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].
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 preorder 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.
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.)
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.
can be written as a polynomial in the defining functions of
[1] Note that Schmüdgen's Positivstellensatz is stated for
and does not hold for arbitrary real closed fields.
[2] Define the quadratic module associated with W as the set Assume there exists L > 0 such that the polynomial