Minkowski Length of 3D Lattice Polytopes |
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Authors: | Olivia Beckwith Matthew Grimm Jenya Soprunova Bradley Weaver |
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Institution: | 1. Department of Mathematics, Harvey Mudd College, 301 Platt Boulevard, Claremont, CA, 91711, USA 2. Department of Mathematics, UCSD, 9500 Gilman Dr., #0112, La Jolla, CA, 92093, USA 3. Department of Mathematics, Kent State University, Summit Street, Kent, OH, 44242, USA 4. Department of Mathematics, Grove City College, 100 Campus Drive, Grove City, PA, 16127, USA
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Abstract: | We study the Minkowski length L(P) of a lattice polytope P, which is defined to be the largest number of non-trivial primitive segments whose Minkowski sum lies in P. The Minkowski length represents the largest possible number of factors in a factorization of polynomials with exponent vectors in P, and shows up in lower bounds for the minimum distance of toric codes. In this paper we give a polytime algorithm for computing L(P) where P is a 3D lattice polytope. We next study 3D lattice polytopes of Minkowski length 1. In particular, we show that if Q, a subpolytope of P, is the Minkowski sum of L=L(P) lattice polytopes Q i , each of Minkowski length 1, then the total number of interior lattice points of the polytopes Q 1,??,Q L is at most 4. Both results extend previously known results for lattice polygons. Our methods differ substantially from those used in the two-dimensional case. |
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