−1}(K_{S}) is compact then T_{λ^{∗}(S)} satisfies (†). Thus, one can conclude that T_{λ^{∗}(S)} satisfies (†) in any case.

Proof: By Theorem 3.1, it suffices to prove that the (closed) preordering T_{λ^{∗}(S)} contains the natural generators of λ^{−1}(K_{S}). To simplify the notation needed in the proof, the coordinates of ℝ^{n+1} are arranged so that L is the Y -axis and λ : ℝ → L ⊆ ℝ^{n+1} is the isomorphism defined by y → (0, y). Thus, λ^{∗} : ℝ[X, Y] → ℝ[Y] is given by X_{i} ↦ 0, i = 1, …, n, Y ↦ Y. If λ^{−1}(K_{S}) = ℝ there is nothing to prove. So assume that λ^{−1}(K_{S}) ≠ ℝ.

Case 1: Suppose that λ^{−1}(K_{S}) contains a smallest element, say a. (The case of a largest element is handled in exactly the same way.) The natural generator belonging to a is the linear polynomial Y − a. It is claimed that Y − a belongs to T_{λ^{−1}(S)}. For each 1 ≤ n ∈ ℕ define b_{n} = a − \frac{1}{n}, c_{n} = b_{n} − n. Then: (b_{n})_{n} increases monotonically with limit a; (c_{n})_{n} decreases monotonically with limit −∞; for all n, b_{n} > c_{n}. The interval [c_{n}, b_{n}] ⊆ L is compact and does not meet the closed set K_{S}. Thus, the distance δ_{n} = dist([c_{n}, b_{n}], K_{S}) is positive.

For each n, let E_{n} be the ellipsoid with center \frac{1}{2}(b_{n} + c_{n}), one axis on the line L, the other perpendicular to L. The half axis on L has length \frac{1}{2}(b_{n} − c_{n}), the perpendicular half axis has length δ_{n}. The ellipsoid is given by the polynomial

P_{n} = (\frac{Y − \frac{1}{2}(b_{n} + c_{n})}{\frac{1}{2}(b_{n} − c_{n})})^{2} + ∑_{i=1}^{n} (\frac{X_{i}}{δ_{n}} )^{2} − 1.

The interior of the ellipsoid is the set

N(P_{n}) = {(x, y) ∈ ℝ^{n+1} : P_{n}(x, y) < 0}.

Every point of N(P_{n}) has a distance less than δ_{n} from the line segment [c_{n}, b_{n}], hence N(P_{n}) ∩ K_{S} = ∅. This means that P_{n} is positive semidefinite on K_{S}. Each linear functional L_{0} on ℝ[Y] non-negative on T_{λ^{∗}(S)} lifts to a linear functional L_{1} = L_{0}∘λ^{∗} on ℝ[X, Y] non-negative on T_{S}. Since T_{S} satisfies (SMP), L_{0}(λ^{∗}(P_{n})) = L_{1}(P_{n}) ≥ 0 for all such L_{0}. It follows that λ^{∗}(P_{n}) ∈ T_{λ^{∗}(S)}^{lin} = T_{λ^{∗}(S)}. One computes

λ^{∗}(P_{n}) = \frac{4}{b_{n} − c_{n}} · (\frac{1}{b_{n} − c_{n}} Y^{2} + (1 − \frac{2b_{n}}{b_{n} − c_{n}} ) Y − (b_{n} − \frac{b_{n}^{2}}{b_{n} − c_{n}} ))

Since \frac{1}{b_{n}−c_{n}} converges to 0 as n → ∞ one concludes that the sequence

(\frac{b_{n} − c_{n}}{4} λ^{∗}(P_{n}))_{n}

of polynomials converges to Y −a. Once again, closedness of T_{λ^{∗}(S)} implies Y −a ∈ T_{λ^{∗}(S)}.

Case 2: Suppose that a, b ∈ λ^{−1}(K_{S}), a < b, (a, b) ∩ λ^{−1}(K_{S}) = ∅. It is claimed that the natural generator (Y − a)(Y − b) belongs to T_{λ^{∗}(S)}. The method of proof is much the same as in Case 1; let n_{0} ∈ ℕ be such that b − a > \frac{2}{n_{0}} . For n ≥ n_{0} one defines a_{n} = a + \frac{1}{n}, b_{n} = b − \frac{1}{n}. Then: a_{n} < b_{n}; a_{n} ↓ a; b_{n} ↑ b; the compact line segment [a_{n}, b_{n}] does not meet K_{S}. Set δ_{n} = dist([a_{n}, b_{n}], K_{S}), and let

P_{n} = (\frac{Y − \frac{1}{2}(b_{n} + a_{n})}{\frac{1}{2}(b_{n} − a_{n})})^{2} + ∑_{i=1}^{n} (\frac{X_{i}}{δ_{n}} )^{2} − 1.

