Polynomial SOS

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 ${\displaystyle g_{1}(x),\ldots ,g_{k}(x)}$ 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, m = 1 or n = 3 and 2m = 4 a form is SOS if and only if is positive. The same is also valid for the analog problem on positive symmetric forms.

Although not every form can be represented as SOS, explicit sufficient conditions for a form to be SOS have been found. Moreover, every real nonnegative form can be approximated as closely as desired (in the ${\displaystyle l_{1}}$-norm of its coefficient vector) by a sequence of forms ${\displaystyle \{f_{\epsilon }\}}$ that are SOS.

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