Fields Institute Communications Volume 00, 0000

Bounds for representations of polynomials positive on compact semi-algebraic sets

Alexander Prestel

Fachbereich Mathematik und Statistik Universität Konstanz 78457 Konstanz Germany

Dedicated to Murray Marshall on the occasion of his 60-th birthday

Abstract. By Schmüdgen’s Theorem, polynomials f ∈ ℝ[X_{1}, …, X_{n}] strictly positive on a bounded basic semialgebraic subset of ℝ^{n}, admit a certain representation involving sums of squares σ_{i} from ℝ[X_{1}, …, X_{n}]. We show the existence of effective bounds on the degrees of the σ_{i} by proving first a suitable version of Schmüdgen’s Theorem over nonarchimedean real closed fields, and then applying the Compactness and Completeness Theorem from Model Theory.

1. Statement of the Results In this paper we deal with ‘effectivity problems’ in connection with the following

theorem proved by K. Schmüdgen in [Sch].

Let ℝ[X] = ℝ[X_{1}, …, X_{n}] be the ring of real polynomials in X_{1}, …, X_{n}. For h_{1}, …, h_{m} ∈ ℝ[X], the set

S(h) = S(h_{1}, …, h_{m}) = {a ∈ ℝ^{n}| h_{1}(a) ≥ 0, …, h_{m}(a) ≥ 0}

is a basic closed semi-algebraic subset of ℝ^{n}. Schmüdgen’s Theorem states that if S(h) is bounded, then every f ∈ ℝ[X] that is strictly positive on S(h) admits a representation

(⋆) f = ∑_{ν∈{0,1}^{m}} h_{1}^{ν_{1}} · . . . · h_{m}^{ν_{m}}σ_{ν}

with σ_{ν} from Σℝ[X]^{2}, the set of sums of squares of polynomials from ℝ[X].

Hilbert’s 17-th Problem dealt with the non-compact case h_{1} = ⋯ = h_{m} = 1. In this case, by E. Artin’s well-known solution, f ∈ Σℝ(X)^{2}, i.e., f is a sum of

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

1991 Mathematics Subject Classification. Primary 14Q20, 12Y05; Secondary 14P10.

This work was done during the author’s stay at the University of Stellenbosch. Financial support from Stellenbosch University is gratefully acknowledged.

squares of rational functions. Using a classical result of Pfister,

f = ∑_{i=1}^{2^{n}} (\frac{p_{i}}{q_{i}} )^{2} with p_{i}, q_{i} ∈ ℝ[X].

Since this result also holds for every real closed field R replacing ℝ, the Compactness Theorem from model theory yields recursive bounds for the degree of p_{i} and q_{i}, depending only on the degree of f (and, of course, on n).

If one tries to find similar results for the Schmüdgen Theorem, one runs into trouble. First of all, for n ≥ 2 there is no bound possible for the length of sums of squares in ℝ[X] (see [CDLR], Theorem 4.1). Secondly, even in case n = 1 (where every sum of squares equals a sum of 2 squares from ℝ[X]), there does not exist a bound on deg σ_{ν} depending only on deg f and deg h_{i} (1 ≤ i ≤ m). This was first published in [St]. The reason for this defect being that Schmüdgen’s Theorem does not hold for non-archimedean real-closed fields R. Thus, if one wants to find upper bounds on the ‘length’ of σ_{ν} (= minimal number of squares such that their sum represents σ_{ν}) and deg σ_{ν} in a representation (⋆), then the bounds have to depend also on invariants of f and the h_{i}’s other than just their degrees.

Our first theorem treats the case where the h_{i}’s are fixed (i.e., we treat all coefficients of h_{1}, …, h_{m} as invariants) and gives a bound depending also on deg f, ||f||, and the minimum of f on the compact set S(h). By ||f|| we mean any norm of the vector of coefficients of f, e.g., the maximal absolute value.

Theorem 1: There exists δ ∈ ℕ depending only on the coefficients of h_{1}, …, h_{m} and on n, d, and N ∈ ℕ such that for all f ∈ ℝ[X] = ℝ[X_{1}, …, X_{n}], if

(i) deg f ≤ d and ||f|| ≤ N, (ii) ||S(h)|| ≤ N, (iii) f(S(h)) > \frac{1}{N}, then f admits a representation (⋆) where δ bounds the length and the degree of each σ_{ν}.

