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 of positivstellensatz (and nichtnegativstellensatz)

• Globally positive polynomials and sum of squares decomposition.
  • Every real polynomial in one variable is non-negative on ${\displaystyle \mathbb {R} }$ 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 ${\displaystyle X^{4}Y^{2}+X^{2}Y^{4}-3X^{2}Y^{2}+1}$ is non-negative on ${\displaystyle \mathbb {R} ^{2}}$ but is not a sum of squares of elements from ${\displaystyle \mathbb {R} [X,Y]}$. (Motzkin showed that it was positive using the AM–GM inequality.)^[3]
  • 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 (see Hilbert's seventeenth problem and Artin's solution^[4]).
  • Suppose that ${\displaystyle p\in \mathbb {R} [X_{1},\dots ,X_{n}]}$ is homogeneous of even degree. 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 elements from ${\displaystyle \mathbb {R} [X_{1},\dots ,X_{n}]}$.^[5]
• Polynomials positive 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 ${\displaystyle f=c_{0}+c_{1}g_{1}+\cdots +c_{k}g_{k}}$.
  • Pólya's theorem:^[6] If ${\displaystyle p\in \mathbb {R} [X_{1},\dots ,X_{n}]}$ is homogeneous and ${\displaystyle p}$ is 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.
  • 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 ${\displaystyle \{g_{i}\}}$.
• Polynomials positive on 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] The point here is that no denominators are needed.
  • For 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]