Krivine–Stengle Positivstellensatz

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


 * $$W=\{x\in R^n\mid\forall f\in F,\,f(x)\ge0;\, \forall g\in G,\,g(x)=0\},$$

and define the preordering associated with W as the set


 * $$P(F,G) = \left\{ \sum_{\alpha \in \{0,1\}^m} \sigma_\alpha f_1^{\alpha_1} \cdots f_m^{\alpha_m} + \sum_{\ell=1}^r \varphi_\ell g_\ell : \sigma_\alpha \in \Sigma^2[X_1,\ldots,X_n];\ \varphi_\ell \in R[X_1,\ldots,X_n] \right\} $$

where Σ2[X1,...,Xn] is the set of sum-of-squares polynomials. In other words, P(F, G) = +, where  is the cone generated by F (i.e., the subsemiring of R[X1,...,Xn] generated by F and arbitrary squares) and  is the ideal generated by G.

Let p &isin; R[X1,...,Xn] be a polynomial. Krivine–Stengle Positivstellensatz states that


 * (i) $$\forall x\in W\;p(x)\ge 0$$ if and only if $$\exists q_1,q_2\in P(F,G)$$ and $$s \in \mathbb{Z}$$ such that $$q_1 p = p^{2s} + q_2$$.
 * (ii) $$\forall x\in W\;p(x)>0$$ if and only if $$\exists q_1,q_2\in P(F,G)$$ such that $$q_1 p = 1 + q_2$$.

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 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.)

Variants
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.

Schmüdgen's Positivstellensatz
Suppose that $$R = \mathbb{R}$$. If the semialgebraic set $$W=\{x\in \mathbb{R}^n\mid\forall f\in F,\,f(x)\ge0\}$$ is compact, then each polynomial $$ p \in \mathbb{R}[X_1, \dots, X_n] $$ that is strictly positive on $$W$$ can be written as a polynomial in the defining functions of $$ W $$ with sums-of-squares coefficients, i.e. $$ p \in P(F, \emptyset) $$. Here P is said to be strictly positive on $$ W $$ if $$p(x)>0$$ for all $$ x \in W $$. Note that Schmüdgen's Positivstellensatz is stated for $$R = \mathbb{R}$$ and does not hold for arbitrary real closed fields.

Putinar's Positivstellensatz
Define the quadratic module associated with W as the set


 * $$Q(F,G) = \left\{ \sigma_0 + \sum_{j=1}^m \sigma_j f_j + \sum_{\ell=1}^r \varphi_\ell g_\ell : \sigma_j \in \Sigma^2 [X_1,\ldots,X_n];\ \varphi_\ell \in \mathbb{R}[X_1,\ldots,X_n] \right\} $$

Assume there exists L > 0 such that the polynomial $$L - \sum_{i=1}^n x_i^2 \in Q(F,G).$$ If $$p(x)>0$$ for all $$x \in W$$, then p &isin; Q(F,G).