Remark: If in addition to the assumptions of the theorem the coefficients of h_{1}, …, h_{m} are integers, the function δ may be chosen to be recursive in n, d, N, and the coefficients of the h_{i}’s.

In the next theorem we also allow the h_{i}’s to vary. As the example below shows, however, this theorem does not cover all h_{i}’s with S(h) bounded.

Theorem 2: There exists a recursive bound δ = δ(n, m, d, N) such that for all f, h_{1}, …, h_{m} ∈ ℝ[X] = ℝ[X_{1}, …, X_{n}], if

(i) deg f, deg h_{i} ≤ d and ||f||, ||h_{i}|| ≤ N (1 ≤ i ≤ m), (ii) for all h′_{1}, …, h′_{m} ∈ ℝ[X] with deg h′_{i} ≤ deg h_{i},

if ||h′_{i} − h_{i}|| < \frac{1}{N} (1 ≤ i ≤ m) then ||S(h′)|| ≤ N, and (iii) f(S(h′)) > \frac{1}{N}, then f has a representation (⋆) with length and degree of each σ_{ν} bounded by δ.

Example 1: (n = 2, m = 3) Let h_{1} = X_{1} −1, h_{2} = X_{2} −1, and h_{3} = 2−X_{1}X_{2}. Then S(h) is bounded. If we now slightly change h_{1} to h′_{1} = X_{1} − 1 + εX_{2} with ε > 0, the set S(h′) is unbounded - it acquires new points coming from +∞ along with X_{2}-line.

In view of this example we call a sequence h_{1}, …, h_{m} ∈ ℝ[X_{1}, …, X_{m}] of polynomials non-critical if there exists N ∈ ℕ such that for all h′_{1}, …, h′_{m} ∈ ℝ[X], if deg h′_{i} ≤ deg h_{i} and ||h′_{i} − h_{i}|| < \frac{1}{N} for all 1 ≤ i ≤ m, then ||S(h)′|| ≤ N. Thus h_{1}, h_{2}, h_{3} from the example are critical.

Theorem 1 is the best we can do in case of a critical sequence h_{1}, …, h_{m} ∈ ℝ[X]. In the non-critical case, however, Theorem 2 applies as we can see from the following proposition which can be easily proved by elementary analytical methods.

Proposition: Let h_{1}, …, h_{m} ∈ ℝ[X_{1}, …, X_{n}] be non-critical. Then for every f ∈ ℝ[X] such that f > 0 on S(h) there exists some N ∈ ℕ such that for all h′_{1}, …, h′_{m} ∈ ℝ[X], if ||h′_{i} − h_{i}|| < \frac{1}{N}, then ||S(h′)|| ≤ N and f > \frac{1}{N} on S(h′).

In order to prove our theorems by model theoretic means, we first have to extend Schmüdgen’s Theorem to non-archimedean real closed fields R.

2. Positive Polynomials over Non-Archimedean Fields In what follows, R always denotes a (non-archimedean) real closed field. The

convex hull O of ℤ in R is a (non-trivial) valuation ring of R with maximal ideal denoted by 𝔪. The residue field \overline{R} = O/𝔪 is a real closed subfield of ℝ (see [P1], Ch.8). The valuation corresponding to O is denoted by v. Its value group is divisible and (R, v) is a henselian valued field.

The residue map ρ : O → \overline{R} therefore admits a section, i.e., \overline{R} embeds into O, say by σ : \overline{R} → O, such that ρ ∘ σ=id. For convenience we shall assume ℝ ⊆ R in this chapter. Thus \overline{R} = ℝ and σ is the identity on ℝ. Moreover, to every a ∈ O there exists a unique α ∈ ℝ such that a − α ∈ 𝔪. We denote this distinguished element α by a^{∘} and call it the standard part of a. For a = (a_{1}, …, a_{n}) ∈ O^{n} we write a^{∘} = (a_{1}^{∘}, …, a_{n}^{∘}).

