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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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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