SUMS OF SQUARES ON REAL ALGEBRAIC SURFACES
CLAUS SCHEIDERER
Abstract. Consider real polynomials g1 , . . . , gr in n variables, and assume
that the subset K = {g1 ≥ 0, . . . , gr ≥ 0} of Rn is compact. We show that a
polynomial f has a representation
X
(∗)
f =
se · g1e1 · · · grer
e∈{0,1}r
in which the se are sums of squares, if and only if the same is true in every
localization of the polynomial ring by a maximal ideal. We apply this result
to provide large and concrete families of cases in which dim(K) = 2 and
every polynomial f with f |K ≥ 0 has a representation (∗). Before, it was not
known whether a single such example exists. Further geometric and arithmetic
applications are given.
Introduction
Let g1 , . . . , gr be real polynomials in n variables for which the subset
K := {x : g1 (x) ≥ 0, . . . , gr (x) ≥ 0}
of Rn is compact. According to a celebrated theorem of Schmüdgen [Sm], any
polynomial f which is strictly positive on K can be written in the form
X
(1)
f=
se · g1e1 · · · grer
e∈{0,1}r
where the se are sums of squares of polynomials. It is well-known that this is
usually not true any more if f is only assumed to be non-negative on K, i.e. is
allowed to have zeros in K. In fact, as soon as K has dimension at least 3, there
exist polynomials f with f |K ≥ 0 which have no representation (1), no matter how
many redundant inequalities gi we add to the description of K (as long as there
are only finitely many). This is true regardless of the compactness of K [Sch1].
On the other hand, there exist both easy and subtle examples of one-dimensional
(compact or non-compact) sets K, together with suitable systems of inequalities gi
describing K, where Schmüdgen’s theorem extends to all non-negative polynomials.
A detailed study of this one-dimensional case was made in [Sch3].
What has been lacking so far is an understanding of the question in dimension
two. Write T for the set of polynomials which admit a representation (1) (T is
called the preordering generated by g1 , . . . , gr ), and let us say that T is saturated if
T contains every polynomial which is non-negative on K. So far, no single example
was known where T is saturated and dim(K) = 2. In this paper we show that
there are plenty of such examples, both with K compact and with K non-compact.
1991 Mathematics Subject Classification. Primary 14P05; secondary 11E25, 26D05.
Key words and phrases. Non-negative polynomials, polynomial inequalities, sums of squares,
preorderings, archimedean, real algebraic geometry, real algebraic surfaces, Hilbert 17th problem.
Support by DFG travel grant KON 1823/2002 and by the European RAAG network HPRNCT-2001-00271 is gratefully acknowledged. Part of this work was done while the author enjoyed
a stay at MSRI Berkeley. He would like to thank the institute for the invitation and the very
pleasant working conditions.
1
2
CLAUS SCHEIDERER
Typical examples are a convex non-degenerate polygon in the plane, or a sphere in
3-space (both described by their natural systems of inequalities).
The key to these results are localization principles. Speaking loosely, they say
that if K is compact and f |K ≥ 0, then it is the behavior of f near its zeros in
K which decides about f ∈ T . One such principle was proved in [Sch3], where it
was mainly applied to one-dimensional cases. In this paper we establish another
such result (Theorem 2.8). It is a localization principle very much in the spirit of
commutative algebra: Assuming that K is compact, f has a representation (1) if
and only if this is true in every localization of the polynomial ring with respect to
a maximal ideal. The proof of this theorem makes essential use of a recent lemma
by Kuhlmann, Marshall and Schwartz [KMS].
Combining this localization principle with recent results about sums of squares
in 2-dimensional regular local rings [Sch2], we can prove saturatedness of T in
many 2-dimensional situations. For example, if we consider a subset of the plane
(case n = 2), it suffices that the curves gi = 0 which bound the compact set K are
sufficiently regular, together with their intersections (Corollary 3.3). Similar results
are proved on general non-singular algebraic surfaces (Theorem 3.2, Corollary 3.4).
These results are shown to apply not only to polynomials with real coefficients, but
also to polynomials with rational or algebraic coefficients, yielding representations
(1) with rational or algebraic coefficients (Corollary 3.9).
Another application concerns Hilbert’s 17th problem. By a theorem of Reznick,
every positive definite form can be written as aPsum of squares of rational functions
whose denominators are uniformly powers of
x2i . This is well-known to be false
in general, if the form in question is only assumed positive semidefinite (psd ).
But surprisingly, it is true for ternary psd forms, as we show by an application of
the results just described (Corollary 3.11). Another application concerns sums of
squares in the so-called rings of real regular functions, and gives a partial answer
to a question of Stengle (3.17, 3.19).
The techniques of this paper are not restricted to compact situations. In fact,
we give an example (3.15) where K is an unbounded 2-dimensional subset of the
plane and T is saturated. A more systematic study of such phenomena will be the
subject of a forthcoming paper. We also like to point out here that the applications
of our method are not limited to regular situations (regular bounding curves, regular
surfaces etc.). Rather, there are interesting applications to singular situations as
well, as we plan to demonstrate elsewhere.
After recalling notations and preliminaries, Section 2 contains the proof of the
localization principle. We decided to present it in a general and “abstract” setting,
rather than only for real polynomials. This makes the proof no harder at all, but
on the other hand, it permits to give applications to other interesting situations.
