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 [fr; de] and then rediscovered by the Canadian Gilbert Stengle [Wikidata].

Statement

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 preorder (in the sense of a prepositive cone) 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.)

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 ${\displaystyle R=\mathbb {R} }$. If the semialgebraic set ${\displaystyle W=\{x\in \mathbb {R} ^{n}\mid \forall f\in F,\,f(x)\geq 0\}}$ is compact, then each polynomial ${\displaystyle p\in \mathbb {R} [X_{1},\dots ,X_{n}]}$ that is strictly positive on ${\displaystyle W}$ can be written as a polynomial in the defining functions of ${\displaystyle W}$ with sums-of-squares coefficients, i.e. ${\displaystyle p\in P(F,\emptyset )}$. Here P is said to be strictly positive on ${\displaystyle W}$ if ${\displaystyle p(x)>0}$ for all ${\displaystyle x\in W}$.^[1] Note that Schmüdgen's Positivstellensatz is stated for ${\displaystyle R=\mathbb {R} }$ and does not hold for arbitrary real closed fields.^[2]

Putinar's Positivstellensatz

Define the quadratic module associated with W as the set

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

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

See also

• Positive polynomial for other positivstellensatz theorems.
• Real Nullstellensatz