For every prime ideal ℘ ⊆ 𝔪 of O, the localization O_{℘} is a valuation ring of R, coarser than O. The maximal ideal of O_{℘} is ℘. Thus for every ℘ ∈ Spec O we get a valuation ring O_{℘} of R. Since Spec O is ordered linearly by inclusion, we obtain an increasing chain of valuation rings in R

O = O_{𝔪} ⊆ O_{℘} ⊆ O_{℘′} ⊆ O_{0} = R

where ℘′ ⊆ ℘. In order to distinguish the different residue maps we write

−_{℘} : O_{℘} → O_{℘}/℘ =: \overline{R}^{℘}.

All residue fields are real closed (cf. [P1], Ch.7, Ch.8), and may be understood as subfields of R. The image \overline{O}^{℘} of O under the residue map of the valuation ring O_{℘} is a valuation ring of \overline{R}^{℘} (it is the convex hull of ℤ in \overline{R}^{℘}) whose residue field is ℝ = \overline{R}^{𝔪}.

We now consider the ring

A := O[X] := O[X_{1}, …, X_{n}]

of polynomials from R[X] = R[X_{1}, …, X_{n}] with “finite” coefficients. The canonical ordering of O induced by R clearly has the positive cone O^{2} = {a^{2}| a ∈ O}. It is archimedean by definition. For h_{1}, …, h_{m} ∈ O[X] we define as usual

S_{R}(h) = S_{R}(h_{1}, …, h_{m}) = {a ∈ R^{n}| h_{1}(a) ≥ 0, …, h_{m}(a) ≥ 0}

and

T_{A}(h) := T_{A}(h_{1}, …, h_{m}) := ∑_{ν∈{0,1}^{m}} h_{1}^{ν_{1}} · . . . · h_{m}^{ν_{m}}ΣA^{2}.

As it will turn out, not only the semi-algebraic set S_{R}(h) is important, but also the semi-algebraic sets S_{R}(\overline{h}^{℘}) where \overline{h}^{℘} = (\overline{h}_{1}^{℘}, …, \overline{h}_{m}^{℘}) is the sequence of polynomials obtained from h_{1}, …, h_{m} by applying the residue map^{−_{℘}} to their coefficients. Note that we now have treated \overline{R}^{℘} as a subfield of R.

In the next lemma we use the notations of [J] and [JP]. This lemma generalizes a theorem proved by Wörmann in the archimedean case (cf. [W]).

Lemma 1: With the notations from above,

(a) T_{A}(h) is a preordering of A iff S_{R}(\overline{h}^{℘}) ≠ ∅ for some ℘ ∈ Spec O

(b) T_{A}(h) is archimedean iff there exists N ∈ ℕ such that ⋃_{℘∈Spec O} S_{R}(\overline{h}^{℘}) is bounded by N, i.e. ∥S_{R}(\overline{h}^{℘})∥ ≤ N for all ℘ ∈ Spec O.

Proof (a): If T = T_{A}(h) is a preordering of A, i.e., T + T ⊆ T, T · T ⊆ T, A^{2} ⊆ T, −1 ∉ T, it can be extended to some ordering P ∈ X_{T}, i.e., P is a preordering with P ∪ −P = A, P ∩ −P is a prime ideal in A, and T ⊆ P. Then ℘ := (P ∩ −P) ∩ O is a prime ideal of O. Thus \overline{O}^{℘} ⊆ \overline{A} := A/(P ∩ −P) and \overline{P} extends the canonical ordering of \overline{O}^{℘} to \overline{A}.

In the real closure of Quot \overline{A} with respect to \overline{P} the following existential statement holds

∃y_{1}, …, y_{n} ⋀_{i=1}^{m} \overline{h_{i}}^{℘}(y) ≥ 0.

In fact, since \overline{h_{i}} = \overline{h_{i}}^{℘}(\overline{X}_{1}, …, \overline{X}_{n}), and \overline{h_{i}} ∈ \overline{P}, we may take y_{i} = \overline{X_{i}}. Now by Tarski’s Transfer, there exists a ∈ (\overline{R}^{℘})^{n} such that \overline{h_{i}}^{℘}(a) ≥ 0 (1 ≤ i ≤ m).

Conversely let S_{R}(\overline{h_{i}}^{℘}) ≠ ∅. Then clearly

−1 ∉ ∑_{ν∈\{0,1\}^{m}} \overline{h_{1}^{ν_{1}}}^{℘} · . . . · \overline{h_{1}^{ν_{m}}}^{℘} ∑ \overline{O}^{℘}[X]^{2}.

