Positive polynomial

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.)

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].^[1]
  • 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^[2])
  • 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].^[3]
• 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(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.
  • Pólya's theorem:^[4] 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:^[5] 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 sets.
  • The most general result is Stengle's Positivstellensatz.
  • For compact semialgebraic sets we have Schmüdgen's positivstellensatz,^[6][7] Putinar's positivstellensatz^[8][9] and Vasilescu's positivstellensatz.^[10] The point here is that no denominators are needed.
  • For nice compact semialgebraic sets of low dimension there exists a nichtnegativstellensatz without denominators.^[11][12][13]

Generalizations

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

References

• Bochnak, Jacek; Coste, Michel; Roy, Marie-Françoise. Real Algebraic Geometry. Translated from the 1987 French original. Revised by the authors. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 36. Springer-Verlag, Berlin, 1998. x+430 pp. ISBN 3-540-64663-9
• Marshall, Murray Positive polynomials and sums of squares. Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008. xii+187 pp. ISBN 978-0-8218-4402-1, ISBN 0-8218-4402-4

Notes