Krivine–Stengle Positivstellensatz Explained

In real algebraic geometry, Krivine–Stengle (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 be a real closed field, and = and = finite sets of polynomials over in variables. Let be the semialgebraic set

W = \{x\in R ^{n\mid\forall} f\in F,f(x)\ge 0 ; \forall g\in G,g(x) =0 \},

and define the preordering associated with as the set

P(F,G) = \left\{ \sum ^{m}} _{\alpha \in \{ 0 , 1 \} \sigma _{\alpha} f ^{\alpha _{1}} _{1} … f ^{\alpha _{m}} _{m} + \sum ^{r} _{\ell =1} \varphi _{\ell} g _{\ell} : \sigma _{\alpha} \in \Sigma ^{2 [X} _{1 ,\ldots,X} _{n];} \varphi _{\ell} \in R[X _{1,\ldots,X} _{n]} \right\}

where Σ²[{{Var|X}}₁,...,{{Var|X}}_{{Var|n}}] is the set of sum-of-squares polynomials. In other words, = +, where is the cone generated by (i.e., the subsemiring of [{{Var|X}}₁,...,{{Var|X}}_{{Var|n}}] generated by and arbitrary squares) and is the ideal generated by .

Let  ∈ [{{Var|X}}₁,...,{{Var|X}}_{{Var|n}}] be a polynomial. Krivine–Stengle Positivstellensatz states that

(i)

\forall x\in W p(x)\ge 0 if and only if \exists q _{1,q} _{2\in} P(F,G) and s \in Z such that q _{1} p = p ^{2s} + q _{2} .

(ii)

\forall x\in W p(x)> 0 if and only if \exists q _{1,q} _{2\in} P(F,G) such that q _{1} p = 1 + q _{2} .

The weak is the following variant of the . Let be a real closed field, and,, and finite subsets of [{{Var|X}}₁,...,{{Var|X}}_{{Var|n}}]. Let be the cone generated by, and the ideal generated by . Then

\{x\in R ^{n\mid\forall} f\in Ff(x)\ge 0 \land\forall g\in Gg(x) =0 \land\forall h\in Hh(x)\ne 0 \} = \emptyset if and only if\exists f \in C,g \in I,n \in N f + g + \left(\prod H\right) ^{2n} = 0 .

(Unlike, 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

R = R . If the semialgebraic set W = \{x\in R ^{n\mid\forall} f\in F,f(x)\ge 0 \} is compact, then each polynomial p \in R [X _{1,} ... , X _{n]} that is strictly positive on W can be written as a polynomial in the defining functions of W with sums-of-squares coefficients, i.e. p \in P(F, \emptyset) . Here is said to be strictly positive on W if p(x)> 0 for all x \in W .^[1] Note that Schmüdgen's Positivstellensatz is stated for R = R and does not hold for arbitrary real closed fields.^[2]

Putinar's Positivstellensatz

Define the quadratic module associated with as the set

Q(F,G) = \left\{ \sigma _{0} + \sum ^{m} _{j =1} \sigma _{j} f _{j} + \sum ^{r} _{\ell =1} \varphi _{\ell} g _{\ell} : \sigma _{j} \in \Sigma ^{2} [X _{1,\ldots,X} _{n];} \varphi _{\ell} \in R [X _{1,\ldots,X} _{n]} \right\}

Assume there exists L > 0 such that the polynomial

L - \sum ^{n} _{i =1} x ^{2} _{i} \in Q(F,G). If p(x)> 0 for all x \in W , then ∈ ().^[3]

See also

• Positive polynomial for other positivstellensatz theorems.
• Real Nullstellensatz

Notes

1. Schmüdgen. Konrad.

   de:Konrad Schmüdgen

   . 1991. The K-moment problem for compact semi-algebraic sets. Mathematische Annalen. 289. 1. 203–206. 10.1007/bf01446568. 0025-5831.
2. Stengle. Gilbert. 1996. Complexity Estimates for the Schmüdgen Positivstellensatz. Journal of Complexity. 12. 2. 167–174. 10.1006/jcom.1996.0011. free.
3. Putinar. Mihai. 1993. Positive Polynomials on Compact Semi-Algebraic Sets. Indiana University Mathematics Journal. 42. 3. 969–984. 10.1512/iumj.1993.42.42045. free.

References

• J. L. . Krivine . Anneaux préordonnés. Journal d'Analyse Mathématique. 12. 1964. 307–326 . 10.1007/bf02807438. 189771756 .
• G. . Stengle . A Nullstellensatz and a Positivstellensatz in Semialgebraic Geometry . . 207 . 1974 . 2 . 87–97 . 10.1007/BF01362149 . 122939347 .
• Book: J. . Bochnak . M. . Coste . M.-F. . Roy . Real algebraic geometry . Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge . 36 . Springer-Verlag . New York . 1999 . 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. 10.1007/s10957-014-0545-3. 0022-3239. 10.1.1.771.2203. 254745314 .