Hence also −1 ∉ T.

(b): If T = T_{A}(h) is archimedean, then by definition N − ΣX_{i}^{2} ∈ T for some N ∈ ℕ. But then for any ℘ ∈ Spec O,

N − ΣX_{i}^{2} ∈ T_{\overline{O}^{℘}[X]}(\overline{h}^{℘}).

Thus ∥S\overline{R}^{℘}(\overline{h}^{℘})∥ ≤ N, and by Tarski’s Transfer we find ∥S_{R}(\overline{h}^{℘})∥ ≤ N. Conversely let ∥S_{R}(\overline{h}^{℘})∥ ≤ N for all ℘ ∈ Spec O We want to show that T is archimedean. Thus let P ∈ X_{T} with prime ideal

P ∩−P. Then ℘ := O ∩(P ∩−P) ∈ Spec O. The integral domain \overline{A} = A/(P ∩−P) contains \overline{O}^{℘}, and \overline{P} extends the canonical ordering of \overline{O}^{℘}. Since in the real closed field R we have

∀y(⋀_{i=1}^{m} \overline{h_{i}}^{℘}(y) ≥ 0 ⇒ ∥y∥^{2} ≤ N^{2})

by Tarski’s Transfer the same must hold in the real closure of Quot \overline{A} with respect to \overline{P}. Thus in particular ∥\overline{X}∥^{2} ≤ N^{2}, as \overline{h_{i}}^{℘}(\overline{X}) = \overline{h_{i}} ≥ 0 with respect to \overline{P}. Therefore f := N^{2} + 1 − ΣX_{i}^{2} ∈ P^{+}. Hence by the ‘Positivstellensatz’ (cf.[BCR], Proposition 4.4.1) there exist t, t_{1} ∈ T such that t_{1}f = 1 + t. Then

(1 + t)f = t_{1}f^{2} ∈ T.

The positive cone T_{0} = ΣO[X]^{2} + fΣO[X]^{2} of A = O[X] is archimedean, since

(N^{2} + 1 + \frac{1}{4}) ± X_{i} = (\frac{1}{2} ± X_{i})^{2} + f + ∑_{j≠i} X_{j}^{2} ∈ T_{0}.

Thus all generators of O[X] are bounded, whence by Proposition 3 of [J], T_{0} is archimedean. Moreover we have

(1 + t)T_{0} ⊆ T.

Writing N_{0} = N^{2} + 1 and picking N_{1} ∈ ℕ such that N_{1} − f ∈ T_{0}, we find the two identities:

f + tN_{0} = f + tf + tΣX_{i}^{2} ∈ T

and

(1 + N_{1})(N_{1} − t) = (1 + t)(N_{1} − t) + (N_{1} − t)^{2} ∈ T.

The last identity implies N_{1} − t ∈ T, since (1 + N_{1})^{−1} ∈ T. Combining this with the first identity, we finally obtain

N_{0}(N_{1} + 1) − ΣX_{i}^{2} = N_{0}N_{1} + f = (f + tN_{0}) + N_{0}(N_{1} − t) ∈ T.

Hence T is archimedean by the same arguments as used for T_{0} above.

It should be noted that in general none of the conditions on the prime ideal ℘ ∈ Spec O is superfluous in the lemma

Example 2: Consider a real closed extension R of ℝ such that Spec O has exactly three elements, so {0} ⊂ ℘ ⊂ 𝔪. Then choose 0 < ϵ′, ϵ′ ∈ ℘, and ϵ′ < ϵ′′, ϵ′′ ∈ 𝔪\℘. Take ω′′ = (ϵ′′)^{−1}. Let h_{1} = (1 − X_{1}^{2} − X_{2}^{2})(1 − ϵ′′X_{1}) and h_{2} = (1 − ϵ′ϵ′′) − ϵ′′X_{1}. Then S_{R}(h_{1}, h_{2}) = {x | ∥x∥ ≤ 1} is bounded, S_{R}(\overline{h_{1}}^{℘}, \overline{h_{2}}^{℘}) = \{x | ∥x∥ ≤ 1\} ∪ \{x|x_{1} = ω′′\} is unbounded, S_{R}(h_{1}^{0}, h_{2}^{0}) = {x | ∥x∥ ≤ 1} is bounded.