Again, N(P_{n}) is the interior of the associated ellipsoid E_{n}, and N(P_{n}) ∩ K_{S} = ∅. Thus, P_{n} ≥ 0 on K_{S}. Again, since T_{S} satisfies (SMP) and T_{λ^{∗}(S)} is closed one gets λ^{∗}(P_{n}) ∈ T_{λ^{∗}(S)}. The sequence

((\frac{1}{2}(b_{n} − a_{n}))^{2} · λ^{∗}(P_{n}))_{n}

converges to

(Y − \frac{1}{2}(b + a))^{2} − (\frac{1}{2}(b − a))^{2}
= Y^{2} − (b + a)Y + ab = (Y − a)(Y − b).

Once again, since T_{λ^{∗}(S)} is closed the limit also belongs to T_{λ^{∗}(S)}. □

For each a ∈ ℝ^{n}, let L_{a} = λ_{a}(ℝ), where λ_{a} : ℝ → ℝ^{n+1} is defined by y ↦ (a, y). The corresponding map λ_{a}^{∗} : ℝ[X, Y] → ℝ[Y] is given by X_{i} ↦ a_{i}, Y ↦ Y. The fiber λ_{a}^{−1}(K_{S}) is the basic closed set in ℝ defined by λ_{a}^{∗}(S).

Theorem 5.1 and [Sc2, Th. 1] (see Theorem 1.5) combine to yield the following:

Corollary 5.2 Let S ⊆ ℝ[X, Y], where ℝ[X, Y] denotes the polynomial ring in n+ 1 variables X_{1}, …, X_{n}, Y. Suppose the image of K_{S} under the projection (x, y) ↦ x is bounded. For each a ∈ ℝ^{n}, define λ_{a} : ℝ → ℝ^{n+1} as above. Then:

(a) T_{S} satisfies (MP).

(b) The following are equivalent:

(i) T_{S} satisfies (SMP). (ii) For each non-compact λ_{a}^{−1}(K_{S}), λ_{a}^{∗}(S) contains the natural generators for λ_{a}^{−1}(K_{S}) (up to scalings by positive reals).

Proof: (a) Since every polynomial non-negative on ℝ is a sum of squares, (MP) holds for each T_{λ_{a}^{∗}(S)}, so (a) is immediate from [Sc2, Th. 1]. (b) The implication (i) ⇒ (ii) follows from Theorem 5.1. (ii) ⇒ (i): To be able to apply [Sc2, Th. 1] we must show that each T_{λ_{a}^{∗}(S)} satisfies (SMP). For λ_{a}^{−1}(K_{S}) compact, this is always the case. For λ_{a}^{−1}(K_{S}) non-compact it is a consequence of assumption (ii). □

See Corollary 6.4 below for a module version of Corollary 5.2.

We are unable to prove that Corollary 5.2(b) holds with (SMP) replaced by (‡). However, we are able to prove the following weaker result:

Theorem 5.3 Let S ⊆ ℝ[X, Y], where ℝ[X, Y] denotes the polynomial ring in n + 1 variables X_{1}, …, X_{n}, Y. Suppose the following conditions hold:

(a) There exists N such that N − ∥X∥^{2} ∈ T_{S} and

(b) For each non-empty λ_{a}^{−1}(K_{S}), λ_{a}^{∗}(S) contains the natural generators for λ_{a}^{−1}(K_{S}) (up to scaling by positive reals).

Then T_{S} satisfies (‡), i.e., for each P ∈ ℝ[X, Y] such that P ≥ 0 on K_{S}, there exists Q ∈ ℝ[X, Y] such that for all real ϵ > 0, P + ϵQ ∈ T_{S}.

See Corollary 5.5 below for a module version of Theorem 5.3.

