# A cone of inhomogeneous second-order polynomials

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## Abstract

^{ n }be the cone of quadratic function

^{ n }that satisfy the additional condition

*i, j*≦

*n*. The extent to which information on the convex structure of ℘

^{ n }can be used to determine the integer solutions of the equation

*f*=0 for

*f*∈ ℘

^{ n }has been studied.

The**root figure** of*f* ∈ ℘^{ n }, denoted*R*_{f}, is the set of*n*-vectors*z* ∈ ℤ^{ n } satisfying the equation*f*(*z*)=0. The root figures relate to the convex structure of ℘^{ n } in an obvious way: if*R* is a root figure, then Open image in new window is a relatively open face with closure {*q*∈℘^{ n }|*q(r)*=0,*r*∈*R*}. However, such formulas do not hold for all the relatively open and closed faces; this relates to some subtleties in the structure of ℘^{ n }.

Enumeration of the possible root figures is the central problem in the theory of ℘^{ n }. The group*G*(ℤ^{ n }), of affine transformations on ℝ^{ n } leaving ℤ^{ n } invariant, is the full symmetry group of ℘^{ 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 ℘^{ 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].

### Keywords

Root Lattice Dual System Deep Hole Integer Vector Extreme Element## Preview

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