On the ideal of the shortest vectors in the Leech lattice and other lattices

Abstract

Let $X \subset {\mathbb R}^m$ 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 $E_6$, $E_7$, $E_8$, $\Lambda_{24}$ (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.