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 and then rediscovered by the Canadian .
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
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[]
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 . 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[]
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[]
- 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". . 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.
- 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. CiteSeerX 10.1.1.771.2203. doi:10.1007/s10957-014-0545-3. ISSN 0022-3239.
- Real algebraic geometry
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- Theorems in algebraic geometry