caa a22 4500 469075031 CHVBK 20180323132933.0 cr unu---uuuuu 170328e19921201xx s 000 0 eng 10.1007/BF02293055 doi (NATIONALLICENCE)springer-10.1007/BF02293055 Erdahl Robert Department of Mathematics and Statistics, Queen's University, K7L 3N6, Kingston, Ontario, Canada aut A cone of inhomogeneous second-order polynomials [Elektronische Daten] [Robert Erdahl] Let ℘ n be the cone of quadratic function % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0xc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaacaWFgb% GaaGymaiaac6cacaWFGaGaamOzaiabg2da9iaadAgadaWgaaWcbaGa% aGimaaqabaGccqGHRaWkdaaeabqaaiaadAgadaWgaaWcbaGaamyAaa% qabaGccaWG4bWaaSbaaSqaaiaadMgaaeqaaaqabeqaniabggHiLdGc% cqGHRaWkdaaeabqaaiaadAgadaWgaaWcbaGaamyAaiaadQgaaeqaaO% GaamiEamaaBaaaleaacaWGPbaabeaaaeqabeqdcqGHris5aOGaamiE% amaaBaaaleaacaWGQbaabeaakiaacYcacaWGMbWaaSbaaSqaaiaadM% gacaWGQbaabeaakiabg2da9iaadAgadaWgaaWcbaGaamOAaiaadMga% aeqaaOGaaiilaaaa!59ED! $$F1. f = f_0 + \sum {f_i x_i } + \sum {f_{ij} x_i } x_j ,f_{ij} = f_{ji} ,$$ on ℝ n that satisfy the additional condition % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0xc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaacaWFgb% Gaa8Nmaiaac6cacaWFGaGaamOzaiaacIcacaWG6bGaaiykaeXafv3y% SLgzGmvETj2BSbacfaGae4xzImRaaGimaiaacYcacaWG6bGaeyicI4% 8efv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiyqacqqFKeIw% daahaaWcbeqaaiaad6gaaaGccaGGSaaaaa!570C! $$F2. f(z) \geqslant 0,z \in \mathbb{Z}^n ,$$ where ℤ denotes the integers. The coefficients and variables are assumed to be real and 1≦i, 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 ∈ 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 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(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]. Springer-Verlag New York Inc., 1992 Discrete & Computational Geometry Springer New York 8/4(1992-12-01), 387-416 0179-5376 8:4<387 1992 8 454 https://doi.org/10.1007/BF02293055 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02293055 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Erdahl Robert Department of Mathematics and Statistics, Queen's University, K7L 3N6, Kingston, Ontario, Canada aut NATIONALLICENCE 773 0- Discrete & Computational Geometry Springer New York 8/4(1992-12-01), 387-416 0179-5376 8:4<387 1992 8 454 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer