---
tag: FN734RQYM6
kind: statement
role: theorem
name: Positivstellensatz
msc: [14P10, "12D15", 13J30]
uses: [BX5WSK8ZCT, 07GNX8GCYP, 08J2M1JYAG, D85VZWS9QK]
about: [BX5WSK8ZCT]
sources:
  - url: https://en.wikipedia.org/wiki/Positive_polynomial
---

## Statement

Let $f_1, \dots, f_m, g \in \R[x_1, \dots, x_n]$ be real [[polynomials|07GNX8GCYP]], and let
$$K = \set{x \in \R^n : f_1(x) \ge 0, \dots, f_m(x) \ge 0}$$
be the basic closed [[semialgebraic set|BX5WSK8ZCT]] they define.
Let $T$ be the preordering generated by $f_1, \dots, f_m$.
It is the set of all sums $\sum_{e \in \set{0,1}^m} \sigma_e f_1^{e_1} \cdots f_m^{e_m}$ in which each $\sigma_e$ is a sum of squares of polynomials.
Then:

1. $K = \emptyset$ if and only if $-1 \in T$.
2. $g > 0$ on $K$ if and only if there are $p, q \in T$ with $pg = 1 + q$.
3. $g \ge 0$ on $K$ if and only if there are $p, q \in T$ and an integer $k \ge 0$ with $pg = g^{2k} + q$.
4. $g = 0$ on $K$ if and only if there is an integer $k \ge 0$ with $-g^{2k} \in T$.

## 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|D85VZWS9QK]].
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.

## Explanations

### L1 human

How can one prove that a polynomial is positive on a region defined by polynomial inequalities?
The Positivstellensatz says that there is always an algebraic proof: an identity built from squares and from the defining inequalities.
Such an identity can be checked by expanding, without solving anything.

### L2 human

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|J45F4F13VB]] 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.