In mathematics, a **positive polynomial** on a particular set is a polynomial whose values are positive on that set.

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

*p*is**positive**on*S*if*p*(*x*) > 0 for every*x*∈*S*.*p*is**non-negative**on*S*if*p*(*x*) ≥ 0 for every*x*∈*S*.*p*is**zero**on*S*if*p*(*x*) = 0 for every*x*∈*S*.

For certain sets *S*, there exist algebraic descriptions of all polynomials that are positive, non-negative, or zero on *S*. Such a description is a **positivstellensatz**, **nichtnegativstellensatz**, or **nullstellensatz**. This article will focus on the former two descriptions. For the latter, see Hilbert's Nullstellensatz for the most known nullstellensatz.

## Examples of positivstellensatz (and nichtnegativstellensatz)

- Globally positive polynomials and sum of squares decomposition.
- Every real polynomial in one variable and with even degree is non-negative on ℝ if and only if it is a sum of two squares of real
*polynomials*in one variable.^{[1]}This equivalence does not generalizes for polynomial with more than one variable: for instance, the Motzkin polynomial*X*^{4}*Y*^{2}+*X*^{2}*Y*^{4}− 3*X*^{2}*Y*^{2}+ 1 is non-negative on ℝ^{2}but is not a sum of squares of elements from ℝ[*X*,*Y*].^{[2]} - A real polynomial in
*n*variables is non-negative on 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 solution^{[3]}) - Suppose that
*p*∈ ℝ[*X*_{1}, ...,*X*_{n}] is homogeneous of even degree. If it is positive on ℝ^{n}\ {0}, then there exists an integer*m*such that (*X*_{1}^{2}+ ... +*X*_{n}^{2})^{m}*p*is a sum of squares of elements from ℝ[*X*_{1}, ...,*X*_{n}].^{[4]}

- Every real polynomial in one variable and with even degree is non-negative on ℝ if and only if it is a sum of two squares of real
- Polynomials positive on polytopes.
- For polynomials of degree ≤ 1 we have the following variant of Farkas lemma: If
*f*,*g*_{1}, ...,*g*_{k}have degree ≤ 1 and*f*(*x*) ≥ 0 for every*x*∈ ℝ^{n}satisfying*g*_{1}(*x*) ≥ 0, ...,*g*_{k}(*x*) ≥ 0, then there exist non-negative real numbers*c*_{0},*c*_{1}, ...,*c*_{k}such that*f*=*c*_{0}+*c*_{1}*g*_{1}+ ... +*c*_{k}*g*_{k}. - Pólya's theorem:
^{[5]}If*p*∈ ℝ[*X*_{1}, ...,*X*_{n}] is homogeneous and*p*is positive on the set {*x*∈ ℝ^{n}|*x*≥ 0, ...,_{1}*x*≥ 0,_{n}*x*_{1}+ ... +*x*_{n}≠ 0}, then there exists an integer*m*such that (*x*_{1}+ ... +*x*_{n})^{m}*p*has non-negative coefficients. - Handelman's theorem:
^{[6]}If*K*is a compact polytope in Euclidean*d*-space, defined by linear inequalities*g*_{i}≥ 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}}.

- For polynomials of degree ≤ 1 we have the following variant of Farkas lemma: If
- Polynomials positive on semialgebraic sets.
- The most general result is Stengle's Positivstellensatz.
- For compact semialgebraic sets we have Schmüdgen's positivstellensatz,
^{[7]}^{[8]}Putinar's positivstellensatz^{[9]}^{[10]}and Vasilescu's positivstellensatz.^{[11]}The point here is that no denominators are needed. - For nice compact semialgebraic sets of low dimension, there exists a nichtnegativstellensatz without denominators.
^{[12]}^{[13]}^{[14]}

## Generalizations of positivstellensatz

Positivstellensatz also exist for trigonometric polynomials, matrix polynomials, polynomials in free variables, various quantum polynomials, etc.^{[citation needed]}

## References

- 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. x+430 pp. ISBN 3-540-64663-9. - Marshall, Murray. "Positive polynomials and sums of squares".
*Mathematical Surveys and Monographs*, 146. American Mathematical Society, Providence, RI, 2008. xii+187 pp. ISBN 978-0-8218-4402-1, ISBN 0-8218-4402-4.

## Notes

**^**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.