Discrete & Computational Geometry

, Volume 8, Issue 4, pp 387–416 | Cite as

A cone of inhomogeneous second-order polynomials

  • Robert Erdahl
Article

Abstract

Let ℘ n be the cone of quadratic function
$$F1. f = f_0 + \sum {f_i x_i } + \sum {f_{ij} x_i } x_j ,f_{ij} = f_{ji} ,$$
on ℝ n that satisfy the additional condition
$$F2. f(z) \geqslant 0,z \in \mathbb{Z}^n ,$$
where ℤ denotes the integers. The coefficients and variables are assumed to be real and 1≦i, jn. The extent to which information on the convex structure of ℘ n can be used to determine the integer solutions of the equationf=0 forf ∈ ℘ n has been studied.

Theroot figure off ∈ ℘ n , denotedRf, is the set ofn-vectorsz ∈ ℤ n satisfying the equationf(z)=0. The root figures relate to the convex structure of ℘ n in an obvious way: ifR is a root figure, then Open image in new window is a relatively open face with closure {q∈℘ n |q(r)=0,rR}. 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 groupG(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 toG(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 ofL-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 

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Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • Robert Erdahl
    • 1
  1. 1.Department of Mathematics and StatisticsQueen's UniversityKingstonCanada

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