Note: It follows from [Sc1, Cor. 3] that condition (a) of Theorem 5.3 holds iff there exist elements f_{1}, …, f_{t} ∈ T_{S} ∩ ℝ[X] such that the subset K_{{f_{1},...,f_{t}}} of ℝ^{n} is compact. Proof: Let B = {a ∈ ℝ^{n} : ∥a∥^{2} ≤ N}. Suppose that P ∈ ℝ[X, Y] belongs to T_{S}^{alg}. Writing P = ∑_{i=0}^{k} p_{i}(X)Y^{i} one has P_{a} := λ_{a}^{∗}(P) = ∑_{i=0}^{k} p_{i}(a)Y^{i} ∈ T_{λ_{a}^{∗}(S)}^{alg} for each a ∈ B. Because of condition (b), T_{λ_{a}^{∗}(S)} = T_{λ_{a}^{∗}(S)}^{alg} for each a. Thus, for each a there is a presentation

(4)  P_{a} = ∑_{i∈{0,1}^{t}} σ_{a,i}λ_{a}^{∗}(s_{1})^{i_{1}} · . . . · λ_{a}^{∗}(s_{t})^{i_{t}}

(where S = {s_{1}, …, s_{t}} and the σ_{a,i} ∈ ℝ[Y] are sums of squares). By assumption (b) we may assume there is a uniform bound for the degrees of the sums of squares occuring in these representations that is independent from a. If λ_{a}^{−1}(K_{S}) ≠ ∅ then the degrees can be bounded by k by Theorem 4.1. For λ_{a}^{−1}(K_{S}) = ∅ a uniform bound exists by the note following Theorem 4.5. Pick any ε > 0. For each a, consider

∑_{i∈{0,1}^{t}} σ_{a,i}s_{1}^{i_{1}} · . . . · s_{t}^{i_{t}}, P − ∑_{i∈{0,1}^{t}} σ_{a,i}s_{1}^{i_{1}} · . . . · s_{t}^{i_{t}}

as polynomials in the variable Y with coefficients from ℝ[X]. There is some large even number D that bounds the Y –degrees of these polynomials and is independent from a. Write

∑_{i∈{0,1}^{t}} σ_{a,i}s_{1}^{i_{1}} · . . . · s_{t}^{i_{t}} = ∑_{d=0}^{D} p_{a,d}Y^{d}, P − ∑_{i∈{0,1}^{t}} σ_{a,i}s_{1}^{i_{1}} · . . . · s_{t}^{i_{t}} = ∑_{d=0}^{D} q_{a,d}Y^{d}.

Note that q_{a,0}(a) = … = q_{a,D}(a) = 0 for each a ∈ B. Hence there is an open semi-algebraic neighborhood U_{a} of a in ℝ^{n} such that

∑_{d=0}^{D} |q_{a,d}(x)| < \frac{ε}{2}

for every x ∈ U_{a}. By compactness of B, there is a finite subset E of B such that B ⊆ ⋃_{a∈E} U_{a}. On B there is a continuous semi-algebraic partition of unity subordinate to this cover, say e_{a}, a ∈ E, with 0 ≤ e_{a} ≤ 1 and ∑_{a∈E} e_{a}(x) = 1 for

each x ∈ B. Each e_{a} has a nonnegative square root f_{a}, which is also a continuous semi-algebraic function. Using Stone–Weierstrass approximation on the compact set B one finds polynomials F_{a}, a ∈ E, so close to the continuous semi-algebraic functions f_{a} that

| ∑_{a∈E} p_{a,d}(f_{a}^{2} − F_{a}^{2})(x)| < \frac{ε}{2}

for every d = 0, …, D and every x ∈ B. The polynomial P can be written in the form P = P_{1} + P_{2} where

P_{1} = ∑_{i∈{0,1}^{t}} ( ∑_{a∈E} σ_{a,i}F_{a}^{2}) s_{1}^{i_{1}} · . . . · s_{t}^{i_{t}} ∈ T_{S},

P_{2} = ∑_{d=0}^{D} r_{d}Y^{d} = ∑_{d=0}^{D} ( ∑_{a∈E} q_{a,d}F_{a}^{2}) Y^{d}

+ ∑_{d=0}^{D} ( ∑_{a∈E} q_{a,d}(f_{a}^{2} − F_{a}^{2})) Y^{d} + ∑_{d=0}^{D} ( ∑_{a∈E} p_{a,d}(f_{a}^{2} − F_{a}^{2})) Y^{d}.

