Positive polynomial

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In mathematics, a positive polynomial (respectively non-negative polynomial) on a particular set is a polynomial whose values are positive (respectively non-negative) on that set. Precisely, 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.

Positivstellensatz (and nichtnegativstellensatz)

For certain sets S, there exist algebraic descriptions of all polynomials that are positive (resp. non-negative) on S. Such a description is a positivstellensatz (resp. nichtnegativstellensatz). The importance of Positivstellensatz theorems in computation arises from its ability to transform problems of polynomial optimization into semidefinite programming problems, which can be efficiently solved using convex optimization techniques.[1]

Examples of positivstellensatz (and nichtnegativstellensatz)

  • Globally positive polynomials and sum of squares decomposition.
    • 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.[2] This equivalence does not generalize for polynomial with more than one variable: for instance, the Motzkin polynomial X 4Y 2 + X 2Y 4 − 3X 2Y 2 + 1 is non-negative on ℝ2 but is not a sum of squares of elements from ℝ[X, Y].[3]
    • 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[4]).
    • 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].[5]
  • 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:[6] 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:[7] 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.

Generalizations of positivstellensatz

Positivstellensatz also exist for signomials,[16] trigonometric polynomials,[17] polynomial matrices,[18] polynomials in free variables,[19] quantum polynomials,[20] and definable functions on o-minimal structures.[21]

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. ↑ Semidefinite optimization and convex algebraic geometry. Grigoriy Blekherman, Pablo A. Parrilo, Rekha R. Thomas. Philadelphia. 2013. ISBN 978-1-61197-228-3. OCLC 809420808. https://www.worldcat.org/oclc/809420808. 
  2. ↑ Benoist, Olivier (2017). "Writing Positive Polynomials as Sums of (Few) Squares" (in en). EMS Newsletter 2017-9 (105): 8–13. doi:10.4171/NEWS/105/4. ISSN 1027-488X. 
  3. ↑ T. S. Motzkin, The arithmetic-geometric inequality. 1967 Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965) pp. 205–224.
  4. ↑ E. Artin, Uber die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Sem. Univ. Hamburg, 5 (1927), 85–99.
  5. ↑ B. Reznick, Uniform denominators in Hilbert's seventeenth problem. Math. Z. 220 (1995), no. 1, 75–97.
  6. ↑ 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.
  7. ↑ D. Handelman, Representing polynomials by positive linear functions on compact convex polyhedra. Pacific J. Math. 132 (1988), no. 1, 35–62.
  8. ↑ K. Schmüdgen. "The K-moment problem for compact semi-algebraic sets". Math. Ann. 289 (1991), no. 2, 203–206.
  9. ↑ T. Wörmann. "Strikt Positive Polynome in der Semialgebraischen Geometrie", Univ. Dortmund 1998.
  10. ↑ M. Putinar, "Positive polynomials on compact semi-algebraic sets". Indiana Univ. Math. J. 42 (1993), no. 3, 969–984.
  11. ↑ T. Jacobi, "A representation theorem for certain partially ordered commutative rings". Math. Z. 237 (2001), no. 2, 259–273.
  12. ↑ 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.
  13. ↑ C. Scheiderer, "Sums of squares of regular functions on real algebraic varieties". Trans. Amer. Math. Soc. 352 (2000), no. 3, 1039–1069.
  14. ↑ C. Scheiderer, "Sums of squares on real algebraic curves". Math. Z. 245 (2003), no. 4, 725–760.
  15. ↑ C. Scheiderer, "Sums of squares on real algebraic surfaces". Manuscripta Math. 119 (2006), no. 4, 395–410.
  16. ↑ Dressler, Mareike; Murray, Riley (2022-12-31). "Algebraic Perspectives on Signomial Optimization" (in en). SIAM Journal on Applied Algebra and Geometry 6 (4): 650–684. doi:10.1137/21M1462568. ISSN 2470-6566. https://epubs.siam.org/doi/10.1137/21M1462568. 
  17. ↑ Dumitrescu, Bogdan (2007). "Positivstellensatz for Trigonometric Polynomials and Multidimensional Stability Tests". IEEE Transactions on Circuits and Systems II: Express Briefs 54 (4): 353–356. doi:10.1109/TCSII.2006.890409. ISSN 1558-3791. https://ieeexplore.ieee.org/document/4155059. 
  18. ↑ Cimprič, J. (2011). "Strict positivstellensätze for matrix polynomials with scalar constraints" (in en). Linear Algebra and Its Applications 434 (8): 1879–1883. doi:10.1016/j.laa.2010.11.046. 
  19. ↑ Helton, J. William; Klep, Igor; McCullough, Scott (2012). "The convex Positivstellensatz in a free algebra" (in en). Advances in Mathematics 231 (1): 516–534. doi:10.1016/j.aim.2012.04.028. 
  20. ↑ Klep, Igor (2004-12-31). "The Noncommutative Graded Positivstellensatz" (in en). Communications in Algebra 32 (5): 2029–2040. doi:10.1081/AGB-120029921. ISSN 0092-7872. http://www.tandfonline.com/doi/abs/10.1081/AGB-120029921. 
  21. ↑ Acquistapace, F.; Andradas, C.; Broglia, F. (2002-07-01). "The Positivstellensatz for definable functions on o-minimal structures". Illinois Journal of Mathematics 46 (3). doi:10.1215/ijm/1258130979. ISSN 0019-2082. 

See also



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