Section 3 contains the geometric applications outlined above.
1. Notation and preliminaries
All rings are commutative and have a unit.
1.1. We use the language of the real spectrum, for which we refer to the wellknown text books, in particular to [BCR], [KS], [PD]. The recent survey [Sch4] on
positivity and sums of squares may perhaps also be found useful. Let us briefly
recall a few key concepts:
Let A be a ring. The real spectrum of A, denoted Sper A, is the set consisting of
all pairs α = (p, ω) where p ∈ Spec A and ω is an ordering of the residue field of p.
For f ∈ A, the notation f (α) ≥ 0 (resp., f (α) > 0) indicates that the residue class
f mod p is non-negative (resp., positive) with respect to ω. We will sometimes use
self-explaining notations like {f ≥ 0} for {α ∈ Sper A : f (α) ≥ 0}. Also, if X is a
SUMS OF SQUARES ON REAL ALGEBRAIC SURFACES
3
subset of Sper A, we may say “f ≥ 0 on X” to mean X ⊂ {f ≥ 0}. The topology
on Sper A is generated by the {f > 0} (f ∈ A) as subbasic open sets.
1.2. A quadratic module in A is a subset M of A which satisfies 1 ∈ M , M +M ⊂ M
and a2 M ⊂ M for every a ∈ A. (We allow the degenerate case −1 ∈ M .) The
support of M is supp(M ) = M ∩ (−M ); this is an ideal of A if 12 ∈ A. If M is
multiplicatively closed, then M is called a preordering of A. More generally, if T is
a preordering, a T -module is a quadratic module M for which T M ⊂ M . Thus a
quadratic module is the same thing as a T0 -module, where T0 := ΣA2 is the cone
of all sums of squares in A.
Given g1 , . . . , gr ∈ A, the quadratic module generated by these elements in A is
QM (g1 , . . . , gr ) = ΣA2 + g1 ΣA2 + · · · + gr ΣA2 .
The preordering of A generated by g1 , . . . , gr ∈ A is denoted P O(g1 , . . . , gr ). It
is the quadratic module generated by the products g1e1 · · · grer (e ∈ {0, 1}r ), i.e. it
consists of all sums f as in (1) with se ∈ ΣA2 (e ∈ {0, 1}r ).
Given a quadratic module M in A, one associates with M the closed subset
XM := {α ∈ Sper(A) : f (α) ≥ 0 for every f ∈ M }
of Sper(A). The saturation of M is the preordering
Sat(M ) := {f ∈ A : f ≥ 0 on XM }
of A. The module M is said to be saturated if M = Sat(M ).
1.3. To speak about algebraic varieties, we shall use the language of schemes, but
need only the most elementary rudiments. As usual, if V is a variety over a base
field k and E is any extension field of V , then V (E) denotes the set of E-rational
points of V .
Let R be a real closed field and A an R-algebra of finite type, let V = Spec A.
One has the well-known Artin-Lang correspondence between V (R) with its semialgebraic subsets and the real spectrum of the ring A = R[V ]. If M is a quadratic
module in A, generated by g1 , . . . , gr , we write
r
\
\
x ∈ V (R) : gi (x) ≥ 0
S(M ) :=
x ∈ V (R) : f (x) ≥ 0 =
f ∈M
i=1
for the basic closed semi-algebraic set in V (R) defined by M . Note that M saturated
means that M consists of all f ∈ A for which f ≥ 0 on S(M ).
The zero set of f ∈ A in V (R) is denoted Z(f ).
1.4. The following notion is of central importance: A quadratic module M in A
is archimedean if A = Z + M . In terms of the partial ordering on A defined by
f ≤M g :⇔ g − f ∈ M , this means that every ring element is bounded in absolute
value by an integer. Schmüdgen’s theorem [Sm] (in Wörmann’s formulation [BW])
asserts that a finitely generated preordering T in R[x1 , . . . , xn ] whose associated set
S(T ) is compact is archimedean.
2. The localization principle
We start by recalling the Basic Lemma 2.1 from [KMS]. With later applications
in mind, we prefer to give a formulation which is slightly different and more general.
2.1. Let A be a ring. Recall that a subset X of Sper(A) is bounded (over Z) if for
every a ∈ A there exists an integer N ≥ 0 with |a| ≤ N on X. ATclosed subset
X of Sper(A) is pro-basic if there exists a subset F of A with X = f ∈F {f ≥ 0}.
For any bounded subset Y of Sper(A) there exists a pro-basic closed subset X of
Sper(A) which contains Y and is also bounded.
4
CLAUS SCHEIDERER
For any f ∈ A we write Z(f ) := {α : f (α) = 0} forTthe zero set of f in Sper A.
More generally, if F is a subset of A we put Z(F ) := f ∈F Z(f ).
Lemma 2.2. Let X be a pro-basic closed subset of Sper(A) which is bounded, and
let a, b ∈ A with a ≥ 0 on X and b < 0 on X ∩ Z(a). Then there exists N ∈ N
with N a > b on X.
