Let $R$ be a real closed field. An example of Positivstellensatz is that for a real polynomial $f\in R[x_1,\ldots, x_n]$, that is strictly positive on $R^n$, then there are sums of squares $s$ and $t$ such that $sf=1+t$. This was first proved by Stengle.

In the case where $R$ is the real number field, we can view this statement as a Positivstellensatz for the coordinate ring of the real algebraic variety $R^n$.

I'm wondering if there are Positivstellensatz for projective real algebraic varieties, eg $RP^n$. For example, Artin's solution of Hilbert's 17th problem tells us we can find for homogeneous $f$, homogeneous sums of squares $s,t$ such that $sf=t$. This is more like a Nichtnegativstellensatz. Can we modify Stengle into a Positivstellensatz for projective case?