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 ∈ 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, ..., 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].[3]
  • Polynomials positive on polytopes.
    • 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.
    • Pólya's theorem:[4] 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:[5] 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 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

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  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.
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