Matrices with Identical Sets of Neighbors

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Discussion Paper

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Given a generic m by n matrix A , a lattice point h in 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 ’ 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.

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