In real algebraic geometry, Krivine–Stengle Positivstellensatz (German for "positive-locus-theorem") 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 a real analogue of Hilbert's Nullstellensatz (which concern complex zeros of polynomial ideals), and this analogy is at the origin of its name. It was proved by French mathematician Jean-Louis Krivine and then rediscovered by the Canadian Gilbert Stengle.
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
W = { x ∈ R n ∣ ∀ f ∈ F , f ( x ) ≥ 0 ; ∀ g ∈ G , g ( x ) = 0 } , {\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 preorder (in the sense of a prepositive cone) associated with W as the set
P ( F , G ) = { ∑ α ∈ { 0 , 1 } m σ α f 1 α 1 ⋯ f m α m + ∑ ℓ = 1 r φ ℓ g ℓ : σ α ∈ Σ 2 [ X 1 , … , X n ] ; φ ℓ ∈ R [ X 1 , … , X n ] } {\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,...,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) ∀ x ∈ W p ( x ) ≥ 0 {\displaystyle \forall x\in W\;p(x)\geq 0} if and only if ∃ q 1 , q 2 ∈ P ( F , G ) {\displaystyle \exists q_{1},q_{2}\in P(F,G)} and s ∈ Z {\displaystyle s\in \mathbb {Z} } such that q 1 p = p 2 s + q 2 {\displaystyle q_{1}p=p^{2s}+q_{2}} . (ii) ∀ x ∈ W p ( x ) > 0 {\displaystyle \forall x\in W\;p(x)>0} if and only if ∃ q 1 , q 2 ∈ P ( F , G ) {\displaystyle \exists q_{1},q_{2}\in P(F,G)} such that q 1 p = 1 + q 2 {\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
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