The reason why we can solve the Membership Problem affirmatively over arbitrary real closed fields if the semialgebraic set S associated to the quadratic module is finite is that the local-global principle is in this case true not just over R but over arbitrary real closed fields. Instead of using the Basic Lemma and the Kadison-Dubois Theorem which forced us to work over R in Theorem 2.9 we now apply the abstract Stellensatz for quadratic modules.
Theorem 2.34
Let f, g1, ..., gs∈R[X] and Q=QM(g1, ..., gs) with S =S(g1, ..., gs)⊆R finite.
If fba∈Qba for every a ∈Z(f)∩S and f|S ≥0 then f ∈Q.
Proof:
An argument similar to the one used in the proof of Lemma 2.6 shows that S =∅ gives Q=R[X] such that f ∈Q is trivially true.
Now we suppose that ∅ 6=S ={a1, ..., am} for some ai ∈R (1≤i≤m).
Then the elements α ∈ SperR[X] with Q ⊆ α are the orderings αai (1 ≤ i ≤ m) which correspond to evaluation in ai.
We consider the polynomial p :=
m
Q
i=1
(X−ai) which is clearly in the support of αai for every 1 ∈ {1, ..., m}. By Proposition 0.3 this means that p ∈supp(β) for every
semiordering β ∈ HSemi(Q). Thus the abstract Stellensatz for quadratic modules (Theorem 0.4 iii)) gives someN ∈N such that
−p2N =−
By the Chinese Remainder Theorem we can now find for every 1 ≤ i ≤ m a poly-nomial qi ∈ R[X] with hi ≡ δij mod (X −aj)2N where δij denotes the Kronecker symbol (1≤j ≤m).
We put these ingredients together and define q:=
m the quadratic module Qbecause −Qm
i=1
(X−ai)2N ∈Q and q∈Q.
Theorem 2.342
Since the description of the structure of the quadratic modules in the local power series rings in Section 2.1 is given for arbitrary real closed fields this local-global principle gives us by looking at the proof of Theorem 2.10 or 2.18 the following characterization of membership in QM(g1, ..., gs)⊆R[X] if S(g1, ..., gs) is finite.
Theorem 2.35
Let f ∈R[X]and G={g1, ..., gs} ⊆R[X] with S(g1, ..., gs) = {a1, ..., am} ⊆R.
Then f ∈ QM(g1, ..., gs) if and only if f(ai) ≥ 0 (1 ≤ i ≤ m) and for every i∈ {1, ..., m} we have ordai(f) even and ai(f) = 1 or
Case 1: ordai(f)≥kai(G) if kai(G)< ka+
i(G) and kai(G)< k−a
i(G).
Case 2: (ordai(f)−k+a
i(G)∈2N0 and ai(f) = 1) or ordai(f)≥min(kai(G), k−a
i(G)) if ka+i(G)≤min(kai(G), ka−i(G)).
Case 3: (ordai(f)−ka−i(G)∈2N0andai(f) = −1) or ordai(f)≥min(kai(G), k+ai(G)) if ka−
i(G)≤min(kai(G), ka+
i(G)).
Proof:
In analogy to the proof of Theorem 2.18 just using Theorem 2.34 instead of 2.9.
Theorem 2.352 Similar to Corollary 2.19 we give another formulation of the theorem where we do distinguish the different types of isolated points.
Corollary 2.36
Let f ∈R[X]and G={g1, ..., gs} ⊆R[X] with S(g1, ..., gs) = {a1, ..., am} ⊆R.
Then f ∈ QM(g1, ..., gs) if and only if f(ai) ≥ 0 (1 ≤ i ≤ m) and for every i∈ {1, ..., m} we have ordai(f) even and ai(f) = 1 or
Type A: ordai(f)≥kai(G)
if kai(G)< k+ai(G)and kai(G)< ka−i(G).
Type B1: (ordai(f)−k+a
i(G)∈2N0 and ai(f) = 1) or ordai(f)≥ka−
i(G) if ka+i(G)≤ka−i(G)< kai(G).
Type B2: (ordai(f)−k−ai(G)∈2N0 and ai(f) =−1) or ordai(f)≥ka+i(G) if ka−
i(G)< k+a
i(G)< kai(G).
Type C: (ordai(f)−k+a
i(G)∈2N0 and ai(f) = 1) or ordai(f)≥kai(G) if ka+i(G)< kai(G)< ka−i(G).
Type D: (ordai(f)−k−ai(G)∈2N0 and ai(f) =−1) or ordai(f)≥kai(G) if ka−
i(G)< kai(G)< ka+
i(G).
The fact that we now have a characterization over arbitrary real closed fields does in fact imply stability of this kind of quadratic modules. We come back to this in the chapter about heirs.
Theorem 2.37
Forg1, ..., gs∈R[X]withS(g1, ..., gs)⊆Rfinite the quadratic moduleQM(g1, ..., gs) is weakly semialgebraic.
Proof:
This is clear by the previous theorem.
