Krivine–Stengle Positivstellensatz

Let R be a real closed field, and F = {f₁, f₂, ..., f_m} and G = {g₁, g₂, ..., g_r} finite sets of polynomials over R in n variables. Let W be the semialgebraic set

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

and define the preordering associated with W as the set

${\displaystyle P(F,G)=\left\{\sum _{\alpha \in \{0,1\}^{m}}\sigma _{\alpha }f_{1}^{\alpha _{1}}\cdots f_{m}^{\alpha _{m}}+\sum _{\ell =1}^{r}\varphi _{\ell }g_{\ell }:\sigma _{\alpha }\in \Sigma ^{2}[X_{1},\ldots ,X_{n}];\ \varphi _{\ell }\in R[X_{1},\ldots ,X_{n}]\right\}}$

where Σ²[X₁,...,X_n] 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,...,X_n] generated by F and arbitrary squares) and I is the ideal generated by G.

Let p ∈ R[X₁,...,X_n] be a polynomial. Krivine–Stengle Positivstellensatz states that

(i) ${\displaystyle \forall x\in W\;p(x)\geq 0}$ if and only if ${\displaystyle \exists q_{1},q_{2}\in P(F,G)}$ and ${\displaystyle s\in \mathbb {Z} }$ such that ${\displaystyle q_{1}p=p^{2s}+q_{2}}$.

(ii) ${\displaystyle \forall x\in W\;p(x)>0}$ if and only if ${\displaystyle \exists q_{1},q_{2}\in P(F,G)}$ such that ${\displaystyle q_{1}p=1+q_{2}}$.

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.)

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.

• Positive polynomial for other positivstellensatz theorems.

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• Bochnak, J.; Coste, M.; Roy, M.-F. (1999). Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. Vol. 36. New York: Springer-Verlag. ISBN 978-3-540-64663-1.
• Jeyakumar, V.; Lasserre, J. B.; Li, G. (2014-07-18). "On Polynomial Optimization Over Non-compact Semi-algebraic Sets". Journal of Optimization Theory and Applications. 163 (3): 707–718. CiteSeerX 10.1.1.771.2203. doi:10.1007/s10957-014-0545-3. ISSN 0022-3239. S2CID 254745314.