Discrete & Computational Geometry

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

A cone of inhomogeneous second-order polynomials

  • Robert Erdahl


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].


Root Lattice Dual System Deep Hole Integer Vector Extreme Element 


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  1. [Col]
    A. J. Coleman, The structure of fermion density matrices,Rev. Modern Phys. 35, 668–687 (1963).MathSciNetCrossRefGoogle Scholar
  2. [CS]
    J. H. Conway and N. J. A. Sloane,Sphere Packing, Lattices and Groups, Grundlehren der mathematischen Wissenschaften, Vol. 290, Springer-Verlag, New York, 1987.Google Scholar
  3. [Cox1]
    H. S. M. Coxeter, Extreme forms,Canad. J. Math. 3, 391–441 (1951).MathSciNetCrossRefMATHGoogle Scholar
  4. [Cox2]
    H. S. M. Coxeter,Regular Polytopes, 3rd edn., Dover, New York, 1973.Google Scholar
  5. [Cox3]
    H. S. M. Coxeter, Regular and semi-regular polytopes, II,Math. Z. 188, 559–591 (1985).MathSciNetCrossRefMATHGoogle Scholar
  6. [Cox4]
    H. S. M. Coxeter, Regular and semi-regular polytopes, III,Math. Z. 200, 3–45 (1988).MathSciNetCrossRefMATHGoogle Scholar
  7. [Da]
    E. R. Davidson, Linear inequalities for density matrices,J. Math. Phys. 10, 725–734 (1969).CrossRefGoogle Scholar
  8. [DM]
    E. R. Davidson and W. B. McRae, LInear inequalities for density matrices, II,J. Math. Phys. 13, 1527–1537 (1972).CrossRefGoogle Scholar
  9. [Del]
    B. Delaunay [Delone], Sur la sphère vide,Proc. International Congress of Mathematicians, Toronto, 1924, University of Toronto Press, Toronto, pp. 695–700, 1928.Google Scholar
  10. [De2]
    B. N. Delone, The geometry of positive quadratic forms,Uspekhi Mat. Nauk 3, 16–62 (1937);4, 102–164 (1938).Google Scholar
  11. [DEL1]
    M. Deza, V. P. Grishukhin, and M. Laurent, Hypermetrics andL-polytopes, Research Report No. R.286, IASI-CNR, Roma Italy. 1990.Google Scholar
  12. [DGL2]
    M. Deza, V. P. Grishukhin, and M. Laurent, Extreme hypermetrics andL-polytopes, Preprint (1991).Google Scholar
  13. [E1]
    R. M. Erdahl, A convex set of second-order inhomogeneous polynomials with applications to quantum mechanical many body theory, Mathematical preprint # 1975-40, Queen's University, Kingston, Ontario.Google Scholar
  14. [E2]
    R. M. Erdahl, Representability,Internat. J. Quant. Chem. 13, 697–718 (1978).CrossRefGoogle Scholar
  15. [E3]
    R. M. Erdahl, On the structure of the diagonal conditions,Internat. J. Quant. Chem. 13, 731–736 (1978).CrossRefGoogle Scholar
  16. [E4]
    R. M. Erdahl, Representability conditions, inDensity Matrices and Density Functionals, D. Reidel, Pardrecht, 1987, pp. 51–75.Google Scholar
  17. [ER]
    R. M. erdahl and S. S. Ryshkov, The empty sphere,Canad. J. Math. 39, 794–824 (1987);40, 1058–1073 (1988).MathSciNetCrossRefMATHGoogle Scholar
  18. [G]
    B. Grünbaum,Convex Polytopes, Interscience Series on Pure and Applied Mathematics, Vol. XVI, Wiley, New York, 1967.Google Scholar
  19. [GL]
    P. M. Gruber and C. G. Lekkerkerker,Geometry of Numbers, North-Holland, Amsterdam, 1987.MATHGoogle Scholar
  20. [RB]
    S. S. Ryshkov and E. P. Baranovskii, TheC-type ofn-dimensional lattices and the five-dimensional primitive parallelohedrons (with applications to the theory of covering),Trudy Mat. Inst. Steklov 137, (1976); English transl.Proc. Steklov Inst. Math. 137, 1–140 (1978).Google Scholar
  21. [Re1]
    S. S. Ryshkov and R. M. Erdahl, the geometry of the integer roots of some quadratic equations with many variables,Soviet Math. Dokl. 26, 668–670 (1982).MATHGoogle Scholar
  22. [RE2]
    S. S. Ryshkov and R. M. Erdahl, The laminar construction ofL-polytopes in lattices,Uspekhi Mat. Nauk 44, 241–242 (1989).MathSciNetMATHGoogle Scholar
  23. [RE3]
    S. S. Ryshkov and R. M. Erdahl, Dual systems of integer vectors and their applications,Dokl. Akad. Nauk SSSR 314, 123–128 (1990).Google Scholar
  24. [RE4]
    S. S. Ryshkov and R. M. Erdahl, Dual systems of integer vectors and their application to the theory of (0–1)-matrices,Trudy Mat. Inst. Steklov 196, 161–173 (1991).MathSciNetGoogle Scholar
  25. [RE5]
    S. S. Ryshkov and R. M. Erdahl, Dual systems of integer vectors (general questions, applications to the geometry of positive quadratic forms), to appear inMat. Sb. (1991).Google Scholar
  26. [RS]
    S. S. Ryshkov and S. S. Susbaev, The structure of theL-partition for the second perfect lattice,Mat. Sb. 116 (1981); Englisy trnasl.Math. USSR-Sb. 44 (1983).Google Scholar
  27. [TD]
    P. Terwiliger and M. Deza, Classification of finite connected hypermetric spaces,Graphs Combin. 3, 293–298 (1987).MathSciNetCrossRefGoogle Scholar
  28. [V1]
    G. F. Voronoi, Nouvelles applications des paramètres continus a là théorie des formes quadratiques. Premier mémoire,J. Reine Angew. Math. 133, 79–178 (1908).MathSciNetGoogle Scholar
  29. [V2]
    G. F. Voronoi, Nouvelles applications des paramètres continus a lá théorie des formes quadratiques. Deuxiéme mémoire,J. Reine Angew. Math. 134, 198–287 (1908);136, 67–178 (1909).MathSciNetMATHGoogle Scholar
  30. [Y]
    C. N. Yang, The concept of off-diagonal long-range order and the quantum phases of liquid He and of superconductors,Rev. Modern Phys. 34, 694–704 (1962).MathSciNetCrossRefGoogle Scholar

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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