Proof. Let Y := X ∩ {b ≥ 0}, and let T = {t ∈ A : t ≥ 0 on Y }, the saturated
preordering associated with Y . Since a > 0 on Y , the Positivstellensatz ([BCR]
§4.4, [KS] III §9, [PD] §4.2) implies that there are s, t ∈ T with as = 1 + t. Choose
m, n ∈ N with b < m and s < n on Y . Then mna > mas ≥ m > b on Y , and
hence mna > b holds on X as well.
2.3. Before proceeding further, let us recall a few well-known facts. Let X be a
closed and bounded subset of Sper A, and denote by X max the topological space of
closed points of X. It is well-known that X max is a compact (Hausdorff) space (e.g.
[KS] III §6, [PD] Exerc. 4.6.3). For every x ∈ X max , the ordered residue field κ(x)
of x is archimedean, since it is relatively archimedean over its subring A/ supp(x)
([KS] III §7) and the latter is archimedean over Z by the boundedness property of
X. Hence κ(x) can be considered as an ordered subfield of R in a unique way.
Given f ∈ A, we denote by Φ(f ) : X max → R the function obtained in this way:
Φ(f ): x 7→ f + supp(x) ∈ κ(x) ⊂ R
max
(x ∈ X
). The function Φ(f ) is continuous, and Φ : A → C(X max , R), f 7→ Φ(f )
is a ring homomorphism. For convenience of notation, let us agree to write f (x),
instead of Φ(f )(x), for f ∈ A and x ∈ X max .
Here is a (slightly generalized) version of Lemma 2.1 from [KMS]. Our proof is
somewhat simpler than the proof in [KMS] and shows that the result is a direct
consequence of Stone-Weierstraß approximation:
Proposition 2.4. Let A be a ring which contains n1 for some integer n > 1, and
let X be a pro-basic closed subset of Sper(A) which is bounded. Let f , g ∈ A with
f , g ≥ 0 on X. Then, given any h ∈ (f, g) with h > 0 on X, there exist s, t ∈ A
with sf + tg = h and s > 0, t > 0 on X.
Proof. Let a, b ∈ A with af + bg = h. Using 2.2 we find integers N1 , N2 ≥ 1 with
N1 g > −a and N2 f > −b on X. We can define continuous R-valued functions ϕ,
ψ on the compact space X max by
n
n
bo
ao
ϕ := max −N1 , −
and ψ := min N2 ,
f
g
(c.f. 2.3). Indeed, on a neighborhood of the zero set of f in X we have b > 0, and
max
hence − fb(x)
approaches the zeros of f . Similarly,
(x) approaches −∞ as x ∈ X
a(x)
g(x)
approaches +∞ as x ∈ X max approaches the zeros of g. By the choices of N1
and N2 we see that ϕ < ψ holds pointwise on X max .
Since n1 ∈ A, we can apply the Stone-Weierstraß Theorem to the subring Φ(A)
of C(X max , R). So there exists c ∈ A such that ϕ < c < ψ holds (pointwise) on
X max . This implies that
−b < cf and cg < a
hold on X. Now it suffices to set s := a − cg and t := b + cf : This gives s, t > 0 on
X and sf + bg = h.
Before we proceed with an essential generalization of 2.4, we need an auxiliary
result of general nature, which is often useful. It generalizes [Sch1] Lemma 5.3:
SUMS OF SQUARES ON REAL ALGEBRAIC SURFACES
5
Lemma 2.5. Let A be a ring, let Y be a closed subset of Sper(A) and I an ideal of
A. Let f ∈ A with f ≥ 0 on Y ∩ Z(I). Assume that for every α ∈ Y ∩ Z(I) ∩ Z(f )
there exists h ∈ I with h ≥ 0 on Y and f + h ≥ 0 on Y around α. Then there exists
h ∈ I with f + h ≥ 0 on Y .
By “f + h ≥ 0 on Y around α” we mean that there is a neighborhood U of α
such that f +h ≥ 0 on U ∩Y . (Recall that Z(I) denotes the zero set of I in Sper A.)
Proof. The proof is a straightforward generalization of the proof of 5.3 in [Sch1].
Assume we have proved that for every β ∈ Y there exists hβ ∈ I with hβ ≥ 0 on Y
and (f + hβ )(β) ≥ 0. Then, by the compactness of Y inSthe constructible topology,
n
thereP
are finitely many β1 , . . . , βn ∈ Y such that Y ⊂ i=1 {f + hβi ≥ 0}. Putting
n
h := i=1 hβi , we have h ∈ I and f + h ≥ 0 on Y .
So let β ∈ Y . If β has a specialization α in Z(I), we are done. (If f (α) > 0,
take hβ := 0; if f (α) = 0, use the hypothesis.) So we can assume {β} ∩ Z(I) = ∅.
As in [Sch1], there exists g ∈ I with g(β) ≥ 1. Then hβ := (1 + f 2 )g 2 ∈ I satisfies
hβ >β |f |, so this element does what we want.
Corollary 2.6. Let Y be a closed subset of Sper(A) and I an ideal of A. Let f ∈ A
with f > 0 on Y ∩ Z(I). Then there exists h ∈ I with f + h > 0 on Y .
Proof. (C.f. [Sch1] Cor. 5.5) The construction in the proof of Lemma 2.5 gives, for
every α ∈ Y , a psd element hα ∈ I with (f + hα )(α) > 0. Conclusion as in the
proof of 2.5.
