On a question of Uri Shapira and Barak Weiss

Leetika Kathuria (), R. J. Hans-Gill () and Madhu Raka ()
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Leetika Kathuria: Panjab University
R. J. Hans-Gill: Panjab University
Madhu Raka: Panjab University

Indian Journal of Pure and Applied Mathematics, 2015, vol. 46, issue 3, 287-307

Abstract: 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.

Keywords: Lattice; covering; non-homogeneous; product of linear forms; critical determinant (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s13226-015-0123-x

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