Sharpness implications for truncated Lasserre hierarchy bounds in spherical code problems

Determine whether, for any truncation of the Lasserre hierarchy applied to spherical code problems, sharpness of the resulting semidefinite programming bound necessarily implies that the associated two-point polynomial p2(u) is not identically zero on its domain (e.g., u ∈ [−1, cos θ]).

Background

In the case of the Bachoc–Vallentin three-point bound, sharpness implies that only finitely many inner products can occur, which immediately yields optimality of the corresponding spherical code. In contrast, for the truncated Lasserre hierarchy introduced in Section 4, the authors cannot conclude an analogous finiteness property from sharpness, because it hinges on whether the two-point polynomial p2(u) can be identically zero.

The authors obtain a sharp bound and use complementary slackness and additional analysis (e.g., Sturm sequences) to deduce optimality and uniqueness in dimension four, but they explicitly note the general uncertainty regarding whether sharpness in the truncated Lasserre hierarchy enforces nontriviality of p2(u).