Stengle's Positivstellensatz

In real semialgebraic geometry, Stengle's Positivstellensatz (German for "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 proved by Jean-Louis Krivine and then rediscovered 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

${\displaystyle W=\{x\in R^{n}\mid \forall f\in F\,f(x)\geq 0\},}$

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

${\displaystyle \forall x\in W\;p(x)>0}$ if and only if ${\displaystyle \exists f_{1},f_{2}\in C\;pf_{1}=1+f_{2}}$.

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

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

if and only if

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

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

References

• Krivine, J.L. (1964). "Anneaux préordonnés". Journal d'analyse mathématique. 12: 307–326. doi:10.1007/bf02807438.
• 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 3-540-64663-9.

See also

• Positive polynomial