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Lattice Octahedra and Sums of Powers of Linear Forms
Author(s) -
Cusick T. W.,
Wolfskill J.
Publication year - 1988
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-38.2.207
Subject(s) - mathematics , minkowski space , combinatorics , octahedron , lattice (music) , quadratic equation , minkowski addition , quadratic form (statistics) , crystal structure , mathematical physics , physics , crystallography , geometry , chemistry , acoustics
In n ‐dimensional real space, let u 1 ,…, u n be n linearly independent points of a lattice L . Assume that the closed octahedron formed by the convex hull of the points ± u 1 ,…, ± u n contains no points of L other than the origin and the octahedron vertices. An old problem, solved by Minkowski for n = 2 and 3, concerns the possible values of the index I in L of the sublattice generated by the u j We prove that there exist lattices L such that I ⩾ n !/2 n for n ⩾ 5. This result is used to prove the existence of n linear forms in n variables with determinant D , the sum of whose absolute values is, asymptotically, at least n | D | 1/ n /2 e . Similar results are proved for the sum of the p th powers of the absolute values of n linear forms, where p is any fixed real number not less than 1. For quadratic forms ( p = 2) our result matches a classical theorem of Minkowski; for p ≠ 2 our result is new.