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Krivine–Stengle Positivstellensatz

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Theorem of real algebraic geometry

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

This section may be confusing or unclear to readers. In particular, firstly, a set of polinomial is not a preordering; secondly, such a huge formula requires an explanation. Please help clarify the section. There might be a discussion about this on the talk page. (January 2024) (Learn how and when to remove this message)

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

W = { x ∈ R n ∣ ∀ f ∈ F , f ( x ) ≥ 0 ; ∀ g ∈ G , g ( x ) = 0 } , {\displaystyle W=\{x\in R^{n}\mid \forall f\in F,\,f(x)\geq 0;\,\forall g\in G,\,g(x)=0\},}

and define the preordering associated with W as the set

P ( F , G ) = { ∑ α ∈ { 0 , 1 } m σ α f 1 α 1 ⋯ f m α m + ∑ ℓ = 1 r φ ℓ g ℓ : σ α ∈ Σ 2 [ X 1 , … , X n ] ;   φ ℓ ∈ R [ X 1 , … , X n ] } {\displaystyle P(F,G)=\left\{\sum _{\alpha \in \{0,1\}^{m}}\sigma _{\alpha }f_{1}^{\alpha _{1}}\cdots f_{m}^{\alpha _{m}}+\sum _{\ell =1}^{r}\varphi _{\ell }g_{\ell }:\sigma _{\alpha }\in \Sigma ^{2};\ \varphi _{\ell }\in R\right\}}

where Σ 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 generated by F and arbitrary squares) and I is the ideal generated by G.

Let p ∈ R be a polynomial. Krivine–Stengle Positivstellensatz states that

(i) ∀ x ∈ W p ( x ) ≥ 0 {\displaystyle \forall x\in W\;p(x)\geq 0} if and only if ∃ q 1 , q 2 ∈ P ( F , G ) {\displaystyle \exists q_{1},q_{2}\in P(F,G)} and s ∈ Z {\displaystyle s\in \mathbb {Z} } such that q 1 p = p 2 s + q 2 {\displaystyle q_{1}p=p^{2s}+q_{2}} .
(ii) ∀ x ∈ W p ( x ) > 0 {\displaystyle \forall x\in W\;p(x)>0} if and only if ∃ q 1 , q 2 ∈ P ( F , G ) {\displaystyle \exists q_{1},q_{2}\in P(F,G)} such that q 1 p = 1 + q 2 {\displaystyle q_{1}p=1+q_{2}} .

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. Let C be the cone generated by F, and I the ideal generated by G. Then

{ x ∈ R n ∣ ∀ f ∈ F f ( x ) ≥ 0 ∧ ∀ g ∈ G g ( x ) = 0 ∧ ∀ h ∈ H h ( x ) ≠ 0 } = ∅ {\displaystyle \{x\in R^{n}\mid \forall f\in F\,f(x)\geq 0\land \forall g\in G\,g(x)=0\land \forall h\in H\,h(x)\neq 0\}=\emptyset }

if and only if

∃ f ∈ C , g ∈ I , n ∈ N f + g + ( ∏ H ) 2 n = 0. {\displaystyle \exists f\in C,g\in I,n\in \mathbb {N} \;f+g+\left(\prod H\right)^{\!2n}=0.}

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

Putinar's Positivstellensatz

Define the quadratic module associated with W as the set

Q ( F , G ) = { σ 0 + ∑ j = 1 m σ j f j + ∑ ℓ = 1 r φ ℓ g ℓ : σ j ∈ Σ 2 [ X 1 , … , X n ] ;   φ ℓ ∈ R [ X 1 , … , X n ] } {\displaystyle Q(F,G)=\left\{\sigma _{0}+\sum _{j=1}^{m}\sigma _{j}f_{j}+\sum _{\ell =1}^{r}\varphi _{\ell }g_{\ell }:\sigma _{j}\in \Sigma ^{2};\ \varphi _{\ell }\in \mathbb {R} \right\}}

Assume there exists L > 0 such that the polynomial L − ∑ i = 1 n x i 2 ∈ Q ( F , G ) . {\displaystyle L-\sum _{i=1}^{n}x_{i}^{2}\in Q(F,G).} If p ( x ) > 0 {\displaystyle p(x)>0} for all x ∈ W {\displaystyle x\in W} , then p ∈ Q(F,G).

See also

Notes

  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

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