Let X ⊂ R m be a spherical code (i.e., a finite subset of the unit sphere) and consider the ideal of all polynomials in m variables which vanish on X. Motivated by a study of cometric (Q-polynomial) association schemes and spherical designs, we wish to determine certain properties of this ideal. After presenting some background material and preliminary results, we consider the case where X is the set of shortest vectors of one of the exceptional lattices E 6 , E 7 , E 8 , Λ 24 (the Leech lattice) and determine for each: (i) the smallest degree of a non-trivial polynomial in the ideal, and (ii) the smallest k for which the ideal admits a generating set of polynomials all of degree k or less. As it turns out, in all four cases mentioned above, these two values coincide, as they also do for the icosahedron, our introductory example. The paper concludes with a discussion of these two parameters, two open problems regarding their equality, and a few remarks concerning connections to cometric association schemes.
These phenomena motivated Martin and Steele REF to consider the ideal of polynomials that vanish on the shortest vectors of certain lattices.
On the ideal of the shortest vectors in the Leech lattice and other lattices
arXiv: Metric Geometry
Mathematics