Polynomial SOS

This article is about representing a non-negative polynomial as sum of squares of polynomials. For representing a 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, although the converse is not always true in general. In the special cases of n = 2 and 2m = 2, or n = 3 and 2m = 4, Hilbert proved that a form is SOS if and only if it is positive. The same is also true for the analogous problem with positive symmetric forms.

Although not every form is SOS, there are efficiently testable sufficient conditions for a form to be SOS. 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.

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