Minkowski Length of 3D Lattice Polytopes

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 Qi, each of Minkowski length 1, then the total number of interior lattice points of the polytopes Q1, · · · , QL 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.