In mathematics, a positive polynomial on a particular set is a polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

 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

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

 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.)

Examples

• 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
    Hilbert's seventeenth problem
    Hilbert's seventeenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails expression of definite rational functions as quotients of sums of squares...
    
     and Artin's solution)
  • Suppose that p ∈ ℝ[X₁, ..., X_n] is homogeneous of even degree. If it is positive on ℝⁿ \ {0}, then there exists an integer m such that (X₁² + ... + X_n²)^m p is a sum of squares of elements from ℝ[X₁, ..., X_n].

• Polynomials positive on polytopes
  Polytope
  In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
  
  .
  • For polynomials of degree ≤ 1 we have the following variant of Farkas lemma: If f,g₁,...,g_k have degree ≤ 1 and f(x) ≥ 0 for every x ∈ ℝⁿ satisfying g₁(x) ≥ 0,...,g_k(x) ≥ 0, then there exist non-negative real numbers c₀,c₁,...,c_k such that f=c₀+c₁g₁+...+c_kg_k.
  • Polya's theorem: If p ∈ ℝ[X₁, ..., X_n] is homogeneous and p is positive on the set {x ∈ ℝⁿ | x₁ ≥ 0,...,x_n ≥ 0,x₁+...+x_n ≠ 0}, then there exists an integer m such that (x₁+...+x_n)^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 {g_i}.

• Polynomials positive on semialgebraic set
  Semialgebraic set
  In mathematics, a semialgebraic set is a subset S of Rn for some real closed field R defined by a finite sequence of polynomial equations and inequalities , or any finite union of such sets. A semialgebraic function is a function with semialgebraic graph...
  
  s.
  • The most general result is Stengle's Positivstellensatz
    Stengle's Positivstellensatz
    In real semialgebraic geometry, Stengle's Positivstellensatz characterizes polynomials that are positive on a semialgebraic set, which is defined by systems of inequalities of polynomials with real coefficients, or more generally, coefficients from any real-closed field.It can be thought of as an...
    
    .
  • 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

Similar results exist for trigonometric polynomials, matrix polynomials, polynomials in free variables, various quantum polynomials, etc.