Positive polynomial

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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 in S.
• p is non-negative on S if p(x) ≥ 0 for every x in S.
• p is zero on S if p(x) = 0 for every x in S.

For certain sets S, there exist algebraic descriptions of all polynomials that are positive, non-negative, or zero on S. Such a description is a positivstellensatz, nichtnegativstellensatz, or nullstellensatz. This article will focus on the former two descriptions. For the latter, see Hilbert's Nullstellensatz for the most known nullstellensatz.

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.^[1] This equivalence does not generalize 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[X, Y].^[2]
  • 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^[3]).
  • 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].^[4]
• 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:^[5] 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:^[6] 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,^[7][8] Putinar's positivstellensatz^[9][10] and Vasilescu's positivstellensatz.^[11] The point here is that no denominators are needed.
  • For nice compact semialgebraic sets of low dimension, there exists a nichtnegativstellensatz without denominators.^[12][13][14]

Generalizations of positivstellensatz[]

Positivstellensatz also exist for trigonometric polynomials, matrix polynomials, polynomials in free variables, various quantum polynomials, etc.^{[citation needed]}

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[]

See also[]

• Polynomial SOS
• Hilbert's seventeenth problem