Stengle's Positivstellensatz

In mathematics, Stengle's Positivstellensatz characterizes polynomials which are positive on a given semialgebraic set over the real numbers, or more generally, any real-closed field. It can be thought of as an ordered analogue of Hilbert's Nullstellensatz. It was discovered by Gilbert Stengle.

tatement

Let "R" be a real-closed field, and "F" a finite set of polynomials over "R" in "n" variables. Let "W" be the semialgebraic set:$W=\{xin\; R^nmidforall\; fin\; F,f(x)ge0\},$and let "C" be the cone generated by "F" (i.e., the subsemiring of "R" ["X"₁,…,"X"_"n"] generated by "F" and arbitrary squares). Let "p" ∈ "R" ["X"₁,…,"X"_"n"] be a polynomial. Then:$forall\; xin\; W;p(x)>0$ if and only if $exists\; f\_1,f\_2in\; C;pf\_1=1+f\_2$.

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" ["X"₁,…,"X"_"n"] . Let "C" be the cone generated by "F", and "I" the ideal generated by "G". Then:$\{xin\; R^nmidforall\; fin\; F,f(x)ge0landforall\; gin\; G,g(x)=0landforall\; hin\; H,h(x)\; e0\}=emptyset$if and only if:$exists\; fin\; C,gin\; I,ninmathbb\; N;f+g+left(prod\; H\; ight)^\{2n\}=0.$

(Unlike Nullstellensatz, the "weak" form actually includes the "strong" form as a special case, so the terminology is a misnomer.)

References

*G. Stengle, "A Nullstellensatz and a Positivstellensatz in Semialgebraic Geometry", Mathematische Annalen 207 (1973), no. 2, pp. 87–97.
*J. Bochnak, M. Coste, M.-F. Roy, "Real algebraic geometry", Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 36, Springer-Verlag, 1999.