Bilinear optimality constraints for the cone of positive polynomials

For a proper cone K ⊂ ℝⁿ and its dual cone K ^* the complementary slackness condition 〈x, s 〉=0 defines an n-dimensional manifold C(K) in the space ℝ²ⁿ . When K is a symmetric cone, points in C(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(K). We examine several well-known cones, in particular the cone of positive polynomials P_2n+1 and its dual, and show that there are exactly four linearly independent bilinear identities which hold for all (x, s) ∈ C(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.