Positivstellensatz
In real algebraic geometry, 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 an real analogue of Hilbert's Nullstellensatz, (which concern complex zeros of polynomial ideals), and this is this analogy that 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
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
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{ \forall x\in W\;p(x)\gt 0 }[/math], then p ∈ Q(F,G).[3]
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.
- ↑ 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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