Now we are able to generalize Schmüdgen’s Theorem to arbitary and real closed fields (containing ℝ).

Theorem 3: Let R be a real closed field containing ℝ and let O be defined as above. Moreover, let h_{1}, …, h_{m} ∈ O[X_{1}, …, X_{n}] such that ∥S_{R}(\overline{h_{1}}^{℘}, …, \overline{h_{m}}^{℘})∥ ≤ N for some N ∈ ℕ and all ℘ ∈ Spec O. Then every f ∈ O[X_{1}, …, X_{n}] satisfying f > \frac{1}{N} on S_{R}(h_{1}^{0}, …, h_{m}^{0}) belongs to T_{O[X]}(h_{1}, …, h_{m}).

Proof If −1 ∈ T = T_{A}(h) then clearly f = (\frac{f+1}{2})^{2} − (\frac{f−1}{2})^{2} ∈ T. If −1 ∉ T, then by the lemma, T is an archimedean preordering of A = O[X]. Thus by Theorem 4 of [J] it suffices to show that f ∈ P^{+} for all P ∈ X_{T}^{max}.

Since T is archimedean, Corollary 1 of [J] yields for such P

α_{P} : A → A/(P ∩ −P) = \overline{A} = ℝ.

Thus (P ∩ −P) ∩ O = 𝔪 and α_{P} (a) = a^{0} for each a ∈ O. Hence we have to show that f^{0}(\overline{X}) = \overline{f} > 0 in ℝ.

Since h_{1}, …, h_{m} ∈ P, we get 0 ≤ \overline{h_{i}} = h_{i}^{0}(\overline{X}). Thus \overline{X} ∈ S_{R}(h^{0}). By assumption f(\overline{X}) > \frac{1}{N}. From f − f^{0} ∈ 𝔪[X] we conclude that the real number f^{0}(\overline{X}) is infinitely close to f(\overline{X}). Thus

f^{0}(\overline{X}) ≥ \frac{1}{N} > 0.

As we can see from Example 2, the condition f > \frac{1}{N} on S_{R}(h) would not be sufficient in Theorem 3. In fact, if we let f = 2−(X_{1}^{2} +X_{2}^{2}) we would have f ≥ 1 on S_{R}(h_{1}, h_{2}) in Example 2. But f ∉ T (h), since f takes negative values on S_{R}(\overline{h}^{℘}).

3. Proof of Theorems 1 and 2 In both cases let R be an ℵ_{1}-saturated real closed extension field of ℝ, e.g., an

ultra-power ℝ^{ℕ}/F of ℝ with respect to some non-principal ultrafilter F on ℕ. Such an extension R always is non-archimedean, i.e., the convex hull O of ℤ in R is a non-trivial valuation ring of R. Moreover, the fact that R is ℵ_{1}-saturated implies that every covering of a semi-algebraic set in R^{n} by countably many semi-algebraic sets admits a finite subcover. This is the property we shall use in the proof.

In the proof of Theorem 1, we fix the coefficients of h_{1}, …, h_{m} ∈ ℝ[X] and assume that ∥S(h)∥ ≤ N for suitably chosen N ∈ ℕ.

We then let

f_{n,d}(C, X) ∈ ℤ[C, X]

be the general polynomial of degree d in X = (X_{1}, …, X_{n}) with coefficients C = (C_{1}, …, C_{M}), where M = \binom{n+d}{n}.

Let Φ be the set of coefficients c ∈ R^{M} such that f_{c} := f_{n,d}(c, X) satisfies

∥f_{c}∥ ≤ N and f_{c} > \frac{1}{N} on S_{R}(h).

Using elimination of quantifiers, we see that Φ is a semi-algebraic subset of R^{M}.

For every δ ∈ ℕ let Ψ_{δ} be the subset of R^{M} such that f_{c} has a representation (⋆) with length and degree of the σ_{ν} bounded by δ. Clearly, the set Ψ_{δ} is again semi-algebraic, and Theorem 3 tells us that

Φ ⊆ ⋃_{δ∈ℕ} Ψ_{δ}.

