Positive polynomial

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

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.
• * 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
• * Suppose that p ∈ ℝ is homogeneous of even degree. If it is positive on ℝⁿ \ , then there exists an integer m such that ^m p is a sum of squares of elements from ℝ.
• Polynomials positive on polytopes.
• * For polynomials of degree ≤ 1 we have the following variant of Farkas lemma: If f, g₁,..., g_k have degree ≤ 1 and f ≥ 0 for every x ∈ ℝⁿ satisfying g₁ ≥ 0,..., g_k ≥ 0, then there exist non-negative real numbers c₀, c₁,..., c_k such that f = c₀ + c₁g₁ + ... + c_kg_k.
• * Pólya's theorem: If p ∈ ℝ is homogeneous and p is positive on the set, then there exists an integer m such that ^m p has non-negative coefficients.
• * Handelman's theorem: If K is a compact polytope in Euclidean d-space, defined by linear inequalities g_i ≥ 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.
• Polynomials positive on semialgebraic sets.
• * The most general result is Stengle's Positivstellensatz.
• * For compact semialgebraic sets we have Schmüdgen's positivstellensatz, Putinar's positivstellensatz and Vasilescu's positivstellensatz. The point here is that no denominators are needed.
• * For nice compact semialgebraic sets of low dimension, there exists a nichtnegativstellensatz without denominators.

  Generalizations of positivstellensatz