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) 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} \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\} }[/math]
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) [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
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 [math]\displaystyle{ R = \mathbb{R} }[/math]. If the semialgebraic set [math]\displaystyle{ W=\{x\in \mathbb{R}^n\mid\forall f\in F,\,f(x)\ge0\} }[/math] is compact, then each polynomial [math]\displaystyle{ p \in \mathbb{R}[X_1, \dots, X_n] }[/math] that is strictly positive on [math]\displaystyle{ W }[/math] can be written as a polynomial in the defining functions of [math]\displaystyle{ W }[/math] with sums-of-squares coefficients, i.e. [math]\displaystyle{ p \in P(F, \emptyset) }[/math]. Here P is said to be strictly positive on [math]\displaystyle{ W }[/math] if [math]\displaystyle{ p(x)\gt 0 }[/math] for all [math]\displaystyle{ x \in W }[/math].[1] Note that Schmüdgen's Positivstellensatz is stated for [math]\displaystyle{ R = \mathbb{R} }[/math] and does not hold for arbitrary real closed fields.[2]
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_{\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\} }[/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{ p(x)\gt 0 }[/math] for all [math]\displaystyle{ x \in W }[/math], then p ∈ Q(F,G).[3]
See also
- Positive polynomial for other positivstellensatz theorems.
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.
- ↑ Stengle, Gilbert (1996). "Complexity Estimates for the Schmüdgen Positivstellensatz". Journal of Complexity 12 (2): 167–174. doi:10.1006/jcom.1996.0011.
- ↑ Putinar, Mihai (1993). "Positive Polynomials on Compact Semi-Algebraic Sets". Indiana University Mathematics Journal 42 (3): 969–984. doi:10.1512/iumj.1993.42.42045.
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 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. doi:10.1007/s10957-014-0545-3. ISSN 0022-3239.
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