Here is the announced generalization of Proposition 2.4:
Proposition 2.7. Let A be a ring with n1 ∈ A for some integer n > 1, and let
X be a pro-basic closed subset of Sper(A) which is bounded. Let f1 , . . . , fr ∈ A
with fi ≥ 0 on X, and let h ∈ (f1 , . . . , fr ) with h > 0 on X. Then there are
a1 , . . . , ar ∈ A with
a1 f1 + · · · + ar fr = h
and ai > 0 on X (i = 1, . . . , r).
Proof. The case r = 1 is trivial and the case r = 2 is Proposition 2.4. Let r > 2
and assume that the proposition has been proved for r − 1. Let A = A/(fr ) and
f i = fi + (fr ) (i = 1, . . . , r − 1). The closed subset X := X ∩ Z(fr ) of Sper(A) is
again bounded. By the inductive hypothesis, there are b1 , . . . , br−1 ∈ A with bi > 0
on X ∩ Z(fr ) and with
b1 f1 + · · · + br−1 fr−1 ≡ h mod (fr ).
By Corollary 2.6 there exist ci ∈ A with ci ≡ bi mod (fr ) and with ci > 0 on X
Pr−1
(i = 1, . . . , r − 1). Now we get the assertion by applying 2.4 to f := i=1 ci fi and
g := fr .
Now we can prove the following powerful localization principle:
Theorem 2.8. Let A be a ring containing n1 for some integer n > 1, and let M be
a module over an archimedean preordering T in A. Let f ∈ A. For every maximal
ideal m of A with supp(M ) ⊂ m, assume that f lies in the quadratic module Mm
generated by M in Am . Then f ∈ M .
Proof. We first claim that f ∈ Mm holds for all maximal ideals m of A. In fact, for
every prime ideal p of A with supp(M ) 6⊂ p one has Mp = Ap .
To see this, note that T archimedean implies T − T = A. Hence supp(M ) is an
ideal of A. Take s ∈ supp(M ) with s ∈
/ p. Then −s2 ∈ M , hence −1 ∈ Mp , and
Tp − Tp = Ap implies Mp = Ap .
6
CLAUS SCHEIDERER
The closed subset XT of Sper(A) is bounded. Let m be a maximal ideal of A.
The quadratic module generated by M in Am is
o
na
:
a
∈
M,
s
∈
A
r
m
.
Mm =
s2
So there exists s ∈ A r m with s2 f ∈ M . Hence there are finitely many elements
2
s1 , . . . , sr ∈ A with (s1 , . . . , sr ) = (1) and with
Pr si f ∈2 M for i = 1, . . . , r. By
Proposition 2.7 there are a1 , . . . , ar ∈ A with i=1 ai si = 1 and with ai > 0 on
XT . Since T is archimedean, we have P
ai ∈ T (i = 1, . . . , r) by the Representation
r
Theorem ([PD] 5.2.7). Therefore f = i=1 ai s2i f lies in M .
As a consequence of 2.8, the property of being saturated localizes:
Corollary 2.9. Let n1 ∈ A for some n > 1, let M be a quadratic module over an
archimedean preordering in A. Then M is saturated if and only if Mm is saturated
for every maximal ideal m of A containing supp(M ).
Proof. Indeed, M saturated implies Mm saturated for every m, even without the
archimedean hypothesis ([Sch3] 5.8). The converse follows from 2.8.
For preorderings we can give the following formulation. It reduces the number
of local conditions to be checked:
Corollary 2.10. Assume 21 ∈ A, let T be an archimedean preordering in A. Let
f ∈ Sat(T ), and assume f ∈ Tm for every maximal ideal m of A which contains
the ideal (f ) + supp(T ). Then f ∈ T .
Proof. It suffices to prove f ∈ Tm for every maximal ideal m with f ∈
/ m, and
then to apply Theorem 2.8. For such m, f is a unit in Am and lies in Sat(Tm ). By
well-known properties of regular quadratic forms in local rings, this implies f ∈ Tm .
(See [Sch6], or see [Sch2] for the particular case where T consists of the sums of
squares.)
Question 2.11. Let M be an archimedean quadratic module in A, and let us assume
1
2 ∈ A. Does f ∈ Mm for every maximal ideal m imply f ∈ M ?
The above proof of 2.8 gives a weaker statement, namely f ∈ M M , where M M
is the quadratic module in A generated by all products xy (x, y ∈ M ). Indeed,
we can (in the notations of the proof) find the ai satisfying ai > 0 on XM , which
implies ai ∈ M by Jacobi’s Positivstellensatz ([J], [PD] 5.3.7). Note anyway that
M M is in general still smaller than the preordering generated by M .
Remark 2.12. In Theorem 2.8 and Corollary 2.10, it is not in general sufficient to
make the assumption f ∈ Mm (resp., f ∈ Tm ) only for the real maximal ideals m of
A. Rather, it is essential that also the maximal ideals with non-real residue field
are included.
This is nicely illustrated by the following example. Consider the polynomial ring
A = R[x, y] and the polynomials p1 = x2 + y 2 − 1 and p2 = x4 + y 2 − 3. Let
T be the preordering generated by −p1 p2 . Then T is archimedean since the set
K := {−p1 p2 ≥ 0} in R2 is compact. The two curves Ci = {pi = 0} (i = 1, 2),
whose real points bound K, are non-singular and have no real intersection points.
