This article is about representing a non-negative polynomial as sum of squares of polynomials. For representing polynomial as a sum of squares of rational functions, see Hilbert's seventeenth problem. For the sum of squares of consecutive integers, see square pyramidal number. For representing an integer as a sum of squares of integers, see Lagrange's four-square theorem.
In mathematics, a form (i.e. a homogeneous polynomial) h(x) of degree 2m in the real n-dimensional vector x is sum of squares of forms (SOS) if and only if there exist forms of degree m such that
Every form that is SOS is also a positive polynomial, and although the converse is not always true, Hilbert proved that for n = 2, 2m = 2, or n = 3 and 2m = 4 a form is SOS if and only if it is positive.[1] The same is also valid for the analog problem on positive symmetric forms.[2][3]
Although not every form can be represented as SOS, explicit sufficient conditions for a form to be SOS have been found.[4][5] Moreover, every real nonnegative form can be approximated as closely as desired (in the -norm of its coefficient vector) by a sequence of forms that are SOS.[6]
^Hilbert, David (September 1888). "Ueber die Darstellung definiter Formen als Summe von Formenquadraten". Mathematische Annalen. 32 (3): 342–350. doi:10.1007/bf01443605. S2CID 177804714.
^Choi, M. D.; Lam, T. Y. (1977). "An old question of Hilbert". Queen's Papers in Pure and Applied Mathematics. 46: 385–405.
^Goel, Charu; Kuhlmann, Salma; Reznick, Bruce (May 2016). "On the Choi–Lam analogue of Hilbert's 1888 theorem for symmetric forms". Linear Algebra and Its Applications. 496: 114–120. arXiv:1505.08145. doi:10.1016/j.laa.2016.01.024. S2CID 17579200.
^Lasserre, Jean B. (2007). "Sufficient conditions for a real polynomial to be a sum of squares". Archiv der Mathematik. 89 (5): 390–398. arXiv:math/0612358. CiteSeerX 10.1.1.240.4438. doi:10.1007/s00013-007-2251-y. S2CID 9319455.
^Powers, Victoria; Wörmann, Thorsten (1998). "An algorithm for sums of squares of real polynomials" (PDF). Journal of Pure and Applied Algebra. 127 (1): 99–104. doi:10.1016/S0022-4049(97)83827-3.
^Lasserre, Jean B. (2007). "A Sum of Squares Approximation of Nonnegative Polynomials". SIAM Review. 49 (4): 651–669. arXiv:math/0412398. Bibcode:2007SIAMR..49..651L. doi:10.1137/070693709.
a form (i.e. a homogeneous polynomial) h(x) of degree 2m in the real n-dimensional vector x is sum of squares of forms (SOS) if and only if there exist...
Greatest common divisior of two polynomials Symmetric function Homogeneous polynomialPolynomialSOS (sum of squares) Polynomial family Quadratic function Cubic...
side length into smaller such squares. PolynomialSOS, polynomials that are sums of squares of other polynomials The Brahmagupta–Fibonacci identity, representing...
In mathematics, a positive polynomial (respectively non-negative polynomial) on a particular set is a polynomial whose values are positive (respectively...
exists for univariate sparse polynomials. It states that the non-negativity of a polynomial can be certified by sospolynomials whose degree only depends...
Kővári–Sós–Turán theorem provides an upper bound on the solution to the Zarankiewicz problem. It was established by Tamás Kővári, Vera T. Sós and Pál...
graphs. The friendship theorem of Paul Erdős, Alfréd Rényi, and Vera T. Sós (1966) states that the finite graphs with the property that every two vertices...
antiderivative, its Taylor series can always be integrated term-by-term like a polynomial, giving the antiderivative function as a Taylor series with the same radius...
board consists of Ronald Graham, Gyula O. H. Katona, Miklós Simonovits, Vera Sós, and Endre Szemerédi. It is published by the János Bolyai Mathematical Society...
further conjectured that there exists a nonconstant integer-coefficient polynomial whose values at the natural numbers form a Sidon sequence. Specifically...
provide a general algorithm which, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns) can...
Bialostocki introduced the EGZ polynomials and contributed to generalize the EGZ theorem for higher degree polynomials. The EGZ theorem is associated...
problems through which it could be attacked. In particular, there exists a polynomial-time reduction from the recognition of small set expanders to the problem...
polynomial equation in the unknown function and its derivatives, its degree of the differential equation is, depending on the context, the polynomial...
k and chromatic index n(k – 1). Its chromatic polynomial can be deduced from the chromatic polynomial of the complete graph and is equal to x ∏ i = 1...
triangular cactus. The largest triangular cactus in any graph may be found in polynomial time using an algorithm for the matroid parity problem. Since triangular...
{\displaystyle \varepsilon \rightarrow -\infty } and goes no faster than polynomially in ε {\displaystyle \varepsilon } as ε → ∞ {\displaystyle \varepsilon...
Global Information
