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
[edit]![]() | This section may be confusing or unclear to readers. In particular, firstly, a set of polinomial is not a preordering; secondly, such a huge formula requires an explanation. (January 2024) |
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
and define the preordering associated with W as the set
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) if and only if and such that .
- (ii) if and only if such that .
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
if and only if
(Unlike Nullstellensatz, the "weak" form actually includes the "strong" form as a special case, so the terminology is a misnomer.)
Variants
[edit]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
[edit]Suppose that . If the semialgebraic set is compact, then each polynomial that is strictly positive on can be written as a polynomial in the defining functions of with sums-of-squares coefficients, i.e. . Here P is said to be strictly positive on if for all .[1] Note that Schmüdgen's Positivstellensatz is stated for and does not hold for arbitrary real closed fields.[2]
Putinar's Positivstellensatz
[edit]Define the quadratic module associated with W as the set
Assume there exists L > 0 such that the polynomial If for all , then p ∈ Q(F,G).[3]
See also
[edit]- Positive polynomial for other positivstellensatz theorems.
- Real Nullstellensatz
Notes
[edit]- ^ Schmüdgen, Konrad [in German] (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
[edit]- 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.