Positive polynomial
Encyclopedia
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[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
      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, ..., 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
    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,...,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.
    • Polya'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 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.
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