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, ..., f_m} and G = {g₁, g₂, ..., g_r} finite sets of polynomials over R in n variables. Let W be the semialgebraic set

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

and define the preorder 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,...,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) $\forall x\in W\;p(x)\geq 0$ 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,...,X_n]. Let C be the cone generated by F, and I the ideal generated by G. Then

$\{x\in R^{n}\mid \forall f\in F\,f(x)\geq 0\land \forall g\in G\,g(x)=0\land \forall h\in H\,h(x)\neq 0\}=\emptyset$

if and only if

$\exists f\in C,g\in I,n\in \mathbb {N} \;f+g+\left(\prod H\right)^{\!2n}=0.$

(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 $W=\{x\in \mathbb {R} ^{n}\mid \forall f\in F,\,f(x)\geq 0\}$ 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 ∈ Q(F,G).

See also

• Positive polynomial for other positivstellensatz theorems.
• Real Nullstellensatz

References

• Krivine, J. L. (1964). "Anneaux préordonnés". Journal d'Analyse Mathématique. 12: 307–326. doi:10.1007/bf02807438. S2CID 189771756.
• Stengle, G. (1974). "A Nullstellensatz and a Positivstellensatz in Semialgebraic Geometry". Mathematische Annalen. 207 (2): 87–97. doi:10.1007/BF01362149. S2CID 122939347.
• 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.