A CONE OF INHOMOGENEOUS 2ND-ORDER POLYNOMIALS

Let P(n) be the cone of quadratic functions F1. f = f0 + SIGMA f(i)x(i) + SIGMA f(ij)x(i)x(j), f(ij) = f(ji), on R(n) that satisfy the additional condition F2. f (z) greater-than-or-equal-to 0, z is-an-element-of Z(n), where Z denotes the integers. The coefficients and variables are assumed to be real and 1 less-than-or-equal-to i,j less-than-or-equal-to n. The extent to which information on the convex structure of P(n) can be used to determine the integer solutions of the equation f = 0 for f is-an-element-of P(n) has been studied. The root figure of f is-an-element-of P(n) denoted R(f), is the set of n-vectors z is-an-element-of Z(n) satisfying the equation f(z) = 0. The root figures relate to the convex structure of P(n) in an obvious way: if R is a root figure, then FR = {q is-an-element-of P(n)\R(q) = R} is a relatively open face with closure {q is-an-element-of P(n)\q(r) = 0, r is-an-element-of R}. However, such formulas do not hold for all the relatively open and closed faces; this relates to some subtleties in the structure of P(n). Enumeration of the possible root figures is the central problem in the theory of P(n). The group G(Z(n)), of affine transformations on R(n) leaving Z(n) invariant, is the full symmetry group of P(n). Classification of the root figures up to G(Z(n))-equivalence provides a complete solution to this problem, and this paper is concerned with some basic questions relating to such a classification. The ideas in this study closely relate to the theory of L-polytopes in lattices as developed by Voronoi [V1], [V2], Delone [De1], [De2], and Ryshkov [RB]; L-polytopes, along with their circumscribing empty spheres (often referred to as holes in lattices), play a central role in the study of optimal lattice coverings of space. In addition, the theory of P(n) makes contact with: (1) the theory of finite metric spaces, in particular hypermetric spaces [DGL1], [DGL2], and (2) a significant problem in quantum mechanical many-body theory related to the theory of reduced density matrices [E2]-[E4].