The polynomials −p1 and p2 are both non-negative on K. We claim that they
lie in Tm for every real maximal ideal m, but are not contained in T . First let m be
a maximal ideal of A, corresponding to a point q ∈ R2 . If q lies on the curve C =
√
{p1 p2 = 0} then, since q is a regular point of C, the local ring B := Am [ −p1 p2 ]
is regular. Since −p1 and p2 are non-negative in B, they are sums of squares in B
([Sch2] Thm. 4.8). This means that they lie in Tm , the preordering generated by
−p1 p2 in Am . If q ∈
/ C, then −p1 , p2 ∈ Tm anyway, c.f. the proof of Corollary 2.10.
SUMS OF SQUARES ON REAL ALGEBRAIC SURFACES
7
On the other hand, p2 ∈ T would mean p2 = s − tp1 p2 with sums of squares
s, t. This would imply p2 | s, hence p22 | s, and after cancelling p2 it would give
1 = s̃p2 − tp1 (with s̃ = ps2 ). This is a contradiction since p1 and p2 do not generate
2
the unit ideal: The curves
C1 and C2 have eight points in C2 in common, for
√
example the point (i, 2). Similarly, one sees that −p1 does not lie in T .
What goes wrong is, of course, the local condition for −p1 and p2 at the eight
(non-real) intersection points of C1 and C2 . One sees easily that the saturation of
T is P O(−p1 , p2 ). (This also follows from Corollary 3.3 below.)
Remark 2.13. Theorem 2.8 is usually false without the archimedean hypothesis.
For example, take the polynomial ring A = R[x, y] and the preordering T = ΣA2 of
sums of squares in A. Consider a psd polynomial f which is not a sum of squares,
such as Motzkin’s polynomial f = x4 y 2 + x2 y 4 − 3x2 y 2 + 1. Such f satisfies f ∈ Tp
for every prime ideal p of A ([Sch2] Thm. 4.8).
Remark 2.14. It is interesting to compare the localization principle 2.10 to the localglobal criteria from [Sch3] (Thm. 3.13, Cor. 3.16). Each considers an archimedean
preordering T and an element f ∈ Sat(T ), and each gives necessary and sufficient
conditions (of “local” nature) for f to lie in T .
For convenience of speaking, let us assume that A = R[x1 , . . . , xn ] and T is
finitely generated, and let K = S(T ) ⊂ Rn (a compact semi-algebraic set). We have
f ≥ 0 on K. The general result 3.13 from [Sch3] localizes the question whether
f ∈ T to an infinitesimal Zariski neighborhood of the real zeros of f in K, that is,
of the Zariski closure of Z(f ) ∩ K. In general, this is still a “global” scheme. The
most favorable case is when Z(f ) ∩ K is a finite set; then the question gets reduced
to the completed local rings at the finitely many points therein.
On the other hand, Cor. 2.10 always reduces the question to local rings, but not
usually to complete local rings, and not just to local rings belonging to points in
K, not even to local rings belonging to real points.
Therefore, while both types of results give the global conclusion from local hypotheses, they are in general not comparable.
Remark 2.15. Example 2.12 shows that a criterion like [Sch3] Cor. 3.17 is definitely
restricted to the case where Z(f ) ∩ K is finite. Indeed, in 2.12 we have f ∈ Tbm for
every maximal ideal m of A, but still f ∈
/ T (for f = −p1 or f = p2 ).
3. Geometric applications
Corollary 2.9 prompts the question of finding criteria for preorderings in local
rings to be saturated. Such criteria can then be used to get global corollaries. The
interesting case is when the local ring has dimension two, and here the first case to
consider is when it is regular.
In the forthcoming paper [Sch6], we plan to investigate this question in greater
detail. Here we will only exploit certain sufficient criteria which are direct consequences of the main results from [Sch2]:
Lemma 3.1. Let (A, m) be a 2-dimensional regular local ring containing 21 . Let
u1 , . . . , un be units in A (n ≥ 0), and let f , g be elements in m r m2 . Then each of
the following preorderings in A is saturated:
(a) T = P O(u1 , . . . , un );
(b) T = P O(u1 , . . . , un , f );
(c) T = P O(u1 , . . . , un , f, g), if f and g are linearly independent modulo m2 .
Proof. Let B be the ring obtained from A by adjoining a formal square root of each
of the generators of T . In each of the three cases, the ring B is a regular semilocal
8
CLAUS SCHEIDERER
ring of dimension two. Hence every psd element in B is a sum of squares in B, by
[Sch2] Thm. 4.8. Via an elementary argument (see [Sch3] Lemma 3.9), this implies
that T is saturated.
Theorem 3.2. Let V be a non-singular affine surface over R, and let T = P O(g1 , . . . , gr )
be a preordering in R[V ]. Assume that the set K = S(T ) is compact, that the curves
gi = 0 (i = 1, . . . , r) on V are non-singular and intersect transversely, and that no
three of them intersect. Then T is saturated.
Proof. For every maximal ideal m of R[V ], the generators g1 , . . . , gr of Tm satisfy
one of the three conditions of Lemma 3.1. Therefore Tm is saturated. Since K is
compact, T is archimedean, and hence T is saturated by Corollary 2.9.