Theorem 2.372
As in Section 2.1 this gives the following two corollaries.
Corollary 2.38
Ifg1, ..., gs ∈R[X]and the input data is computable then the Membership Problem is solvable affirmatively forQM(g1, ..., gs)if S(g1, ..., gs) is finite.
Corollary 2.39
For f(X, Y) ∈ Z[X, Y], g1(X, Z), ..., gs(X, Z) ∈ Z[X, Z] there are L-formulas ψ(Z) andϕ(Y, Z)such that we have for every real closed field R and anyc∈RY, b∈RZ: If
R|=ψ(b) then
f(X, c)∈QM(g1(X, b), ..., gs(X, b))⇔R|=ϕ(c, b).
Proof:
The formula
ψ(Z) :=∀X
s
^
i=1
gi(X, Z)≥0→
s
_
i=1
gi(X, Z) = 0
!
expresses the finiteness of the basic closed set S(g1, ..., gs) and the formula ϕ(Y, Z) can be obtained with the considerations from Remark 2.12.
Corollary 2.392 Theorem 2.35 together with Lemma 2.6 imply that forf, g1, ..., gs ∈R[X] the mem-bership off in the quadratic module generated by g1, ..., gs,−f2 is given by finitely many local conditions which can be expressed by a semialgebraic formula as ex-plained in Section 2.1.
Corollary 2.40
Let f, g1, ..., gs∈R[X].
Then we have with Q=QM(g1, ..., gs)and S =S(g1, ..., gs)
fba ∈Qba ∀a∈Z(f)∩S⇔f ∈Q+f2R[X] =QM(g1, ..., gs,−f2) Proof:
The implication ⇒ is just Lemma 2.6.
For the other inclusion let Qe:=QM(g1, ..., gs,−f2) =Q+f2R[X].
Then S(Q) =e Z(f)∩S which is empty or finite.
IfS(Q) is empty thene H(g1, ...gs,−f2) =∅and thus by Proposition 0.3 we also have
∅ = HSemi(g1, ...gs,−f2) = Hsemi(Q) which implies by the abstract Stellensatz fore quadratic modules (Theorem 0.4 iv)) that −1∈Qe and thus Qe =R[X]. Hence the assumption as well as the conclusion is in this case true for every f ∈R[X].
Now we suppose that S(Q)e 6= ∅. As f ∈ Qe we know by Theorem 2.35 that f ful-fills for every a ∈S(Q) =e Z(f)∩S the order conditions determined by the values ka(G), ke a+(G) ande ka−(G) wheree Ge := {g1, ..., gs,−f2}. Since orda(−f2) > orda(f) for every zero of f this means thatf also satisfies the order conditions from Theo-rem 2.35 for every a∈Z(f)∩S with respect to the valueska(G), ka+(G) andka−(G) where G:={g1, ..., gs}. This means nothing else than fba∈Qba ∀a∈Z(f)∩S.
Corollary 2.40 2 As in Section 2.1 we transfer with the help of the local-global principle the multi-plicative closure of the quadratic modules in the formal power series ring to the ring of polynomials.
Theorem 2.41
Let g1, ..., gs ∈R[X] with S(g1, ..., gs)⊆R finite.
Then the quadratic module Q = QM(g1, ..., gs) is closed under multiplication and thus Q=P O(g1, ..., gs).
Proof:
We use exactly the same argument as in the proof of Theorem 2.17 with Theorem 2.9 replaced by Theorem 2.34.
Theorem 2.412 Now we describe the support of finitely generated quadratic modulesQ=QM(G) of R[X] with finiteS =S(G). In general it is hard to determine the supportQ∩ −Qof a quadratic moduleQ. With the help of our explicit characterization of membership
we can do better. Actually what the support concerns the case where S consists of finitely many points is the interesting one. Since in the other case int(S)6=∅which forces the support ofQ to be{0}.
Corollary 2.42
LetG={g1, ..., gs} ⊆R[X] and ∅ 6=S(G) ={a1, ..., am} ⊆R. Then the support of the quadratic module Q=QM(G) is given by
supp(Q) =
The condition that f and −f has to be in Q implies that f has to fulfill the order conditions from Theorem 2.35 which do not depend onai(f) in everyai(1≤i≤m). in the statement of the Corollary. This gives⊆.
The other inclusion is true because
m
Q
i=1
(X−ai)ki ∈ supp(Q) by Theorem 2.35.
Corollary 2.422
With the help of this characterization of the support we can derive the following.
Corollary 2.43
LetQ=QM(g1, ..., gs)⊆R[X] and S =S(g1, ..., gs)6=∅. Then supp(Q) ={0} ⇔int(S)6=∅.
Proof:
If the interior of S is not empty then every polynomial f ∈ supp(Q) has to be zero because it must fulfillf|S = 0.
For the other implication we use the preceding corollary. Suppose that the interior of S is empty. ThenS is of the formS ={a1, ..., am}for some ai ∈R (1≤i≤m).
By Corollary 2.42 there is some polynomial f which is not identically zero and be-longs to the support ofQ.
