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) and then rediscovered by the Canadian Gilbert Stengle ({{{2}}}).
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
- [math]\displaystyle{ W=\{x\in R^n\mid\forall f\in F,\,f(x)\ge0;\, \forall g\in G,\,g(x)=0\}, }[/math]
and define the preordering associated with W as the set
- [math]\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 \mathbb{R}[X_1,\ldots,X_n] \right\} }[/math]
where Σ[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) [math]\displaystyle{ \forall x\in W\;p(x)\ge 0 }[/math] if and only if [math]\displaystyle{ \exists q_1,q_2\in P(F,G) }[/math] and [math]\displaystyle{ s \in \mathbb{Z} }[/math] such that [math]\displaystyle{ q_1 p = p^{2s} + q_2 }[/math].
- (ii) [math]\displaystyle{ \forall x\in W\;p(x)\gt 0 }[/math] if and only if [math]\displaystyle{ \exists q_1,q_2\in P(F,G) }[/math] such that [math]\displaystyle{ q_1 p = 1 + q_2 }[/math].
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
- [math]\displaystyle{ \{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 }[/math]
if and only if
- [math]\displaystyle{ \exists f\in C,g\in I,n\in\mathbb N\;f+g+\left(\prod H\right)^{2n}=0. }[/math]
(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 [math]\displaystyle{ W=\{x\in R^n\mid\forall f\in F,\,f(x)\ge0\} }[/math] is compact and let p ∈ R[X1,…,Xn] be a polynomial. If [math]\displaystyle{ \forall x\in W\;p(x)\gt 0 }[/math], then p∈ P(F, ∅).[1]
Putinar's Positivstellensatz
Define the quadratic module associated with W as the set
[math]\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\} }[/math]
Assume there exists L > 0 such that the polynomial [math]\displaystyle{ L - \sum_{i=1}^n x_i^2 \in Q(F,G). }[/math] If [math]\displaystyle{ \forall x\in W\;p(x)\gt 0 }[/math], then p∈ Q(F,G).[2]
See also
Notes
- ↑ Schmüdgen, Konrad (1991). "The K-moment problem for compact semi-algebraic sets". Mathematische Annalen 289 (1): 203–206. doi:10.1007/bf01446568. ISSN 0025-5831. https://link.springer.com/article/10.1007%2FBF01446568.
- ↑ Putinar, Mihai (1993). "Positive Polynomials on Compact Semi-Algebraic Sets". Indiana University Mathematics Journal 42 (3): 969–984.
References
- Krivine, J. L. (1964). "Anneaux préordonnés". Journal d'analyse mathématique 12: 307–326. doi:10.1007/bf02807438. http://hal.archives-ouvertes.fr/hal-00165658/.
- Stengle, G. (1974). "A Nullstellensatz and a Positivstellensatz in Semialgebraic Geometry". Mathematische Annalen 207 (2): 87–97. doi:10.1007/BF01362149.
- Bochnak, J.; Coste, M.; Roy, M.-F. (1999). Real algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge. 36. New York: Springer-Verlag. ISBN 3-540-64663-9.
- 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. doi:10.1007/s10957-014-0545-3. ISSN 0022-3239. https://link.springer.com/article/10.1007%2Fs10957-014-0545-3.
 |
