Positive polynomial

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 $\mathbb {R} ^{n}$ . We say that:

$p$ is positive on $S$ if $p(x)>0$ for every $x$ in $S$ .
$p$ is non-negative on $S$ if $p(x)\geq 0$ 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] In the Hermitian case, Putinar observed that the resulting semidefinite programming problems converge asymptotically and reduce to calculating the largest eigenvalues of explicitly given matrices, which can be solved more efficiently than general semidefinite programming problems.^[2]

Examples

Positive polynomials on Euclidean space

A real univariate polynomial is non-negative on $\mathbb {R}$ if and only if it is a sum of two squares of real univariate polynomials.^[3] 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 $X^{4}Y^{2}+X^{2}Y^{4}-3X^{2}Y^{2}+1$ is not a sum of squares of polynomials but is non-negative on $\mathbb {R} ^{2}$ , which follows from the AM-GM inequality.^[4]

In higher dimensions, a real polynomial in $n$ variables is non-negative on $\mathbb {R} ^{n}$ if and only if it is a sum of squares of real rational functions in $n$ variables. This was originally posed as Hilbert's seventeenth problem in 1900, and later solved by Emil Artin in 1927.^[5]

For homogeneous polynomials, more information can be determined about the denominator. Suppose that $p\in \mathbb {R} [X_{1},\dots ,X_{n}]$ is homogeneous of degree 2k. If it is positive on $\mathbb {R} ^{n}\setminus \{0\}$ , then there exists an integer $m$ such that $(X_{1}^{2}+\cdots +X_{n}^{2})^{m}p$ is a sum of squares of homogeneous polynomials of degree $m+2k$ .^[6]

Positive polynomials on polytopes

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

For higher degree polynomials on the simplex, Pólya showed that if $p\in \mathbb {R} [X_{1},\dots ,X_{n}]$ is homogeneous and positive on the set $\{x\in \mathbb {R} ^{n}\mid x_{1}\geq 0,\dots ,x_{n}\geq 0,x_{1}+\cdots +x_{n}\neq 0\}$ , then there exists an integer $m$ such that $(x_{1}+\cdots +x_{n})^{m}p$ has non-negative coefficients.^[7]

For higher degree polynomials on general compact polytopes, we have Handelman's theorem:^[8] If $K$ is a compact polytope in Euclidean $d$ -space, defined by linear inequalities $g_{i}\geq 0$ , 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 $\{g_{i}\}$ .

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,^[9]^[10] Putinar's positivstellensatz^[11]^[12] and Vasilescu's positivstellensatz.^[13] In particular, no denominators are needed.

For sufficiently nice compact semialgebraic sets of low dimension, there exists a nichtnegativstellensatz without denominators.^[14]^[15]^[16]

Positive Hermitian polynomials

A polynomial $p$ in complex variables $z_{1},\dots ,z_{n}$ and their conjugates $z_{1}^{*},\dots ,z_{n}^{*}$ is Hermitian if it takes on only real values for all choices of $z$ . It is a hermitian sum-of-squares (HSOS) if it can be written as $p=\sum _{i=1}^{k}g_{i}^{*}g_{i}$ for some polynomials $g_{1},\dots ,g_{k}$ in only the variables $z_{1},\dots ,z_{n}$ . A result due to Quillen states that any strictly positive, homogeneous Hermitian polynomial is a Hermitian sum-of-squares of rational functions whose denominator is the squared norm $z_{1}^{*}z_{1}+\dots +z_{n}^{*}z_{n}$ .^[17] This was later generalized by Putinar to a much larger class of spaces, including all complex projective varieties.^[2] In the Hermitian case the Hermitian sum-of-squares representation is unique if it exists and can be found by diagonalizing an explicitly given Hermitian matrix, which was first observed by Putinar.^[2]

Generalizations of positivstellensatz

Positivstellensatz also exist for signomials,^[18] trigonometric polynomials,^[19] polynomial matrices,^[20] polynomials in free variables,^[21] quantum polynomials,^[22] and definable functions on o-minimal structures.^[23]

Notes

^ 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)
^ ^a ^b ^c Putinar, Mihai (2012). "Chapter 9: Sums of Hermitian Squares: Old and New". Semidefinite Optimization and Convex Algebraic Geometry. Philadelphia, PA: Society for Industrial and Applied Mathematics. p. 407–446. doi:10.1137/1.9781611972290.ch9. ISBN 978-1-61197-228-3.
^ 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.
^ T. S. Motzkin, The arithmetic-geometric inequality. 1967 Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965) pp. 205–224.
^ E. Artin, Uber die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Sem. Univ. Hamburg, 5 (1927), 85–99.
^ B. Reznick, Uniform denominators in Hilbert's seventeenth problem. Math. Z. 220 (1995), no. 1, 75–97.
^ 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.
^ D. Handelman, Representing polynomials by positive linear functions on compact convex polyhedra. Pacific J. Math. 132 (1988), no. 1, 35–62.
^ K. Schmüdgen. "The $K$ -moment problem for compact semi-algebraic sets". Math. Ann. 289 (1991), no. 2, 203–206.
^ T. Wörmann. "Strikt Positive Polynome in der Semialgebraischen Geometrie", Univ. Dortmund 1998.
^ M. Putinar, "Positive polynomials on compact semi-algebraic sets". Indiana Univ. Math. J. 42 (1993), no. 3, 969–984.
^ T. Jacobi, "A representation theorem for certain partially ordered commutative rings". Math. Z. 237 (2001), no. 2, 259–273.
^ 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.
^ C. Scheiderer, "Sums of squares of regular functions on real algebraic varieties". Trans. Amer. Math. Soc. 352 (2000), no. 3, 1039–1069.
^ C. Scheiderer, "Sums of squares on real algebraic curves". Math. Z. 245 (2003), no. 4, 725–760.
^ C. Scheiderer, "Sums of squares on real algebraic surfaces". Manuscripta Math. 119 (2006), no. 4, 395–410.
^ Quillen, Daniel G. (1968). "On the representation of hermitian forms as sums of squares". Invent. Math. 5 (4): 237–242. Bibcode:1968InMat...5..237Q. doi:10.1007/bf01389773. S2CID 119774934. Zbl 0198.35205.
^ 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.
^ 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.
^ 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.
^ 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.
^ 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.
^ 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.