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 space ℝn. 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 (resp. non-negative, zero) on S. Any such description is called a positivstellensatz (resp. nichtnegativstellensatz, nullstellensatz.)
Globally positive polynomials
Every real polynomial in one variable is non-negative on ℝ if and only if it is a sum of two squares of real polynomials in one variable.
The Motzkin polynomial X4Y2 + X2Y4 − 3X2Y2 + 1 is non-negative on ℝ2 but is not a sum of squares of elements from ℝ[X, Y].
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.
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.
Similar results exist for trigonometric polynomials, matrix polynomials, polynomials in free variables, various quantum polynomials, etc.
