Positivstellensatz

Remarks

The result is due to Krivine (1964) and Stengle (1974). The "if" directions are direct: for instance, if $pg = 1 + q$ and $x \in K$, then $p(x) \ge 0$ and $q(x) \ge 0$, so $p(x) g(x) \ge 1$ and $g(x) > 0$. The "only if" directions are the substance. They follow from the theory of orderings of rings together with the Tarski–Seidenberg transfer principle. That principle says that a first-order statement about polynomials holds over $\R$ if and only if it holds over every real closed field. Example: $K = \set{x \in \R : -x^2 - 1 \ge 0}$ is empty, and the certificate is $-1 = (-x^2 - 1) + x^2 \in T$. The theorem needs the real field: over $\C$ sign conditions make no sense, and its analogue for equations is Hilbert's Nullstellensatz. The denominators $p$ cannot be dropped in general, which is why stronger forms, such as Schmüdgen's theorem that $g \in T$ when $K$ is compact and $g > 0$ on $K$, need extra hypotheses.

Going deeper

For a student

The theorem is the real counterpart of Hilbert's Nullstellensatz, with sums of squares playing the role of the ideal. It contains Artin's solution of Hilbert's seventeenth problem as the case $m = 0$ of part 3. Bounded-degree versions of these certificates can be searched for by semidefinite programming, which is the basis of sum-of-squares methods in polynomial optimization.