Let $δ (\mathcal{P} )=(δ _0,δ _1,\ldots,δ _d)$ be the $δ$ -vector of an integral convex polytope $\mathcal{P}$ of dimension $d$. First, by using two well-known inequalities on $δ$ -vectors, we classify the possible $δ$ -vectors with $\sum_{i=0}^d δ _i ≤3$. Moreover, by means of Hermite normal forms of square matrices, we also classify the possible $δ$ -vectors with $\sum_{i=0}^d δ _i = 4$. In addition, for $\sum_{i=0}^d δ _i ≥5$, we characterize the $δ$ -vectors of integral simplices when $\sum_{i=0}^d δ _i$ is prime.

Source : oai:HAL:hal-01283113v1

Volume: DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)

Section: Proceedings

Published on: January 1, 2012

Submitted on: January 31, 2017

Keywords: Ehrhart polynomial, δ -vector, integral convex polytope, integral simplex.,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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