Abstract
For a proper cone
$${{\mathcal K}\subset\mathbb{R}^n}$$
and its dual cone
$${{\mathcal K}^*}$$
the complementary slackness condition
$${\langle{\rm {\bf x}},{\rm {\bf s}}\rangle=0}$$
defines an n-dimensional manifold
$${C({\mathcal K})}$$
in the space
$${{\mathbb R}^{2n}}$$
. When
$${{\mathcal K}}$$
is a symmetric cone, points in
$${C({\mathcal K})}$$
must satisfy at least n linearly independent bilinear identities. This fact proves to be useful when optimizing over such cones, therefore it is natural to look for similar bilinear relations for non-symmetric cones. In this paper we define the bilinearity rank of a cone, which is the number of linearly independent bilinear identities valid for points in
$${C({\mathcal K})}$$
. We examine several well-known cones, in particular the cone of positive polynomials
$${{\mathcal P}_{2n+1}}$$
and its dual, and show that there are exactly four linearly independent bilinear identities which hold for all
$${({\rm {\bf x}},{\rm {\bf s}})\in C({\mathcal P}_{2n+1})}$$
, regardless of the dimension of the cones. For nonnegative polynomials over an interval or half-line there are only two linearly independent bilinear identities. These results are extended to trigonometric and exponential polynomials. We prove similar results for Müntz polynomials.