In fact, since h_{i} ∈ ℝ[X], we get \overline{h_{i}}^{℘} = h_{i} for all ℘ ∈ Spec O, and hence we obtain ∥S_{R}(\overline{h_{i}}^{℘})∥ ≤ N from (ii), using Tarski’s Transfer. From ∥f_{c}∥ ≤ N we infer that f_{c} ∈ O[X]. Thus all the conditions of Theorem 3 are satisfied, hence f_{c} has a representation (⋆) over R (even over O).

Since R is ℵ_{1}-saturated, we get

Φ ⊆ Ψ_{δ}

for some fixed δ ∈ ℕ. If we identify Φ and Ψ_{δ} with formulas defining these sets, in R the formula

∀c(Φ(c) ⇒ Ψ_{δ}(c))

holds. Thus by Tarski’s Transfer, it also holds in ℝ. This, however, is the conclusion of Theorem 1.

In the proof of Theorem 2, we let Φ be the set of coefficients (c, c^{(1)}, …, c^{(m)}) ∈ R^{(m+1)M} such that f = f_{n,d}(c, X), h_{i} = f_{n,d}(c^{(i)}, X) (1 ≤ i ≤ m) satisfy the conditions (i),(ii),(iii) of Theorem 2. As in the first case we let Ψ_{δ} be the set of coefficients (c, c^{(1)}, …, c^{(m)}) such that f has a representation (⋆) with the length and the degree of each σ_{ν} bounded by δ. (Note that now the coefficients of the h_{i}’s are not fixed.) Again we obtain

Φ ⊆ ⋃_{δ∈ℕ} Ψ_{δ}

from Theorem 3. In fact, from ∥f∥, ∥h_{i}∥ ≤ N we find f, h_{1}, …, h_{m} ∈ O[X]. For every ℘ ∈ Spec O we have \overline{h_{i}}^{℘} − h_{i} ∈ 𝔪[X], thus in particular ∥\overline{h_{i}}^{℘} − h_{i}∥ < \frac{1}{N}. Hence (ii) implies ∥S_{R}(\overline{h_{i}}^{℘})∥ ≤ N, and (iii) implies f > \frac{1}{N} on S_{R}(h^{0}). Therefore f has a representation (⋆) over R (even over O).

Now we conclude as in the first case.

The fact that the bound δ may be chosen as a recursive function in the data given, follows from the completeness of the theory of real closed fields and general remarks from recursion theory. For more details the reader is refered to [P2], Ch.6.

4. Remarks on “Linear” Representations

At the end of this note let us consider “linear” representations of f.

(⋆⋆) f = σ_{0} + h_{1} + σ_{1} + ⋯ + h_{m}σ_{m}

with σ_{i} ∈ Σℝ[X]^{2}. Such linear representations do not always exist even when S(h) is bounded. Example 1, for instance, does not allow a representation (⋆⋆) for every f, strictly positive on S(h) (see [JP], Example 4.5). Under certain conditions, however, linear representations are always possible. One such condition is (compare [JP], Theorem 4.1):

∀x ≠ 0 (h̃_{1}(x) < 0 or … or h̃_{m}(x) < 0)

(where h̃_{i} denotes the homogeneous part of h_{i} of highest degree) together with

deg h_{1} ≡ … ≡ deg h_{m} mod 2.

This condition in particular holds if h_{1}, …, h_{m} are linear polynomials and S(h) is non-empty and bounded.

Similarly to Theorem 3 we obtain in the non-archimedean case

Theorem 4: Let R be a real closed field containing ℝ, and let O be defined as above. Moreover, let h_{1}, …, h_{m} ∈ O[X_{1}, …, X_{n}] such that (h̃_{1})^{0}(x) < 0 or … or (h̃_{m})^{0}(x) < 0 for all x ∈ R^{n}\{0}, and assume that

deg h_{1} ≡ … ≡ deg h_{m} mod 2.

Then every f ∈ O[X] with f > \frac{1}{N} on S_{R}(h_{1}^{0}, …, h_{m}^{0}) for some N ∈ ℕ has a representation f = σ_{0} + Σ_{i=1}^{m}h_{i}σ_{i} with σ_{i} ∈ ΣO[X]^{2}.

The proof uses tools developed in [JP]. It is not difficult to deduce from Theorem 4 results about bounds in representations (⋆⋆) similar to the case (⋆). The details of the corresponding proofs may be found in the forthcoming book [PD].

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