Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T09:02:10.046Z Has data issue: false hasContentIssue false
  • English
  • Français

The Empty Sphere

Published online by Cambridge University Press:  20 November 2018

R. M. Erdahl  and
S. S. Ryshkov
R. M. Erdahl
Affiliation:
Queen's University, Kingston, Ontario
S. S. Ryshkov
Affiliation:
Steklov Institute, Moscow, U.S.S.R.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In 1924 at the Toronto meeting of the International Congress of Mathematicians, B. N. Delone introduced his empty sphere method for lattices. We have titled our paper after this method as a tribute to his memory.

We have studied the sets of integer solutions of equations of the form

1

where f satisfies the following condition in which Z denotes the integers,

2

and have resolved this problem using the theory of L-types of lattices [3, 4, 11]. We have been able to give a complete description of all such integer solutions when n ≦ 4.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 4 , 01 August 1987 , pp. 794 - 824
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Erdahl, R. M., A convex set of second order inhomogeneous polynomials with applications to quantum mechanical many-polytope theory, Queen's Math. Preprint 40 (Queen's University, 1975).Google Scholar
2. Erdahl, R. M., A cone of inhomogeneous second order polynomials, to appear.Google Scholar
3. Delaunay, B. [Delone], Sur la sphére vide, Proc. Internat. Congr. Math. (Toronto, 1924) (Univ. of Toronto Press, 1928), 695–700.Google Scholar
4. Delone, B. N., The geometry of positive quadratic forms, Uspehi Mat. Nauk 3 (1937) 16–62; 4 (Russian) (1938), 102–164.Google Scholar
5. Gale, D., Neighbourly and cyclic poly topes, Proc. Symp. Pure Math. (A. M. S.) 7 (1963), 225.Google Scholar
6. Ryshkov, S. S., The structure of an n-dimensional parallelohedron of the first kind, Dokl. Akad. Nauk SSSR 146 (1962), 1027–1030; English transi. Soviet Math. Dokl. 3 (1962).Google Scholar
7. Ryshkov, S. S., Some remarks on the types of n-dimensional parallelohedra and on the density of lattice coverings of the space En by equal spheres, Dokl. Akad. Nauk. SSSR 162 (1965).Google Scholar
8. Ryshkov, S. S. and Baranovskii, E. P., The C-types of n-dimensional lattices and the five dimensional primitive parallelohedrons (with applications to the theory of covering), Trudy Mat. Inst. Steklov 137 (1976); English transi. Proc. Steklov Inst. Math. 137 (1976).Google Scholar
9. Ryshkov, S. S. and Erdahl, R. M., The geometry of the integer roots of some quadratic equations with many variables, Soviet Math. Dokl. 26 (1982).Google Scholar
10. Voronoi, G. F., Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier mémoire, J. Reine Angew. Math. 133 (1908), 79–178.Google Scholar
11. Voronoi, G. F., Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire, J. Reine Angew. Math. 134 (1908), 198–287; 136 (1909), 67–178.Google Scholar