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On Eutactic Forms

Published online by Cambridge University Press:  20 November 2018

Avner Ash
Avner Ash*
Affiliation:
Columbia University, New York, New York
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Let (aij) = A be a positive definite n × n symmetric matrix with real entries. To it corresponds a positive definite quadratic form ƒ on Rn: ƒ(x) = txAx = ∑ aijXiXj for x any column vector in Rn. The set of values ƒ(y) for y in Zn — {0} has a minimum m (A) > 0 and the number of “minimal vectors“ y1, … , yr in Zn for which ƒ(yi) = m (A) is finite. By definition, ƒ and A are called eutactic if and only if there are positive numbers s1 ,… , sr such that

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 5 , 01 October 1977 , pp. 1040 - 1054
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Coxeter, H. S. M., Extreme forms, Can. J. Math. 3 (1951), 391–441.Google Scholar
2. Milnor, J. and Husemoller, D., Symmetric bilinear forms (Springer-Verlag, New York, 1973), 29–39.Google Scholar
3. Delone, R. N. and Ryskov, S. S., Extremal problems in the theory of positive definite quadratic forms, Proc. Steklov Inst, of Math. 112 (1971), 211–231.Google Scholar
4. Stogrin, M. I., Locally quasi-densest lattice packings of spheres, Soviet Math. Dokl. 15 (1974), 1288–1292.Google Scholar
5. Brann, H. and Koecher, M., Jordan-Algebren (Springer-Verlag, Berlin, 1966).Google Scholar
6. Ash, A., Mumford, D., Rapport, M., and Tai, Y., Smooth compactifications of locally symmetric varieties (Math-Sci.. Press, Brookline, Ma., 1975).Google Scholar
7. E. S. Barnes and M. J. Cohn, On the inner product of positive quadratic forms, J. London Math. Soc. (2) 12, (1975).Google Scholar
8. Morse, M., Topologically non-degenerate functions on a compact n-manifold M, J. Analyse Math. 7 (1959), 189.Google Scholar
9. Stogrin, A. I. I., personal communication.Google Scholar
10. Kneser, M., Two remarks on extreme forms, Can. J. Math. 7 (1955), 145–9.Google Scholar
11. Barnes, E. S., On a theorem of Voronot, Proc. Camb. Phil. Soc. 53 (1957), 537–9.Google Scholar
12. Ash, A., Cohomology of subgroups of finite index o/SL(3, Z) and SL(4, Z), Bull. Amer. Math. Soc, 83 (1977), 367.Google Scholar