Krivine–Stengle Positivstellensatz

Template:Short description In real algebraic geometry, Krivine–Stengle Template:Lang (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 Template:Ill and then rediscovered by the Canadian Template:Ill.

Statement

Template:Confusing section Let Template:Var be a real closed field, and Template:Var = {f₁, f₂, ..., f_m} and Template:Var = {g₁, g₂, ..., g_r} finite sets of polynomials over Template:Var in Template:Var variables. Let Template:Var be the semialgebraic set

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

and define the preorder (in the sense of a prepositive cone) associated with Template:Var as the set

$P(F,G)=\{\sum _{\alpha \in \{0,1{\}}^m}{\sigma }_{\alpha }{f}_1^{{\alpha }_1}\cdots f_m^{{\alpha }_m}+\sum _{\ell =1}^r{\phi }_{\ell }{g}_{\ell }:{\sigma }_{\alpha }\in {\mathrm{\Sigma }}^2[X_1,\dots ,X_n];{\phi }_{\ell }\in R[X_1,\dots ,X_n]\}$

where Σ²[[[:Template:Var]]₁,...,Template:Var_Template:Var] is the set of sum-of-squares polynomials. In other words, Template:Var(Template:Var, Template:Var) = Template:Varserif + Template:Varserif, where Template:Varserif is the cone generated by Template:Var (i.e., the subsemiring of Template:Var[[[:Template:Var]]₁,...,Template:Var_Template:Var] generated by Template:Var and arbitrary squares) and Template:Varserif is the ideal generated by Template:Var.

Let Template:Var ∈ Template:Var[[[:Template:Var]]₁,...,Template:Var_Template:Var] be a polynomial. Krivine–Stengle Positivstellensatz states that

(i) $\mathrm{\forall }x\in Wp(x)\ge 0$ if and only if $\mathrm{\exists }{q}_1,q_2\in P(F,G)$ and $s\in \mathbb{Z}$ such that $q_{1}p=p^{2s}+q_2$.

(ii) $\mathrm{\forall }x\in Wp(x)>0$ if and only if $\mathrm{\exists }{q}_1,q_2\in P(F,G)$ such that $q_{1}p=1+q_2$.

The weak Template:Lang is the following variant of the Template:Lang. Let Template:Var be a real closed field, and Template:Var, Template:Var, and Template:Var finite subsets of Template:Var[[[:Template:Var]]₁,...,Template:Var_Template:Var]. Let Template:Var be the cone generated by Template:Var, and Template:Varserif the ideal generated by Template:Var. Then

$\{x\in R^n\mid \mathrm{\forall }f\in Ff(x)\ge 0\wedge \mathrm{\forall }g\in Gg(x)=0\wedge \mathrm{\forall }h\in Hh(x)\ne 0\}=\mathrm{\varnothing }$

if and only if

$\mathrm{\exists }f\in C,g\in I,n\in \mathbb{N}f+g+{(\prod H)}^{2n}=0.$

(Unlike Template:Lang, 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 $W=\{x\in {ℝ}^n\mid \mathrm{\forall }f\in F,f(x)\ge 0\}$ is compact, then each polynomial $p\in ℝ[X_1,\dots ,X_n]$ 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. $p\in P(F,\mathrm{\varnothing })$. Here Template:Var is said to be strictly positive on $W$ if $p(x)>0$ for all $x\in W$. Note that Schmüdgen's Positivstellensatz is stated for $R=ℝ$ and does not hold for arbitrary real closed fields.

Putinar's Positivstellensatz

Define the quadratic module associated with Template:Var as the set

$Q(F,G)=\{{\sigma }_0+\sum _{j=1}^m{\sigma }_j{f}_j+\sum _{\ell =1}^r{\phi }_{\ell }{g}_{\ell }:{\sigma }_j\in {\mathrm{\Sigma }}^2[X_1,\dots ,X_n];{\phi }_{\ell }\in ℝ[X_1,\dots ,X_n]\}$

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

See also

• Positive polynomial for other positivstellensatz theorems.
• Real Nullstellensatz

References

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