For a proper cone $\K\subset\mathbb{R}^n$ and its dual cone $\K^*$
the complementary slackness condition $\vx^T\vs=0$ defines an
$n$-dimensional manifold $C(\K)$ in the space
$\left\{(\vx,\vs)|\vx\in\K,\vs\in\K^*\right\}$. When $\K$ is a
symmetric cone, this fact translates to a set of $n$ linearly
independent bilinear identities (optimality conditions) satisfied
by every $(\vx,\vs) \in C(\K)$. This proves to be very useful when
optimizing over such cones, therefore it is natural to look for
similar optimality conditions for non-symmetric cones. In this
paper we define the \emph{bilinearity rank} of a cone, which is the
number of linearly independent bilinear identities valid for the
cone, and describe a linear algebraic technique to bound this
quantity. We examine several well-known cones, in particular the
cone of positive polynomials $\Po_{2n+1}$ and its dual, the closure
of the moment cone $\M_{2n+1}$, and compute their bilinearity
ranks. We show that there are exactly four linearly independent
bilinear identities which hold for all $(\vx,\vs)\in
C(\Po_{2n+1})$, 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.