Stengle's Positivstellensatz

In real semialgebraic geometry, Stengle's Positivstellensatz (German: "positive-locus-theorem" – see Satz) characterizes polynomials that are positive on a semialgebraic set, which is defined by systems of inequalities of polynomials with real coefficients, or more generally, coefficients from any real-closed field.

It can be thought of as an ordered analogue of Hilbert's Nullstellensatz. It was discovered by Gilbert Stengle.

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.)

References

• Stengle, G. (1974). "A Nullstellensatz and a Positivstellensatz in Semialgebraic Geometry". Mathematische Annalen 207 (2): 87–97. doi:10.1007/BF01362149. 
• Bochnak, J.; Coste, M.; Roy, M.-F. (1999). Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. 36. New York: Springer-Verlag. ISBN 3540646639.