The choice of the approximations yields

|r_{d}(x)| = | ∑_{a∈E} q_{a,d}f_{a}^{2}(x) + ∑_{a∈E} p_{a,d}(f_{a}^{2} − F_{a}^{2})(x)|
≤ | ∑_{a∈E} q_{a,d}f_{a}^{2}(x)| + | ∑_{a∈E} p_{a,d}(f_{a}^{2} − F_{a}^{2})(x)|
≤ | ∑_{a∈E} q_{a,d}(x)| + | ∑_{a∈E} p_{a,d}(f_{a}^{2} − F_{a}^{2})(x)| < ε

for each x ∈ B. But then [Sc1, Cor. 3] implies that ε ± r_{d} ∈ T_{{N−∥X∥^{2}}} for every d = 0, …, D. The argument in [K–M, Th. 5.1] shows that

εY^{2} + (ε + r_{d})Y + ε ∈ (Y + 1)^{2}T_{{N−∥X∥^{2}}} + (Y^{2} + 1)T_{{N−∥X∥^{2}}}.

As in loc.cit., one adds εQ where

Q = 2 + Y + 3Y^{2} + Y^{3} + 3Y^{4} + … + 3Y^{D−2} + Y^{D−1} + 2Y^{D}

to both sides of P = P_{1} + P_{2} and obtains

\begin{aligned}P + εQ = & ∑_{i=\{0,1\}^{t}} ( ∑_{a∈E} σ_{a,i}F_{a}^{2}) s_{1}^{i_{1}} · . . . · s_{t}^{i_{t}} \\ & + ∑_{{d=0 \atop d even}}^{D}(ε + r_{d})Y^{d} + ∑_{{d=0 \atop d odd}}^{D} εY^{d+1} + (ε + r_{d})Y^{d} + εY^{d−1} ∈ T_{S}. \end{aligned}

Since the polynomial Q is independent from ε, the proof is finished. □

There is one case for which the results proved so far give even a necessary and sufficient condition for (‡) to hold:

Corollary 5.4 Let S ⊆ ℝ[X, Y], where ℝ[X, Y] denotes the polynomial ring in n + 1 variables X_{1}, …, X_{n}, Y. Suppose the following conditions hold:

(a) There exists N such that N − ∥X∥^{2} ∈ T_{S} and

(b) Each nonempty λ_{a}^{−1}(K_{S}) is unbounded. Then the following conditions are equivalent:

(i) T_{S} satisfies condition (‡). (ii) For each non-empty λ_{a}^{−1}(K_{S}), λ_{a}^{∗}(S) contains the natural generators for λ_{a}^{−1}(K_{S}) (up to scaling by positive reals).

If we look carefully at the proof of Theorem 5.3, we see that in fact we have proved more than is stated: We have shown that, for each ϵ > 0, P = P_{1} + P_{2},

P_{1} ∈ T_{S} and P_{2} + ϵQ ∈ ∑_{k} T_{{N−∥X∥^{2}}}Y^{2k} + ∑_{k} T_{{N−∥X∥^{2}}}Y^{2k}(1 + Y)^{2}.

We may not need all the various sorts of terms in T_{S} to represent P_{1}. If S = {s_{1}, …, s_{k}} and e_{i} ∈ {0, 1}, ∑ e_{i} > 1, we need the term involving s_{1}^{e_{1}} … s_{k}^{e_{k}} only if there exists a such that the λ_{a}^{∗}(s_{i}) with e_{i} = 1 actually occur as (distinct) natural generators of λ_{a}^{−1}(K_{S}). For example, if the hypothesis of Corollary 4.3 holds for each non-empty fiber, then we can arrange things so P_{1} ∈ M_{S}.

Corollary 5.5 Let S ⊆ ℝ[X, Y], where ℝ[X, Y] denotes the polynomial ring in n + 1 variables X_{1}, …, X_{n}, Y. Suppose the following conditions hold:

(a) There exists N such that N − ∥X∥^{2} ∈ M_{S}.

(b) For each non-empty λ_{a}^{−1}(K_{S}), λ_{a}^{∗}(S) contains the natural generators for λ_{a}^{−1}(K_{S}) (up to scaling by positive reals).

(c) Each non-empty λ_{a}^{−1}(K_{S}) has the form (−∞, ∞) or (−∞, p], [q, ∞), (−∞, p]∪ [q, ∞) (perhaps with a discrete point added) or [p, q].

Then M_{S} satisfies (‡), i.e., for each P ∈ ℝ[X, Y] such that P ≥ 0 on K_{S}, there exists Q ∈ ℝ[X, Y] such that for all real ϵ > 0, P + ϵQ ∈ M_{S}.

