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 be a polynomial in variables with real coefficients and let be a subset of the -dimensional Euclidean space . We say that:

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Positivstellensatz and nichtnegativstellensatz

For certain sets , there exist algebraic descriptions of all polynomials that are positive (resp. non-negative) on . 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

Positive polynomials on Euclidean space

Every real polynomial in one variable is non-negative on 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 is non-negative on but is not a sum of squares of elements from . (Motzkin showed that it was positive using the AM–GM inequality.) [3]

In higher dimensions, a real polynomial in variables is non-negative on if and only if it is a sum of squares of real rational functions in variables (see Hilbert's seventeenth problem and Artin's solution [4] ).

In the case of homogeneous polynomials, more information can be determined about the denominator. Suppose that is homogeneous of even degree. If it is positive on , then there exists an integer such that is a sum of squares of elements from . [5]

Positive polynomials on polytopes

For polynomials of degree we have the following variant of Farkas lemma: If have degree and for every satisfying , then there exist non-negative real numbers such that .

For higher degree polynomials on the simplex, we have Pólya's theorem: [6] If is homogeneous and is positive on the set , then there exists an integer such that has non-negative coefficients.

For higher degree polynomials on general compact polytopes, we have Handelman's theorem: [7] If is a compact polytope in Euclidean -space, defined by linear inequalities , and if is a polynomial in variables that is positive on , then can be expressed as a linear combination with non-negative coefficients of products of members of .

Positive polynomials on semialgebraic sets

For general semialgebraic sets, the most general result is Stengle's Positivstellensatz.

For compact semialgebraic sets we have Schmüdgen's positivstellensatz, [8] [9] Putinar's positivstellensatz [10] [11] and Vasilescu's positivstellensatz. [12] In particular, no denominators are needed.

For sufficiently nice compact semialgebraic sets of low dimension, there exists a nichtnegativstellensatz without denominators. [13] [14] [15]

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]

See also

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.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: others (link)
  2. Benoist, Olivier (2017). "Writing Positive Polynomials as Sums of (Few) Squares". 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". SIAM Journal on Applied Algebra and Geometry. 6 (4): 650–684. arXiv: 2107.00345 . doi:10.1137/21M1462568. ISSN   2470-6566. S2CID   235694320.
  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. S2CID   38131072.
  18. Cimprič, J. (2011). "Strict positivstellensätze for matrix polynomials with scalar constraints". Linear Algebra and Its Applications. 434 (8): 1879–1883. arXiv: 1011.4930 . doi: 10.1016/j.laa.2010.11.046 . S2CID   119169153.
  19. Helton, J. William; Klep, Igor; McCullough, Scott (2012). "The convex Positivstellensatz in a free algebra". Advances in Mathematics . 231 (1): 516–534. arXiv: 1102.4859 . doi: 10.1016/j.aim.2012.04.028 .
  20. Klep, Igor (2004-12-31). "The Noncommutative Graded Positivstellensatz" . Communications in Algebra. 32 (5): 2029–2040. doi:10.1081/AGB-120029921. ISSN   0092-7872. S2CID   120795025.
  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. S2CID   122451112.

Further reading