positive polynomial

{{About|positive polynomials and positivstellensatz-like theorems|the Krivine–Stengle Positivstellensatz|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 $p$ 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 $p(x)\ge 0$ 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.{{Cite book |url=https://www.worldcat.org/oclc/809420808 |title=Semidefinite optimization and convex algebraic geometry |date=2013 |others=Grigoriy Blekherman, Pablo A. Parrilo, Rekha R. Thomas |isbn=978-1-61197-228-3 |location=Philadelphia |oclc=809420808}}

Examples of positivstellensatz (and nichtnegativstellensatz)

Globally positive polynomials and sum of squares decomposition.
Every real polynomial in one variable is non-negative on $\mathbb{R}$ if and only if it is a sum of two squares of real polynomials in one variable.{{Cite journal|last=Benoist|first=Olivier|date=2017|title=Writing Positive Polynomials as Sums of (Few) Squares|journal=EMS Newsletter|language=en|volume=2017-9|issue=105|pages=8–13|doi=10.4171/NEWS/105/4|issn=1027-488X|doi-access=free}} 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 $\mathbb{R}^2$ but is not a sum of squares of elements from $\mathbb{R}[X,Y]$ . (Motzkin showed that it was positive using the AM–GM inequality.)T. S. Motzkin, The arithmetic-geometric inequality. 1967 Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965) pp. 205–224.
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 (see Hilbert's seventeenth problem and Artin's solutionE. Artin, Uber die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Sem. Univ. Hamburg, 5 (1927), 85–99.).
Suppose that $p\in\mathbb{R}[X_1,\dots,X_n]$ is homogeneous of even degree. If it is positive on $\mathbb{R}^n\setminus\{0\}$ , then there exists an integer $m$ such that $(X_1^2+\cdots+X_n^2)^mp$ is a sum of squares of elements from $\mathbb{R}[X_1,\dots,X_n]$ .B. Reznick, Uniform denominators in Hilbert's seventeenth problem. Math. Z. 220 (1995), no. 1, 75–97.
Polynomials positive on polytopes.
For polynomials of degree ${}\le 1$ we have the following variant of Farkas lemma: If $f,g_1,\dots,g_k$ have degree ${}\le 1$ and $f(x)\ge 0$ for every $x\in\mathbb{R}^n$ satisfying $g_1(x)\ge 0,\dots,g_k(x)\ge 0$ , then there exist non-negative real numbers $c_0,c_1,\dots,c_k$ such that $f=c_0+c_1g_1+\cdots+c_kg_k$ .
Pólya's theorem: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. If $p\in\mathbb{R}[X_1,\dots,X_n]$ is homogeneous and $p$ is positive on the set $\{x\in\mathbb{R}^n\mid x_1\ge 0,\dots,x_n\ge 0,x_1+\cdots+x_n\ne 0\}$ , then there exists an integer $m$ such that $(x_1+\cdots+x_n)^mp$ has non-negative coefficients.
Handelman's theorem:D. Handelman, Representing polynomials by positive linear functions on compact convex polyhedra. Pacific J. Math. 132 (1988), no. 1, 35–62. If $K$ is a compact polytope in Euclidean $d$ -space, defined by linear inequalities $g_i\ge 0$ , 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 $\{g_i\}$ .
Polynomials positive on semialgebraic sets.
The most general result is Stengle's Positivstellensatz.
For compact semialgebraic sets we have Schmüdgen's positivstellensatz,K. Schmüdgen. "The {{mvar|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. Putinar's positivstellensatzM. 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. and Vasilescu's positivstellensatz.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. The point here is that no denominators are needed.
For nice compact semialgebraic sets of low dimension, there exists a nichtnegativstellensatz without denominators.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.

Generalizations of positivstellensatz

Positivstellensatz also exist for signomials,{{Cite journal |last1=Dressler |first1=Mareike |last2=Murray |first2=Riley |date=2022-12-31 |title=Algebraic Perspectives on Signomial Optimization |url=https://epubs.siam.org/doi/10.1137/21M1462568 |journal=SIAM Journal on Applied Algebra and Geometry |language=en |volume=6 |issue=4 |pages=650–684 |doi=10.1137/21M1462568 |arxiv=2107.00345 |s2cid=235694320 |issn=2470-6566}} trigonometric polynomials,{{Cite journal |last=Dumitrescu |first=Bogdan |date=2007 |title=Positivstellensatz for Trigonometric Polynomials and Multidimensional Stability Tests |url=https://ieeexplore.ieee.org/document/4155059 |journal=IEEE Transactions on Circuits and Systems II: Express Briefs |volume=54 |issue=4 |pages=353–356 |doi=10.1109/TCSII.2006.890409 |s2cid=38131072 |issn=1558-3791}} polynomial matrices,{{Cite journal |last=Cimprič |first=J. |date=2011 |title=Strict positivstellensätze for matrix polynomials with scalar constraints |journal=Linear Algebra and Its Applications |language=en |volume=434 |issue=8 |pages=1879–1883 |doi=10.1016/j.laa.2010.11.046|s2cid=119169153 |doi-access=free |arxiv=1011.4930 }} polynomials in free variables,{{Cite journal |last1=Helton |first1=J. William |last2=Klep |first2=Igor |last3=McCullough |first3=Scott |date=2012 |title=The convex Positivstellensatz in a free algebra |journal=Advances in Mathematics |language=en |volume=231 |issue=1 |pages=516–534 |doi=10.1016/j.aim.2012.04.028 |doi-access=free|arxiv=1102.4859 }} quantum polynomials,{{Cite journal |last=Klep |first=Igor |date=2004-12-31 |title=The Noncommutative Graded Positivstellensatz |url=http://www.tandfonline.com/doi/abs/10.1081/AGB-120029921 |journal=Communications in Algebra |language=en |volume=32 |issue=5 |pages=2029–2040 |doi=10.1081/AGB-120029921 |s2cid=120795025 |issn=0092-7872}} and definable functions on o-minimal structures.{{Cite journal |last1=Acquistapace |first1=F. |last2=Andradas |first2=C. |last3=Broglia |first3=F. |date=2002-07-01 |title=The Positivstellensatz for definable functions on o-minimal structures |journal=Illinois Journal of Mathematics |volume=46 |issue=3 |doi=10.1215/ijm/1258130979 |s2cid=122451112 |issn=0019-2082|doi-access=free }}

positive polynomial

Positivstellensatz (and nichtnegativstellensatz)

Examples of positivstellensatz (and nichtnegativstellensatz)

Generalizations of positivstellensatz

Notes

Further reading

See also