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 French mathematiciam Jean-Louis Krivine [fr; de] and then rediscovered by the Canadian Gilbert Stengle [Wikidata].

Statement

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

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

and define the preordering associated with W as the set

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

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) ${\displaystyle \forall x\in W\;p(x)\geq 0}$ 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,…,Xn]. 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.)

Variants

Stengle's Positivstellensatz have also 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

Assume that the set ${\displaystyle W=\{x\in R^{n}\mid \forall f\in F,\,f(x)\geq 0\}}$ is compact and let p ∈ R[X1,…,Xn] be a polynomial. If ${\displaystyle \forall x\in W\;p(x)>0}$, then pP(F, ).[1] Note that Schmüdgen's Positivstellensatz requires ${\displaystyle R=\mathbb {R} }$ and does not hold for arbitrary real closed fields.[2]

Putinar's Positivstellensatz

Define the quadratic module associated with W as the set

${\displaystyle Q(F,G)=\left\{\sigma _{0}+\sum _{j=1}^{m}\sigma _{j}f_{j}+\sum _{l=1}^{r}\phi _{l}g_{l}:\sigma _{j}\in \Sigma ^{2}[X_{1},\ldots ,X_{n}];\ \phi _{l}\in \mathbb {R} [X_{1},\ldots ,X_{n}]\right\}}$

Assume there exists L > 0 such that the polynomial ${\displaystyle L-\sum _{i=1}^{n}x_{i}^{2}\in Q(F,G).}$ If ${\displaystyle \forall x\in W\;p(x)>0}$, then pQ(F,G).[3]