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

1. T. S. Motzkin, The arithmetic-geometric inequality. 1967 Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965) pp. 205–224.
2. E. Artin, Uber die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Sem. Univ. Hamburg, 5 (1927), 85-99.
3. B. Reznick, Uniform denominators in Hilbert's seventeenth problem. Math. Z. 220 (1995), no. 1, 75–97.
4. G. Pólya, Über positive Darstellung von Polynomen Vierteljschr, Naturforsch. Ges. Zürich 73 (1928) 141--145, in: R.P. Boas (Ed.), Collected Papers Vol. 2, MIT Press, Cambridge, MA, 1974, pp. 309--313
5. D. Handelman, Representing polynomials by positive linear functions on compact convex polyhedra. Pacific J. Math. 132 (1988), no. 1, 35--62.
6. K. Schmüdgen, The K-moment problem for compact semi-algebraic sets. Math. Ann. 289 (1991), no. 2, 203–206.
7. T. Wörmann Strikt Positive Polynome in der Semialgebraischen Geometrie, Univ. Dortmund 1998.
8. M. Putinar, Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42 (1993), no. 3, 969–984.
9. T. Jacobi, A representation theorem for certain partially ordered commutative rings. Math. Z. 237 (2001), no. 2, 259–273.
10. Vasilescu, F.-H. Spectral measures and moment problems. Spectral analysis and its applications, 173--215, Theta Ser. Adv. Math., 2, Theta, Bucharest, 2003. See Theorem 1.3.1.
11. C. Scheiderer, Sums of squares of regular functions on real algebraic varieties. Trans. Amer. Math. Soc. 352 (2000), no. 3, 1039–1069.
12. C. Scheiderer, Sums of squares on real algebraic curves. Math. Z. 245 (2003), no. 4, 725–760.
13. C. Scheiderer, Sums of squares on real algebraic surfaces. Manuscripta Math. 119 (2006), no. 4, 395–410.