Abstract
Let \({\mathcal{P} \subset \mathbb{R}^{d}}\) and \({\mathcal{Q} \subset \mathbb{R}^{e}}\) be integral convex polytopes of dimension d and e which contain the origin of \({\mathbb{R}^{d}}\) and \({\mathbb{R}^{e}}\), respectively. We say that an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^{d}}\) possesses the integer decomposition property if, for each \({n\geq1}\) and for each \({\gamma \in n\mathcal{P}\cap\mathbb{Z}^{d}}\), there exist \({\gamma^{(1)}, . . . , \gamma^{(n)}}\) belonging to \({\mathcal{P}\cap\mathbb{Z}^{d}}\) such that \({\gamma = \gamma^{(1)} +. . .+\gamma^{(n)}}\). In the present paper, under some assumptions, the necessary and sufficient condition for the free sum of \({\mathcal{P}}\) and \({\mathcal{Q}}\) to possess the integer decomposition property will be presented.
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References
Atiyah M.F., Macdonald I.G.: Introduction to Commutative Algebra. Addison-Wesley, Reading (1969)
Beck M., Jayawant P., McAllister T.B.: Lattice-point generating functions for free sums of convex sets. J. Combin. Theory Ser. A 120, 1246–1262 (2013)
Bruns W., Gubeladze J.: Polytopes, Rings and K-Theory. Springer-Verlag, Heidelberg (2009)
Hibi T.: Algebraic Combinatorics on Convex Polytopes. Carslaw Publications, Glebe (1992)
Schrijver A.: Theory of Linear and Integer Programming. JohnWiley & Sons, Chichester (1986)
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Hibi, T., Higashitani, A. Integer Decomposition Property of Free Sums of Convex Polytopes. Ann. Comb. 20, 601–607 (2016). https://doi.org/10.1007/s00026-016-0314-0
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DOI: https://doi.org/10.1007/s00026-016-0314-0