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.
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The author acknowledges the support of CSIR, India
The author acknowledges the support of Indian National Science Academy during the period when the results of this paper were obtained
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Kathuria, L., Hans-Gill, R.J. & Raka, M. On a question of Uri Shapira and Barak Weiss. Indian J Pure Appl Math 46, 287–307 (2015). https://doi.org/10.1007/s13226-015-0123-x
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DOI: https://doi.org/10.1007/s13226-015-0123-x