6 Module Version of Schmüdgen’s Theorem

We prove the following module version of Schmüdgen’s Theorem 1.5 quoted in the introduction:

Theorem 6.1 Let h_{1}, …, h_{d} ∈ ℝ[X]. Suppose there exists N such that N − ∑_{i=1}^{d} h_{i}^{2} ∈ M_{S}. Denote by V_{h} ⊆ ℝ^{d} the algebraic set associated to the ℝ-algebra ℝ[h] := ℝ[h_{1}, …, h_{d}]. Suppose that the image of K_{S} under the map ℝ^{n} → V_{h}, x ↦ (h_{1}(x), …, h_{d}(x)), is dense in the set {z ∈ V_{h} : g(z) ≥ 0 for all g(h) ∈ ℝ[h] ∩ M_{S}}. If (SMP) (resp., (MP)) holds for M_{S_{λ}} for each λ in the closed ball of radius \sqrt{N} about the origin in ℝ^{d}, then (SMP) (resp., (MP)) holds for M_{S}.

The density assumption in Theorem 6.1 is somewhat restrictive. At the same time, this assumption holds in a variety of interesting cases: Consider the case d = 1. Suppose h ∈ ℝ[X] is non-constant and bounded on K_{S}. Then V_{h} is identified with ℝ and the closure of the image of K_{S} in V_{h} is some disjoint union of closed intervals, say ∪_{i=1}^{k}[a_{i}, b_{i}], b_{i−1} < a_{i}. Adding the natural generators b_{k} − h, h − a_{1}, (h − b_{i−1})(h − a_{i}), i = 2, …, k to S does not change K_{S}. Once this is done, N − h^{2} ∈ M_{S} for N sufficiently large, and the density assumption of Theorem 6.1 holds. Similar remarks apply if d > 1 and h_{1}, …, h_{d} ∈ ℝ[X] are bounded on K_{S}, provided the closure of the image of K_{S} in V_{h} is a basic closed semi-algebraic set in V_{h} (this latter condition is automatic if d = 1).

The proof of Theorem 6.1 follows along the same lines as the proof given in [Sc2]. We need two lemmas.

Lemma 6.2 Suppose p ∈ ℝ[h], p ≥ 0 on K_{S}. Then p + ϵ ∈ M_{S} holds for all real ϵ > 0.

Proof: The hypothesis N −∑_{i=1}^{d} h_{i}^{2} ∈ M_{S} implies the quadratic module ℝ[h]∩M_{S} is Archimedean in ℝ[h]. Let z ∈ V_{h} be such that g(z) ≥ 0 for all g(h) ∈ ℝ[h] ∩ M_{S}. Write p = f(h), f ∈ ℝ[Z_{1}, …, Z_{d}]. By hypothesis, there is a sequence x^{(j)} in K_{S}, (h_{1}(x^{(j)}), …, h_{d}(x^{(j)}) → z, so p(x^{(j)}) = f(h_{1}(x^{(j)}), …, h_{d}(x^{(j)})) → f(z) as j → ∞. It follows that f(z) ≥ 0 for any such z. By Jacobi’s generalization of the Kadison-Dubois Theorem [J, Th. 4], this implies p + ϵ = f(h) + ϵ ∈ M_{S} for any real ϵ > 0. □

We use the notation of [Sc2]: L is a linear functional on ℂ[X], L(M_{S}) ≥ 0. Denote by N the ideal in ℂ[X] defined by N = \{p ∈ ℂ[X] : L(p\overline{p}) = 0\}. The algebra D_{L} = ℂ[X]/N comes equipped with an involution p + N ↦ \overline{p} + N and a scalar product ⟨p_{1} + N , p_{2} + N ⟩ := L(p_{1}\overline{p}_{2}). Each p ∈ ℂ[X] defines a linear operator π_{L}(p) on D_{L} by π_{L}(p)(q + N) = pq + N. We use the following easy version of [Sc2, Prop. 2]:

Lemma 6.3 If p ∈ ℂ[h] then the operator π_{L}(p) on D_{L} is bounded and ∥π_{L}(p)∥ ≤ ∥p∥_{K} where ∥p∥_{K} := sup{|p(x)| : x ∈ K_{S}}.

