Stengle's Positivstellensatz

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}=emptysetif 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.


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