For certain sets
, there exist algebraic descriptions of all polynomials that are positive (resp. non-negative) on
. 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]
- Globally positive polynomials and sum of squares decomposition.
- Every real polynomial in one variable is non-negative on
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
is non-negative on
but is not a sum of squares of elements from
.[3]
- A real polynomial in
variables is non-negative on
if and only if it is a sum of squares of real rational functions in
variables (see Hilbert's seventeenth problem and Artin's solution[4]).
- Suppose that
is homogeneous of even degree. If it is positive on
, then there exists an integer
such that
is a sum of squares of elements from
.[5]
- Polynomials positive on polytopes.
- For polynomials of degree
we have the following variant of Farkas lemma: If
have degree
and
for every
satisfying
, then there exist non-negative real numbers
such that
.
- Pólya's theorem:[6] If
is homogeneous and
is positive on the set
, then there exists an integer
such that
has non-negative coefficients.
- Handelman's theorem:[7] If
is a compact polytope in Euclidean
-space, defined by linear inequalities
, and if
is a polynomial in
variables that is positive on
, then
can be expressed as a linear combination with non-negative coefficients of products of members of
.
- Polynomials positive on semialgebraic sets.
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]
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.
- 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.