Proof: Using ∥π_{L}(p)∥^{2} = ∥π_{L}(p\overline{p})| and ∥p∥^{2} = ∥p\overline{p}∥, we are reduced to the case p ∈ ℝ[h]. Fix ϵ > 0 and let ρ^{2} = ∥p∥_{K}^{2} + ϵ. By Lemma 6.2, ρ^{2} − p^{2} ∈ M_{S} so, for q ∈ ℂ[X], ρ^{2}q\overline{q} − p^{2}q\overline{q} = (ρ^{2} − p^{2})q\overline{q} ∈ M_{S}. It follows that ρ^{2}L(q\overline{q}) ≥ L(p^{2}q\overline{q}), i.e., ρ^{2}∥q + N ∥^{2} ≥ ∥π_{L}(p)(q + N)∥^{2}. Since q ∈ ℂ[X] is arbitrary, this implies ρ ≥ ∥π_{L}(p)∥. Since ϵ > 0 is arbitrary, the result follows. □

It follows from Lemma 6.3 that the operators π_{L}(h_{1}), …, π_{L}(h_{d}) are bounded. One finishes the proof of Theorem 6.1 now, exactly as in [Sc2].

In terms of basic closed semi-algebraic subsets of cylinders with compact crosssection, set-up as in Section 5, Theorem 6.1 yields the following:

Corollary 6.4 Let S ⊆ ℝ[X, Y], where ℝ[X, Y] denotes the polynomial ring in n + 1 variables X_{1}, …, X_{n}, Y. Suppose N − ∥X∥^{2} ∈ M_{S} and, for each x ∈ ℝ^{n}, x ∉ \overline{π(K_{S})}, there exists g ∈ M_{S} ∩ ℝ[X] such that g(x) < 0, where π : ℝ^{n+1} → ℝ^{n} denotes the projection (x, y) ↦ x. Then:

(a) (MP) holds for M_{S}.

(b) If (SMP) holds for M_{λ_{a}^{∗}(S)} for each non-compact fiber λ_{a}^{−1}(K_{S}) then (SMP) holds for M_{S}.

7 Applications

In [K–M] the authors asked whether [K–M, Cor. 5.4] generalizes to sets of the form K × L, K compact, L non-compact closed semi-algebraic in ℝ (with the natural description). We are now able to prove this. See [Po, Cor. 3] for another proof.

Denote by ℝ[X, Y] the polynomial ring in n + 1 variables X_{1}, …, X_{n}, Y. Consider a subset S = {g_{1}, …, g_{s}} of ℝ[X, Y] where the polynomials g_{1}, …, g_{s} involve only the variables X_{1}, …, X_{n}, so K_{S} has the form K × ℝ, K ⊆ ℝ^{n}. We further assume that K is compact. Given L closed semi-algebraic in ℝ, let N ⊂ ℝ[Y] be the natural set of generators for L. In this situation we have the following:

Corollary 7.1 Let S_{L} := S ∪ N (so K_{S_{L}} = K × L). Then (‡) holds for T_{S_{L}}.

Proof: Immediate from Theorem 5.3. □

Our next application is to generalized polyhedra. Assume that K_{S} is the basic closed semi-algebraic set in ℝ^{m}, m ≥ 1, defined by S = {ℓ_{1}, …, ℓ_{s}}, where ℓ_{1}, …, ℓ_{s} are linear, so K_{S} is a closed polyhedron. If K_{S} is compact then, by [J–P, Th. 4.2], (†) holds for M_{S}. What if K_{S} is not compact? If K_{S} contains a cone of dimension 2 then, by Theorem 1.2, (MP) fails for T_{S}. In [K–M] the authors ask whether (‡) holds in the remaining case, i.e., when K_{S} is not compact and does not contain a cone of dimension 2. We are now in a position to settle this question completely.

Lemma 7.2 Let P be a closed polyhedron in ℝ^{m} and let a ∈ P. Then P is compact if and only if there is no half line starting at a and contained in P.

Lemma 7.3 Let P be a non-compact closed polyhedron in ℝ^{m} and let a ∈ P. Then the following are equivalent:

(i) There are at most two half lines starting at a and contained in P. (ii) There is a subset S_{1} of the set of linear polynomials defining P such that K_{S_{1}} is a cylinder with compact cross-section.

The proofs of Lemmas 7.2 and 7.3 are elementary and will be omitted.

Theorem 7.4 Let P be a closed polyhedron in ℝ^{m} defined by a finite set S of linear polynomials.

(i) If P is compact then (†) holds for M_{S}. (ii) If P is not compact but does not contain a 2-dimensional cone then (‡) holds for M_{S}. (iii) If P contains a 2-dimensional cone then (MP) fails for T_{S}.