Corollary 2.432
Theorem 2.35 and Corollary 2.42 also allow us to describe the finitely generated quadratic modules Q ⊆ R[X] with finite associated semialgebraic set in the form Q=Q0+ supp(Q) whereQ0 is a definable subset of a finite dimensional vector space R[X]≤D for some D∈N.
Corollary 2.44
Let G={g1, ..., gs} ⊆R[X] and ∅ 6=S(G) = {a1, ..., am} ⊆R.
If supp(Q) = m
Q
i=1
(X−ai)ki
R[X] for some k1, ..., km ∈ 2N is the support of the quadratic module Q:=QM(G)and D :=k1+...+km−1 then we have
Q=Q0+supp(Q)
with Q0 ⊆ R[X]≤D characterized by p ∈ Q0 if and only if for every 1 ≤ i ≤ m ordai(p)even and ai(p) = 1 or
Case 1: ordai(p)≥kai(G) if kai(G)< k+a
i(G) and kai(G)< k−a
i(G).
Case 2: (ordai(p)−ka+i(G) ∈2N0 and ai(p) = 1) or ordai(p)≥ min(kai(G), k−ai(G)) if ka+i(G)≤min(kai(G), ka−i(G)).
Case 3: (ordai(p)−ka−
i(G)∈2N0 andai(p) =−1) or ordai(p)≥min(kai(G), k+a
i(G)) if ka−i(G)≤min(kai(G), ka+i(G)).
Proof:
For abbreviation we write g(X) :=
m
Q
i=1
(X−ai)ki.
We consider some f ∈Q and get by division through g that f =qg+r
with unique q, r ∈R[X] such that either r = 0 or deg(r)≤D.
If r= 0 then f ∈ supp(Q).
In the other case let a=ai be some point of S with k =ki. We have
orda(r) = orda(f−qg)≥min{orda(q) + orda(g),orda(f)} (∗) with equality if orda(q)+orda(g)6= orda(f).
We note that orda(g) =k and distinguish the different possibilities for k according to Corollary 2.42.
First let k =ka(G) (i.e. ka(G)< k+a(G), ka−(G))
If orda(f)≥k then we also have orda(r)≥k because of (∗) .
If otherwise ν = orda(f)< k even and a(f) = 1 then we have equality in (∗) and hence orda(r) = orda(f) also even. Fromf = (X−a)νfe= (X−a)ν(qeg+er) we see
that ˜r(a) = ˜f(a)>0 because of ˜g(a) = 0 asν < k. Hence a(r) =a(f) = 1.
Now suppose that k = min(ka(G), ka+(G)) (i.e. ka−(G)≤min(ka(G), ka+(G))) If orda(f)≥k then we also have orda(r)≥k because of (∗) .
If otherwise ν = orda(f) < k we have equality in (∗) and hence orda(r) = orda(f).
Similar to the first case we also see thata(f) =a(r) which means that the condi-tions for orda(r) and a(r) are the same as the conditions for orda(f) and a(f).
The final case k = min(ka(G), k−a(G)) (i.e. ka+(G)≤min(ka(G), ka−(G))) can be done similarly.
Altogether we have proved that r∈Q0,i.e. Q⊆Q0+ supp(Q).
The other inclusion is clear.
Corollary 2.442 An important class of quadratic modules with finite associated semialgebraic set are preorderings of the formP
R[X]2+IwhereI ⊆R[X] is an ideal. IfI is generated by g1, ..., glthenP
R[X]2+I =P O(g1, ..., gl,−g1, ...,−gl) =QM(g1, ..., gl,−g1, ...,−gl) with associated semialgebraic setZ(I) = {x∈R|gi(x) = 0 (1 ≤i≤l)}. This kind of preordering which is composed of the two stable parts sums of squares and ideal is itself again stable as we will see in the chapter about heirs. Although one might guess that the support of P
R[X]2+I is I this is not true in general, the support can strictly contain I, which we show with the following example.
Example 2.45
The support of the preordering
P =QM(X(X2+ 1),−X(X2+ 1)) =X
R[X]2+ X(X2+ 1)
R[X]⊆R[X]
is by Corollary 2.42 given by
supp(P) =XR[X]⊃ X(X2+ 1) R[X].
The fact that X ∈supp(P) can be seen explicitly as follows:
Since ±X(X2+ 1) ∈P we have pX(X2+ 1) = (p+ 1
2 )2X(X2+ 1) + (p−1
2 )2(−X(X2+ 1))∈P
for every p ∈ R[X]. Thus in particular (−X)X(X2 + 1) = −X4 −X2 ∈ P which gives that −X2 ∈P. Hence
X =X(X2+ 1)−X3 =X(X2+ 1) + (X+ 1
2 )2(−X2) + (X−1
2 )2X2 ∈P.
Similarly
−X =−X(X2+ 1) +X3 =−X(X2+ 1) + (X+ 1
2 )2(X2) + (X−1
2 )2(−X2)∈P which implies that X ∈supp(P).