Positive polynomial

This article is about positive polynomials and positivstellensatz-like theorems. For the Krivine–Stengle Positivstellensatz, see Krivine–Stengle Positivstellensatz.

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 ${\displaystyle p}$ be a polynomial in ${\displaystyle n}$ variables with real coefficients and let ${\displaystyle S}$ be a subset of the ${\displaystyle n}$-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$. We say that:

• ${\displaystyle p}$ is positive on ${\displaystyle S}$ if ${\displaystyle p(x)>0}$ for every ${\displaystyle x}$ in ${\displaystyle S}$.
• ${\displaystyle p}$ is non-negative on ${\displaystyle S}$ if ${\displaystyle p(x)\geq 0}$ for every ${\displaystyle x}$ in ${\displaystyle S}$.

Positivstellensatz and nichtnegativstellensatz

For certain sets ${\displaystyle S}$, there exist algebraic descriptions of all polynomials that are positive (resp. non-negative) on ${\displaystyle S}$. 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

Positive polynomials on Euclidean space

A real univariate polynomial is non-negative on ${\displaystyle \mathbb {R} }$ if and only if it is a sum of two squares of real univariate polynomials. ^[2] This equivalence does not generalize to multivariate polynomials, which was originally shown by Hilbert. An explicit example of such a polynomial was not known until Theodore Motzkin showed in 1967 that ${\displaystyle X^{4}Y^{2}+X^{2}Y^{4}-3X^{2}Y^{2}+1}$ is not a sum of squares of polynomials but is non-negative on ${\displaystyle \mathbb {R} ^{2}}$, which follows from the AM-GM inequality. ^[3]

In higher dimensions, a real polynomial in ${\displaystyle n}$ variables is non-negative on ${\displaystyle \mathbb {R} ^{n}}$ if and only if it is a sum of squares of real rational functions in ${\displaystyle n}$ variables. This was originally posed as Hilbert's seventeenth problem in 1900, and later solved by Emil Artin in 1927. ^[4]

For homogeneous polynomials, more information can be determined about the denominator. Suppose that ${\displaystyle p\in \mathbb {R} [X_{1},\dots ,X_{n}]}$ is homogeneous of degree 2k. If it is positive on ${\displaystyle \mathbb {R} ^{n}\setminus \{0\}}$, then there exists an integer ${\displaystyle m}$ such that ${\displaystyle (X_{1}^{2}+\cdots +X_{n}^{2})^{m}p}$ is a sum of squares of homogeneous polynomials of degree ${\displaystyle m+2k}$. ^[5]

Positive polynomials on polytopes

For polynomials of degree${\displaystyle {}\leq 1}$ we have the following variant of Farkas lemma: If ${\displaystyle f,g_{1},\dots ,g_{k}}$ have degree${\displaystyle {}\leq 1}$ and ${\displaystyle f(x)\geq 0}$ for every ${\displaystyle x\in \mathbb {R} ^{n}}$ satisfying ${\displaystyle g_{1}(x)\geq 0,\dots ,g_{k}(x)\geq 0}$, then there exist non-negative real numbers ${\displaystyle c_{0},c_{1},\dots ,c_{k}}$ such that ${\displaystyle f=c_{0}+c_{1}g_{1}+\cdots +c_{k}g_{k}}$.

For higher degree polynomials on the simplex, Pólya showed that if ${\displaystyle p\in \mathbb {R} [X_{1},\dots ,X_{n}]}$ is homogeneous and positive on the set ${\displaystyle \{x\in \mathbb {R} ^{n}\mid x_{1}\geq 0,\dots ,x_{n}\geq 0,x_{1}+\cdots +x_{n}\neq 0\}}$, then there exists an integer ${\displaystyle m}$ such that ${\displaystyle (x_{1}+\cdots +x_{n})^{m}p}$ has non-negative coefficients. ^[6]

For higher degree polynomials on general compact polytopes, we have Handelman's theorem: ^[7] If ${\displaystyle K}$ is a compact polytope in Euclidean ${\displaystyle d}$-space, defined by linear inequalities ${\displaystyle g_{i}\geq 0}$, and if ${\displaystyle f}$ is a polynomial in ${\displaystyle d}$ variables that is positive on ${\displaystyle K}$, then ${\displaystyle f}$ can be expressed as a linear combination with non-negative coefficients of products of members of ${\displaystyle \{g_{i}\}}$.

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

• Polynomial SOS
• Positivstellensatz
• Sum-of-squares optimization
• Hilbert's seventeenth problem
• Hilbert's Nullstellensatz for algebraic descriptions of polynomials that are zero on a set S.

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.
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

• 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. ISBN   3-540-64663-9.
• Marshall, Murray. "Positive polynomials and sums of squares". Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008. ISBN   978-0-8218-4402-1, ISBN   0-8218-4402-4.