In real algebraic geometry, KrivineâÂÂStengle (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 .
Let be a real closed field, and = {f<sub>1</sub>, f<sub>2</sub>, ..., f<sub>m</sub>} and = {g<sub>1</sub>, g<sub>2</sub>, ..., g<sub>r</sub>} finite sets of polynomials over in variables. Let be the semialgebraic set
and define the preorder (in the sense of a prepositive cone) associated with as the set
where ã<sup>2</sup>[<sub>1</sub>,...,<sub></sub>] is the set of sum-of-squares polynomials. In other words, (, ) = + , where is the cone generated by (i.e., the subsemiring of [<sub>1</sub>,...,<sub></sub>] generated by and arbitrary squares) and is the ideal generated by .
Let ∈ [<sub>1</sub>,...,<sub></sub>] be a polynomial. KrivineâÂÂStengle Positivstellensatz states that
The weak is the following variant of the . Let be a real closed field, and , , and finite subsets of [<sub>1</sub>,...,<sub></sub>]. Let be the cone generated by , and the ideal generated by . Then
if and only if
(Unlike , the "weak" form actually includes the "strong" form as a special case, so the terminology is a misnomer.)
The KrivineâÂÂStengle 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.
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 is said to be strictly positive on if for all . Note that Schmüdgen's Positivstellensatz is stated for and does not hold for arbitrary real closed fields.
Define the quadratic module associated with as the set
Assume there exists L > 0 such that the polynomial If for all , then ∈ (,').