Krivine–Stengle Positivstellensatz

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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; de] and then rediscovered by the Canadian Gilbert Stengle [Wikidata].

Statement[edit]

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 pQ(F,G).[3]

See also[edit]

Notes[edit]

  1. ^ 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.
  2. ^ Stengle, Gilbert (1996). "Complexity Estimates for the Schmüdgen Positivstellensatz". Journal of Complexity. 12 (2): 167–174. doi:10.1006/jcom.1996.0011.
  3. ^ 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]