Even the very particular case of a convex polygon in the plane is new and unexpected! Note how the theorem provides large families of saturated preorderings
whose associated sets are two-dimensional. Before, not a single such example was
known.
In the affine plane we can prove the following variant, in which the non-real
points do not matter:
Corollary 3.3. Let p1 , . . . , pr be irreducible polynomials in R[x, y], let Ci be the
plane affine curve pi = 0 (i = 1, . . . , r). Assume:
(1) K := {x ∈ R2 : p1 (x) ≥ 0, . . . , pr (x) ≥ 0} is compact;
(2) Ci has no real singular points (i = 1, . . . , r);
(3) the real points of intersection of any two of the Ci are transversal, and no
three of the Ci intersect in a real point.
Then the preordering T = P O(p1 , . . . , pr ) is saturated.
Proof. For any boundary point q of K there are local analytic coordinates (u, v)
around q such that K = {u ≥ 0} or K = {u ≥ 0, v ≥ 0}, locally around q. This
shows that K has local dimension two everywhere.
Let 0 6= f ∈ R[x, y] with f ≥ 0 on K, and write f = f1 g 2 where f1 is squarefree. Then f1 ≥ 0 on K, so we can assume that f itself is square-free. The same
argument shows that if pi divides f , for some index i, then also pfi ≥ 0 on K. Hence
we can assume in addition that f is not divisible by any of the pi .
It follows that f has only finitely many zeros in K. Indeed, assume that p is an
irreducible polynomial dividing f such that K ∩ Z(f ) is infinite. Then K contains
a regular point q of the curve p = 0 for which (p1 · · · pr )(q) 6= 0. In particular, q is
an interior point of K. Since p changes sign locally around q, the same is true for
f , a contradiction.
We can now use Cor. 3.17 from [Sch3], combined with Lemma 3.1, to conclude
f ∈ T . Alternatively, we may apply Corollary 2.10: For any real maximal ideal m
of R[x, y] we have f ∈ Tm by 3.1. If m is a non-real maximal ideal then f > 0 on
e ∩Sper(Am ), and this implies f ∈ Tm by general reasons ([Sch2] Cor. 2.4 combined
K
with the elementary argument in [Sch3] 3.9).
We isolate the case r = 0 of Theorem 3.2, since it solves Open Problem 3 from
[Sch1]:
Corollary 3.4. Let V be a non-singular affine surface over R for which V (R) is
compact. Then every f ∈ R[V ] which is non-negative on V (R) is a sum of squares
in R[V ].
Here is another particular case of Theorem 3.2 (for sets of rectangular type):
Corollary 3.5. Let C, D be non-singular affine curves over R. Let f1 , . . . , fr ∈
R[C] and g1 , . . . , gs ∈ R[D] have the following properties:
SUMS OF SQUARES ON REAL ALGEBRAIC SURFACES
9
(1) All zeros of f1 · · · fr on C are simple;
(2) all zeros of g1 · · · gs on D are simple;
(3) the subset {f1 ≥ 0, . . . , fr ≥ 0} of C(R) is compact;
(4) the subset {g1 ≥ 0, . . . , gs ≥ 0} of D(R) is compact.
Then the preordering generated by
f1 ⊗ 1, . . . , fr ⊗ 1, 1 ⊗ g1 , . . . , 1 ⊗ gs
in R[C] ⊗ R[D] = R[C × D] is saturated.
Remark 3.6. Particular cases of 3.4 were found and announced in fall 1998, and
a general proof was found in early 2002. This proof was much more complicated
than the one presented here, being based on difficult algebraic-geometric arguments.
Fortunately, Kuhlmann, Marshall and Schwartz found their “basic lemma” [KMS],
which comes as a great help and allows us to give the easy proof presented here.
Remark 3.7. In all the results above, the complexity of the representations (of sums
of squares, or of preordering type) whose existence is asserted behaves badly. More
specifically, it is never possible to bound the degrees of the polynomials which constitute these representations in terms of the degree of the represented polynomial.
This is a general feature, the reasons for which are the compactness of the situation
and its dimension being at least two. See [Sch5], where a precise sense is given to
this remark.
3.8. We can get similar corollaries over base fields different from R. Evidently,
R can be replace by a real closed subfield of R in all the geometric results stated
here (replacing compactness by semi-algebraic compactness). More interestingly,
we get similar results for fields which are not real closed. Typically, the necessary
archimedean conditions will force us to consider fields with archimedean orderings.
The most interesting examples of such fields are number fields. We content ourselves
with giving a version of Theorem 3.2 for number fields; from the statement (and
the proof) one sees how other results can be modified to hold over such fields:
Corollary 3.9. Let k be a number field, let V be a non-singular affine surface over
k and let g1 , . . . , gr ∈ k[V ]. Assume that the curves gi = 0 on V are non-singular
and intersect transversely, and that no three of them intersect.
Further, let σν : k ,→ R (ν = 1, . . . , m) be real places of k with the property that
for each ν = 1, . . . , m the subset
Kν := {x : g1 (x) ≥ 0, . . . , gr (x) ≥ 0}
of V (R, σν ) is compact.
