Matrices with Identical Sets of Neighbors

Given a generic m by n matrix A, a lattice point h in {bold Z} is a neighbor of the origin if the body {x : Ax <= b}, with b_{i} = max{0, a_{i}h}, i = 1, ..., m, contains no lattice point other than 0 and h. The set of neighbors, N(A), is finite and Asymmetric. We show that if A' is another matrix of the same size with the property that sign a_{i}h = sign a'_{i} in h for every i and every h in N(A), then A' has precisely the same set of neighbors as A. The collection of such matrices is a polyhedral cone, described by a finite set of linear inequalities, each such inequality corresponding to a generator of one of the cones C_{i} = pos(h in N(A) : a_{i}h < 0}. Computational experience shows that C_{i} has "few" generators. We demonstrate this in the first nontrivial case n = 3, m = 4.