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On a question of Uri Shapira and Barak Weiss

Author

Listed:
  • Leetika Kathuria

    (Panjab University)

  • R. J. Hans-Gill

    (Panjab University)

  • Madhu Raka

    (Panjab University)

Abstract

Here it is proved that if Q(x 1,..., x n) is a positive definite quadratic form which is reduced in the sense of Korkine and Zolotareff and has outer coefficients B 1,..., B n satisfying B 1 ≥ 1) B n ≤ 1 and B 1 ⋯ B n = 1, then its inhomogeneous minimum is at most n/4 for n ≤ 7. This result implies a positive answer to a question of Shapira and Weiss for stable lattices and thereby provides another proof of Minkowski’s Conjecture on the product of n real non-homogeneous linear forms in n variables for n ≤ 7. Our result is an analogue of Woods’ Conjecture which has been proved for n ≤ 9. The analogous problem when B 11 is also investigated.

Suggested Citation

  • Leetika Kathuria & R. J. Hans-Gill & Madhu Raka, 2015. "On a question of Uri Shapira and Barak Weiss," Indian Journal of Pure and Applied Mathematics, Springer, vol. 46(3), pages 287-307, June.
    • Handle: RePEc:spr:indpam:v:46:y:2015:i:3:d:10.1007_s13226-015-0123-x
    DOI: 10.1007/s13226-015-0123-x
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