Positive polynomial explained
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
be a polynomial in
variables with
real coefficients and let
be a
subset of the
-dimensional
Euclidean space
. We say that:
is
positive on
if
for every
in
.
is
non-negative on
if
for every
in
.
Positivstellensatz (and nichtnegativstellensatz)
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] Examples of positivstellensatz (and nichtnegativstellensatz)
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]
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]).
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
.
is homogeneous and
is positive on the set
\{x\inRn\midx1\ge0,...,xn\ge0,x1+ … +xn\ne0\}
, 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.
- 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]
Further reading
- 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. .
- Marshall, Murray. "Positive polynomials and sums of squares". Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008., .
See also
Notes and References
- Book: Semidefinite optimization and convex algebraic geometry . 2013 . Grigoriy Blekherman, Pablo A. Parrilo, Rekha R. Thomas . 978-1-61197-228-3 . Philadelphia . 809420808.
- Benoist. Olivier. 2017. Writing Positive Polynomials as Sums of (Few) Squares. EMS Newsletter. en. 2017-9. 105. 8–13. 10.4171/NEWS/105/4. 1027-488X. free.
- T. S. Motzkin, The arithmetic-geometric inequality. 1967 Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965) pp. 205–224.
- [Emil Artin|E. Artin]
- 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 -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.
- Dressler . Mareike . Murray . Riley . 2022-12-31 . Algebraic Perspectives on Signomial Optimization . SIAM Journal on Applied Algebra and Geometry . en . 6 . 4 . 650–684 . 10.1137/21M1462568 . 2107.00345 . 235694320 . 2470-6566.
- Dumitrescu . Bogdan . 2007 . Positivstellensatz for Trigonometric Polynomials and Multidimensional Stability Tests . IEEE Transactions on Circuits and Systems II: Express Briefs . 54 . 4 . 353–356 . 10.1109/TCSII.2006.890409 . 38131072 . 1558-3791.
- Cimprič . J. . 2011 . Strict positivstellensätze for matrix polynomials with scalar constraints . Linear Algebra and Its Applications . en . 434 . 8 . 1879–1883 . 10.1016/j.laa.2010.11.046. 119169153 . free . 1011.4930 .
- Helton . J. William . Klep . Igor . McCullough . Scott . 2012 . The convex Positivstellensatz in a free algebra . . en . 231 . 1 . 516–534 . 10.1016/j.aim.2012.04.028 . free. 1102.4859 .
- Klep . Igor . 2004-12-31 . The Noncommutative Graded Positivstellensatz . Communications in Algebra . en . 32 . 5 . 2029–2040 . 10.1081/AGB-120029921 . 120795025 . 0092-7872.
- Acquistapace . F. . Andradas . C. . Broglia . F. . 2002-07-01 . The Positivstellensatz for definable functions on o-minimal structures . Illinois Journal of Mathematics . 46 . 3 . 10.1215/ijm/1258130979 . 122451112 . 0019-2082. free .