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## From Wikipedia, the free encyclopedia

In mathematics, a positive polynomial on a particular set is a polynomial whose values are positive on that set.

Let p be a polynomial in n variables with real coefficients and let S be a subset of the n-dimensional Euclidean spacen. We say that:

• p is positive on S if p(x) > 0 for every x ∈ S.
• p is non-negative on S if p(x) ≥ 0 for every x ∈ S.
• p is zero on S if p(x) = 0 for every x ∈ S.

For certain sets S, there exist algebraic descriptions of all polynomials that are positive, non-negative, or zero on S. Such a description is a positivstellensatz, nichtnegativstellensatz, or nullstellensatz. This article will focus on the former two descriptions. For the latter, see Hilbert's Nullstellensatz for the most known nullstellensatz.

## Examples of positivstellensatz (and nichtnegativstellensatz)

• Globally positive polynomials and sum of squares decomposition.
• Every real polynomial in one variable and with even degree is non-negative on ℝ if and only if it is a sum of two squares of real polynomials in one variable. This equivalence does not generalizes for polynomial with more than one variable: for instance, the Motzkin polynomial X4Y2 + X2Y4 − 3X2Y2 + 1 is non-negative on ℝ2 but is not a sum of squares of elements from ℝ[XY].
• A real polynomial in n variables is non-negative on ℝn if and only if it is a sum of squares of real rational functions in n variables (see Hilbert's seventeenth problem and Artin's solution)
• Suppose that p ∈ ℝ[X1, ..., Xn] is homogeneous of even degree. If it is positive on ℝn \ {0}, then there exists an integer m such that (X12 + ... + Xn2)m p is a sum of squares of elements from ℝ[X1, ..., Xn].
• Polynomials positive on polytopes.
• For polynomials of degree ≤ 1 we have the following variant of Farkas lemma: If f, g1, ..., gk have degree ≤ 1 and f(x) ≥ 0 for every x ∈ ℝn satisfying g1(x) ≥ 0, ..., gk(x) ≥ 0, then there exist non-negative real numbers c0, c1, ..., ck such that f = c0 + c1g1 + ... + ckgk.
• Pólya's theorem: If p ∈ ℝ[X1, ..., Xn] is homogeneous and p is positive on the set {x ∈ ℝn | x1 ≥ 0, ..., xn ≥ 0, x1 + ... + xn ≠ 0}, then there exists an integer m such that (x1 + ... + xn)m p has non-negative coefficients.
• Handelman's theorem: If K is a compact polytope in Euclidean d-space, defined by linear inequalities gi ≥ 0, and if f is a polynomial in d variables that is positive on K, then f can be expressed as a linear combination with non-negative coefficients of products of members of {gi}.
• Polynomials positive on semialgebraic sets.

## Generalizations of positivstellensatz

Positivstellensatz also exist for trigonometric polynomials, matrix polynomials, polynomials in free variables, various quantum polynomials, etc.[citation needed]

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