Stengle's Positivstellensatz

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In mathematics, Stengle's Positivstellensatz characterizes polynomials which are positive on a given semialgebraic set over the real numbers, or more generally, any real-closed field. It can be thought of as an ordered analogue of Hilbert's Nullstellensatz. It was discovered by Gilbert Stengle.

[edit] Statement

Let R be a real-closed field, and F a finite set of polynomials over R in n variables. Let W be the semialgebraic set

$W=\{x\in R^n\mid\forall f\in F\,f(x)\ge0\},$

and let C be the cone generated by F (i.e., the subsemiring of R[X₁,…,X_n] generated by F and arbitrary squares). Let p ∈ R[X₁,…,X_n] be a polynomial. Then

$\forall x\in W\;p(x)>0$ if and only if $\exists f_1,f_2\in C\;pf_1=1+f_2$ .

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,…,X_n]. Let C be the cone generated by F, and I the ideal generated by G. Then

$\{x\in R^n\mid\forall f\in F\,f(x)\ge0\land\forall g\in G\,g(x)=0\land\forall h\in H\,h(x)\ne0\}=\emptyset$

if and only if

$\exists f\in C,g\in I,n\in\mathbb N\;f+g+\left(\prod H\right)^{2n}=0.$

(Unlike Nullstellensatz, the "weak" form actually includes the "strong" form as a special case, so the terminology is a misnomer.)

[edit] References

G. Stengle, A Nullstellensatz and a Positivstellensatz in Semialgebraic Geometry, Mathematische Annalen 207 (1973), no. 2, pp. 87–97.
J. Bochnak, M. Coste, M.-F. Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 36, Springer-Verlag, 1999.