Then, given any f ∈ k[V ] with f ≥ 0 on Kν (ν = 1, . . . , m), there is an identity
X X
f=
ce,i p2e,i · g1e1 · · · grer
(2)
e∈{0,1}r
i
with pe,i ∈ k[V ] and scalars ce,i ∈ k satisfying σν (ce,i ) ≥ 0 (ν = 1, . . . , m).
(Of course, V (R, σν ) denotes the set of R-points on the R-variety V ⊗k,σν R.)
Note that the ce,i may be omitted if σ1 , . . . , σm is the set of all real places of k.
Proof. Choose scalars a1 , . . . , as ∈ k such that σ1 , . . . , σm are precisely those real
places of k which make a1 , . . . , as positive. Let T be the preordering in k[V ] generated by g1 , . . . , gr and a1 , . . . , as . Then T is archimedean ([Sch3] Thm. 3.6). For
f ∈ k[V ], the conditions f |Kν ≥ 0 (ν = 1, . . . , m) mean that f lies in Sat(T ).
By Lemma 3.1, Tm is saturated for any maximal ideal m of k[V ]. Therefore, T is
saturated by Cor. 2.9. The assertion follows since the general element of T has the
form of the right hand side in (2).
10
CLAUS SCHEIDERER
3.10. We next give an application to Hilbert’s seventeenth problem. Let f ∈
R[x0 , . . . , xn ] be a form (= homogeneous polynomial) in n + 1 variables which is
positive semidefinite (psd). By Artin’s solution, there exists an identity
h2 f = g12 + · · · + gr2
(3)
where r ≥ 1 and g1 , . . . , gr , h are non-zero forms. Recall that f is said to be positive
definite if f (p) > 0 for every 0 6= p ∈ Rn+1 . It has been proved by Reznick [R1]
that for positive definite f , the denominator h can be chosen uniformly to be a
power of x20 + · · · + x2n . This fact can also be deduced as a direct consequence of
Schmüdgen’s theorem (see [Sch4] 2.2.4).
The restriction to positive definite f is necessary: For any n ≥ 3 there exists a
psd form f in n + 1 variables, together with a point 0 6= p ∈ Rn+1 , such that, in
any representation (3), the form h vanishes in p. Such a point p has been called a
bad point for f . While the existence of these bad points has long been known for
forms in more than three variables, it is a consequence of the main result in [Sch2]
that they do not exist for ternary forms. More strikingly, for ternary forms we can
extend (and even generalize) Reznick’s uniform denominators result to arbitrary
positive semidefinite forms, without requiring that they are definite. This is an
application of the previous results:
Corollary 3.11. Let f , h ∈ R[x, y, z] be two positive semidefinite forms, where h
is positive definite. Then there exists an integer N ≥ 1 such that hN · f is a sum
of squares of forms.
In particular, if f (x, y, z) is any positive semidefinite form, then
(x2 + y 2 + z 2 )N · f (x, y, z)
is a sum of squares of forms for all N 0.
Remark 3.12. At the end of the formulation of his 17th problem, Hilbert [H] asked
for a refinement: Assuming that the positive semidefinite form f has coefficients in
the subfield k of R, is it true that f is a sum of squares of rational functions with
coefficients in k?
Strictly speaking, the answer is no, but it is well-known how to rectify the situa2
tion: There does exist an identity f = c1 g12 + · · · + cm gm
, where the gν are rational
functions with coefficients in k and the cν are scalars in k ∩ R+ . (See [KS] II §12,
for example.)
Reasoning as in Corollary 3.9, we can sharpen Corollary 3.11 in a similar spirit:
Let k be a subfield of R which contains the
P coefficients of f and h. Then there
exists N ≥ 1 such that hN f has the form ν cν pν (x, y, z)2 with cν ∈ k ∩ R+ and
pν ∈ k[x, y, z].
Proof of Corollary 3.11. Let V be the complement of the curve h = 0 in the projective plane P2 over R. Then V is a non-singular affine surface, and V (R) is compact.
The ring R[V ] of regular functions consists of all fractions hgd , where d ≥ 0 and g is
a form of the same degree as hd . Since f and h have even degree, there are integers
2m
d ≥ 1 and m ≥ 0 such that deg(hd ) = deg(x2m f ). Thus x hd f is a non-negative
regular function on V . As such it is a sum of squares of regular functions (Corollary
3.4). Clearing denominators, we get an identity
hN x2m f = g12 + · · · + gr2
with N ≥ 0 and forms gi . Each summand gi2 on the right must be divisible by
x2m ([Sch1] Lemma 0.2), and hence the pi := xgmi are also forms. We have hN f =
p21 + · · · + p2r .
SUMS OF SQUARES ON REAL ALGEBRAIC SURFACES
11
Remark 3.13. Corollary 3.11 says that every definite ternary form h is a “weak
common denominator” for Hilbert’s 17th problem, in the sense that, for every psd
ternary form f , a suitable power of h can be used as a denominator in a rational
sums of squares decomposition of f . This fact is nicely complemented by a recent
result of Reznick. It says that there cannot be any common denominator h in the
“strong” sense (without the need of raising h to powers). The actual result is even
a bit stronger; see [R2] for the details.
