Algebraic Volume for Polytope Arise from Ehrhart Theory

Guoce Xin, Xinyu Xu, Yingrui Zhang, Zihao Zhang

Published 2023-06-21Version 1

Volume computation for $d$-polytopes $\mathcal{P}$ is fundamental in mathematics. There are known volume computation algorithms, mostly based on triangulation or signed-decomposition of the polytope. We consider the cone $ \mathrm{cone}(\mathcal{P})$ over $\mathcal{P}$ in view of Ehrhart theory. By using technique from algebraic combinatorics, we obtain an algorithm using only signed simplicial cone decompositions of $ \mathrm{cone}(\mathcal{P})$. Each cone is associated with a simple algebraic volume formula. Summing them gives the volume of the polytope. Our volume formula applies to various kind of cases. In particular, we use it to explain the traditional triangulation method and Lawrence's signed decomposition method.