Proof: It only remains to prove (ii). We can assume m ≥ 2. Set n = m − 1. According to Lemmas 7.2 and 7.3, after a suitable affine change in coordinates, there exists a subset S_{1} ⊆ S ∩ ℝ[X_{1}, …, X_{n}] such that K_{S_{1}} is a cylinder with compact cross-section. The result follows, applying Corollary 5.5. The fact that condition (a) of Corollary 5.5 holds follows by applying [J–P, Theorem 4.2] to the compact polyhedron K_{S_{1}} ∩ ℝ^{n}. □

8 Examples in the Plane

Example 8.1 Consider {Y − X^{2}} ⊆ ℝ[X, Y]. K_{{Y −X^{2}}} is the region above the parabola Y = X^{2} in ℝ^{2}. K_{{Y −X^{2}}} does not contain a 2–dimensional cone. On the other hand, the isomorphism

ϕ : ℝ^{2} → ℝ^{2}, (x, y) ↦ (x, y − x^{2}),

carries K_{{Y −X^{2}}} onto K_{{Y }} which does contain a 2–dimensional cone. By [K–M, Th. 3.5] and Proposition 1.2, T_{{Y }} = T_{{Y }}^{lin} ⊉ T_{∅}^{alg}. Applying the induced algebra isomorphism

ϕ^{∗} : ℝ[X, Y] → ℝ[X, Y], X ↦ X, Y ↦ Y − X^{2},

yields T_{{Y −X^{2}}} = T_{{Y −X^{2}}}^{lin} ⊉ T_{∅}^{alg}, so T_{{Y −X^{2}}} is closed and (MP) fails for T_{{Y −X^{2}}}.

Example 8.2 Let S_{1} = {X, 1 − X, Y − 1, 1 − XY }, S_{2} = {X, 1 − X, XY − 1}. By Corollary 5.5, M_{S_{1}} and M_{S_{2}} satisfy condition (‡).

Example 8.3 Let S_{1} = {X, 1 − X, Y^{2} − X^{3}}, S_{2} = {X, 1 − X, Y^{3} − X^{2}}. By Corollary 5.5, M_{S_{1}} satisfies (‡). By Theorem 5.1, T_{S_{2}} does not satisfy (SMP). By Corollary 6.4, M_{S_{2}} does satisfy (MP).

In [K–S] the authors ask for examples where (SMP) holds but (‡) does not hold, and for examples where T_{S}^{lin} ≠ T_{S}^{‡}. Although we are unable to settle this issue yet, we note that T_{S}^{lin} = T_{S}^{‡} does hold when n = 1. Also, in the special case where S is a set of linear polynomials, T_{S}^{lin} = T_{S}^{‡} holds when n = 2 (using Theorem 7.4 and [K–M, Th. 3.5]), and also when n = 3, except possibly in the case when K_{S} contains a 2-dimensional cone but does not contain a 3-dimensional cone. What happens in this exceptional case is not clear.

Also, in this regard, the following two examples are interesting.

Example 8.4 Let S = {X, Y, 1 − XY }. The polynomial XY is bounded on K_{S} and one checks that (SMP) holds for T_{S∪{XY −λ,−(XY −λ)}}, for each 0 ≤ λ ≤ 1. Thus by Theorem 1.5, (SMP) holds for T_{S}. It is not known if (‡) holds for T_{S}. This example is due to K. Schmüdgen.

We note the following:

Proposition 8.5 Let S and S′ ⊆ ℝ[X] (any number of variables) and assume that K_{S} = K_{S′}. Assume that (SMP) holds for T_{S} and that S ⊆ T_{S′}^{lin}. Then (SMP) holds for T_{S′}.

Proof: If S ⊆ T_{S′}^{lin} then T_{S} ⊆ T_{S′}^{lin} (since T_{S′}^{lin} is a preordering), therefore T_{S}^{lin} ⊆ T_{S′}^{lin}. But T_{S′}^{alg} = T_{S}^{alg} = T_{S}^{lin}. It follows that T_{S′}^{alg} = T_{S}^{lin}. □

Example 8.6 In Example 8.2, the set K_{S_{2}} was defined using the set S_{2} = {X, 1 − X, XY − 1}. But it can also be defined by the set S_{3} = {1 − X, Y, XY − 1}. According to Theorem 1.5, (SMP) holds for T_{S_{3}}. Is it true that (‡) holds for T_{S_{3}}? This would be true by Theorem 5.3 if we could show that X + N ∈ T_{S_{3}} for N sufficiently large. Unfortunately, we have the following:

