Abstract
Let ℘n be the cone of quadratic function
on ℝn that satisfy the additional condition
where ℤ denotes the integers. The coefficients and variables are assumed to be real and 1≦i, j≦n. 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, denotedR f, is the set ofn-vectorsz ∈ ℤn 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 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 groupG(ℤn), of affine transformations on ℝn leaving ℤn invariant, is the full symmetry group of ℘n. Classification of the root figures up toG(ℤn)-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].
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Communicated by H. S. M. Coxeter
This research was supported by the Natural Sciences and Engineering Research Council of Canada and the Advisory Research Committee of Queen's University.
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Erdahl, R. A cone of inhomogeneous second-order polynomials. Discrete Comput Geom 8, 387–416 (1992). https://doi.org/10.1007/BF02293055
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DOI: https://doi.org/10.1007/BF02293055