Suppose I have a polynomial $p\in R[x\_1,\ldots,x\_n]$ and I look at the set $S:=\{ x\in R^n : p(x)\geq 0\}$. Are there algebraic certificates on $p$ that will certify that $S$ is compact? 



Compactness of a semi algebraic set
Let $p \in \mathbb{R}[x\_1, \ldots, x\_n]$. The set $S=\{x \in \mathbb{R}^n: p(x) \geq 0\}$ is compact if and only if there is a natural number $N$ and polynomials $g\_i, h\_i \in \mathbb{R}[x\_1, \ldots, x\_n]$, such that $N-\sum\_{i=1}^n x\_i^2 = \sum\_{i=1}^rg\_i^2+p \sum\_{i=1}^s h\_i^2$.
It is easy to see that this criterion is sufficient. Schmüdgen's Positivstellensatz says that it is also necessary.