Remark 3.14. The results of [Sch5] show that, keeping the positive definite form
h fixed, it is not possible to bound the exponent N of Corollary 3.11 in terms of
the degree of f . They also show that Reznick’s uniform denominator result for
positive definite forms does not extend to arbitrary real closed fields: For any nonarchimedean real closed field R and any n ≥ 2, there exists a positive definite form
f in R[x0 , . . . , xn ] such that f · (x20 + · · · + x2n )N is not a sum of squares for any
N ≥ 1.
Remark 3.15. The results proved in this section can be extended to certain noncompact situations as well. Here we content ourselves with illustrating this by an
example:
Let T be the preordering in R[x, y] generated by x, 1 − x, y and 1 − xy. We claim
that T is saturated. Note that the subset K = S(T ) of the plane is unbounded (of
dimension two).
Let f ∈ R[x, y] with f |K ≥ 0. Choose n ≥ 0 so large that x2n f ∈ R[x, xy], and
write x2n f = g(x, xy) with g ∈ R[u, v] (i.e., g = u2n f (u, uv )). Then g ≥ 0 on the
closed square [0, 1]2 . By 3.2 or 3.3, the preordering P O(u − u2 , v − v 2 ) is saturated
in R[u, v]. So there are sums of squares sν in R[u, v] (ν = 0, . . . , 3) with
g = s0 + s1 u(1 − u) + s2 v(1 − v) + s3 uv(1 − u)(1 − v).
Substituting back we get
x2n f = t0 + t1 x(1 − x) + t2 xy(1 − xy) + t3 x2 y(1 − x)(1 − xy)
with sums of squares tν in R[x, y]. It follows that x2n divides each of the four
summands on the right ([Sch1] Lemma 0.2). Therefore,
t̃ν := tν /x2n (ν = 0, 1, 2), t̃3 := t3 x2 /x2n
are sums of squares in R[x, y], and we have
f = t̃0 + t̃1 x(1 − x) + t̃2 xy(1 − xy) + t̃3 y(1 − x)(1 − xy).
This implies the claim. In fact, we have even shown that the quadratic module
generated by x(1 − x), xy(1 − xy) and y(1 − x)(1 − xy) — which is a priori smaller
than T — is saturated.
3.16. We conclude with an application to real regular functions. Recall that if V
is a scheme of finite type over R, the ring of real regular functions on V is defined
to be
R[V ] := lim OV (U ),
−→
U
inductive limit over the Zariski open subsets U of V with U (R) = V (R). If V is
affine then R[V ] is the localization of R[V ] with respect to the multiplicative set of
denominators 1 + ΣR[V ]2 .
Proposition 3.17. Let V be a quasi-projective R-variety of dimension ≤ 2 which
has no real singular points. Then every non-negative real regular function on V is
a sum of squares of real regular functions.
12
CLAUS SCHEIDERER
Proof. If dim(V ) ≤ 1, this is already known ([Sch3] Cor. 4.22). So we can assume
dim(V ) = 2. We can replace V by its regular locus since Vsing (R) = ∅. So we
can assume that V is a non-singular and irreducible surface. The R-points of V
are contained in an affine open subvariety, so we can in addition assume that V is
affine. By choosing a projective closure of V , resolving its singularities and removing
a suitable hypersurface without R-points, we find an open immersion V ,→ W into
an affine non-singular surface W for which W (R) is compact.
Let C1 , . . . , Cm be those irreducible curves in W − V which have at least one
real point. There exists a real regular function s on W which vanishes along each
Ci and does not vanish along any other curve C ⊂ W with C(R) 6= ∅; this follows
from [BCR] Prop. 12.4.4, for example. In particular, the restriction of s to V is a
unit in R[V ].
Now let f ∈ R[V ] be psd. Then s2n f lies in R[W ] for n sufficiently large. Choose
t ∈ R[W ] without a zero in W (R) such that t2 s2n f ∈ R[W ]. By Theorem 3.4, this
function is a sum of squares in R[W ]. Dividing by t2 s2n , we see that f is a sum of
squares in R[V ].
In particular, the case where V is affine gives a Nichtnegativstellensatz of a very
special form:
Corollary 3.18. Let V be an affine R-variety of dimension ≤ 2 without real singular points, and let f ∈ R[V ]. Then f is non-negative on V (R) if and only if there
are sums of squares s, t in R[V ] with
(1 + s)f = t.
Remark 3.19. In his talk on the 1999 Saskatoon conference on valuation theory, Gil
Stengle posed the following question: Given a psd polynomial f ∈ R[x1 , . . . , xn ],
do there always exist finitely many polynomials g1 , . . . , gr such that
f ∈ P O(ε1 g1 , . . . , εr gr )
r
for all 2 choices of signs ε1 , . . . , εr ∈ {±1}? Note that any system g1 , . . . , gr with
this property is a witness for the non-negativity of f .
Stengle writes c(f ) for the smallest number r of such polynomials, and puts
c(f ) = ∞ if no such system of finitely many gi exists. He further asked for examples
with c(f ) > 1, in particular in the case n = 2.
Our last corollary answers Stengle’s questions for n = 2, and more generally, for
the coordinate ring of any affine R-variety V of dimension ≤ 2 without real singularities. Indeed, it asserts that f ∈ P O(−f ) holds for every positive semidefinite
f ∈ R[V ]. In particular, this implies c(f ) = 1.
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Institut für Mathematik, Fakultät 4, Universität Duisburg, 47048 Duisburg, Germany
E-mail address: [email protected]