Algebraic Volume for Polytope Arise from Ehrhart Theory

TL;DR

This work develops an algebraic, Ehrhart-theoretic framework to compute the volume of rational polytopes by lifting them to a cone and decomposing that cone into signed simplicial cones. A central volume formula expresses $\mathrm{vol}(\mathcal{P})$ as a signed sum of simple algebraic volumes $\mathrm{Vol}_d^{\beta}$ over a cone decomposition, unifying simplex-triangulation and Lawrence-type formulas, and naturally extending to non-full-dimensional cases and to polytopes defined by linear systems $A\alpha=\mathbf{b}$. It introduces a practical primal–dual approach and the SimpCone algorithm for combinatorial cone decomposition, including determinant-free variants that improve efficiency in fixed dimensions. Computer experiments on perturbed cubes, Birkhoff polytopes, magic-square polytopes, and random polytopes demonstrate favorable performance of the new methods for simple and simplicial polytopes, often reducing the cone count and computation time compared to traditional triangulation-based strategies. Overall, the paper provides a cohesive, algebraic pathway from Ehrhart theory to practical volume computation with broad applicability and promising room for optimization.

Abstract

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 $\mathcal{P}$. We consider $ \mathrm{cone}(\mathcal{P})$ as a lift of $\mathcal{P}$ in view of Ehrhart theory. By using technique from algebraic combinatorics, we obtain a volume algorithm using only signed simplicial cone decompositions of $ \mathrm{cone}(¶)$. 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. Moreover, we give a completely new primal-dual method for volume computation. This solves the traditional problem in this area: All existing methods are hopelessly impractical for either the class of simple polytopes or the class of simplicial polytopes. Our method has a good performance in computer experiments.

Figures (3)

• Figure 1: A perturbed square and its polyhedron decomposition. The two polyhedrons on the right are $\textrm{conv}(A,B)+K((1,0))$ and $\textrm{conv}(C,D)+K((-1,0))$.
• Figure 2: Line segment from $(0,0)$ to $(4,2)$ and its affine space.
• Figure 3: The pentagon $P$ with vertices $(0,0),(1,0),(0,1),(3,1),(1,3)$.

Theorems & Definitions (34)

• Definition 1
• Theorem 2
• Definition 3
• Theorem 4
• Definition 5
• Proposition 6
• proof : Proof of Proposition \ref{['p-denominator-volume']}
• proof : Proof of Theorem \ref{['theo-vol-formula']}
• Remark 7
• proof : Proof of Theorem \ref{['general-full-volcompute']}