Claim 1: If N is a positive integer, then X + N ∉ T_{S_{3}}. Assuming by way of contradiction that X + N ∈ T_{S_{3}}, there is a representation

X + N = ∑_{i,j,k=0}^{1} σ_{ijk}(1 − X)^{i}(XY − 1)^{j}Y^{k}

with σ_{ijk} sums of squares in ℝ[X, Y]. Since the variable Y does not occur on the left hand side one concludes that σ_{ijk} = 0 if j + k ≥ 1, hence

X + N = σ_{0} + σ_{1}(1 − X),

where σ_{0}, σ_{1} are sums of squares in ℝ[X]. But then (1+σ_{1})(X+N) = σ_{0}+(N +1)σ_{1}, hence X + N is a sum of squares in ℝ(X) – which is false.

On the other hand, we have:

Claim 2: X + εY ∈ T_{S_{3}} for every 0 < ε ∈ ℝ. Note that X + εY = (X + \frac{1}{n}Y) + (ε − \frac{1}{n})Y ∈ T_{S_{3}} if 1 ≤ n ∈ ℕ such that \frac{1}{n} ≤ ε and X + \frac{1}{n}Y ∈ T_{S_{3}}. Thus it suffices to prove that there are arbitrarily large numbers n ∈ ℕ with nX + Y − 1 ∈ T_{S_{3}}. The proof is by induction on n.

n = 1: (XY − 1)(1 − X) = XY − 1 − X^{2}Y + X ∈ T_{S_{3}}. Adding X^{2}Y ∈ T_{S_{3}} and Y (1 − X) ∈ T_{S_{3}} one gets X + Y − 1 ∈ T_{S_{3}}.

n ⇒ n + 1: (n + 1)X + Y − 1 − nX^{2} − XY = (nX + Y − 1)(1 − X) ∈ T_{S_{3}}. Adding nX^{2} ∈ T_{S_{3}}, XY − 1 ∈ T_{S_{3}} and 1 ∈ T_{S_{3}} one obtains (n + 1)X + Y − 1 ∈ T_{S_{3}}.

By Claim 2, X ∈ T_{S_{3}}^{‡}, so T_{S_{2}} ⊆ T_{S_{3}}^{‡} ⊆ T_{S_{3}}^{lin}. Using Proposition 8.5 this provides a second proof that (SMP) holds for T_{S_{3}}. The question of whether or not (‡) holds for T_{S_{3}} remains open.

9 Some Open Problems

1. In Theorem 5.3, is it possible to replace assumption (a) with the (apparently weaker) assumption that there exists N such that N − ∥X∥^{2} ≥ 0 on K_{S}? Note: If the answer is ‘no’ then we have an example where (SMP) holds for T_{S} but (‡) fails for T_{S}. This would answer Open Problem 3 in [K–M, Sect. 6].

2. In Theorem 5.3, are the strong assumptions (b) on the fibers actually necessary? If they are necessary then again we have the answer to Open Problem 3 in [K–M].

3. Does the converse of Theorem 1.5 hold? This seems unlikely. By Theorem 5.1, it is true when the fibers are linear. To the extent that the converse of Theorem 1.5 is true, it provides a way of studying the question of when (SMP) and (MP) hold by induction on the dimension.

4. Suppose n = 2, S = {X, Y, 1−(X −1)(Y −1)}. Does (SMP) hold for T_{S}? Note: The only polynomials which are bounded on K_{S} are the constants, so Theorem 1.5 does not apply in this case. This problem is due to K. Schmüdgen.

5. Is it true that if K_{S} = K_{S′}, (‡) holds for T_{S} and S ⊆ T_{S′}^{‡} then (‡) holds for T_{S′}. (Compare to Proposition 8.5.) This seems unlikely.

6. For n = 1, does there exist a finite set S in ℝ[X] with K_{S} compact and M_{S} ≠ T_{S}? In particular, does there exist a finite set S in ℝ[X] with K_{S} compact, T_{S} saturated, and M_{S} ≠ T_{S}? These questions are of interest already in the case |S| = 2. Note: If K_{S} is compact, then M_{S}^{lin} = T_{S}^{alg}, so M_{S} = T_{S} holds whenever M_{S} is closed.

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Kuhlmann and Marshall: Research Unit Algebra & Logic, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK S7N 5E6, Canada email: skuhlman@math.usask.ca, marshall@math.usask.ca Schwartz: Fakultät für Mathematik und Informatik, Universität Passau, D-94030 Passau, Germany email: schwartz@mathematik.uni-passau.de