Let R be a real closed field, and F = {f1, f2, ..., fm} and G = {g1, g2, ..., gr} finite sets of polynomials over R in n variables. Let W be the semialgebraic set
and define the preordering associated with W as the set
where Σ2[X1,...,Xn] is the set of sum-of-squares polynomials. In other words, P(F, G) = C + I, where C is the cone generated by F (i.e., the subsemiring of R[X1,...,Xn] generated by F and arbitrary squares) and I is the ideal generated by G.
Let p ∈ R[X1,...,Xn] be a polynomial. Krivine–Stengle Positivstellensatz states that
- (i) if and only if and such that .
- (ii) if and only if such that .
The weak Positivstellensatz is the following variant of the Positivstellensatz. Let R be a real closed field, and F, G, and H finite subsets of R[X1,...,Xn]. Let C be the cone generated by F, and I the ideal generated by G. Then
if and only if
(Unlike Nullstellensatz, the "weak" form actually includes the "strong" form as a special case, so the terminology is a misnomer.)
The Krivine–Stengle Positivstellensatz also has the following refinements under additional assumptions. It should be remarked that Schmüdgen's Positivstellensatz has a weaker assumption than Putinar's Positivstellensatz, but the conclusion is also weaker.
Schmüdgen's Positivstellensatz
Suppose that . If the semialgebraic set is compact, then each polynomial that is strictly positive on can be written as a polynomial in the defining functions of with sums-of-squares coefficients, i.e. . Here P is said to be strictly positive on if for all .[1] Note that Schmüdgen's Positivstellensatz is stated for and does not hold for arbitrary real closed fields.[2]
Putinar's Positivstellensatz
Define the quadratic module associated with W as the set
Assume there exists L > 0 such that the polynomial If for all , then p ∈ Q(F,G).[3]