A combinatorial simplicial cone decomposition

TL;DR

This paper presents a novel algebraic combinatorial method for decomposing simplicial cones, improving efficiency in solving Diophantine systems and counting lattice points, with applications to volume computation and unimodular cone decompositions.

Contribution

It introduces the exttt{SimpCone[S]} algorithm for versatile simplicial cone decomposition, extending existing algorithms to more general cases and parametric polyhedra.

Findings

The exttt{SimpCone[S]} algorithm efficiently decomposes polyhedra into simplicial cones.

The method extends to parametric polyhedra and improves volume computation.

Application to unimodular cone decompositions enhances the exttt{DecDenu} algorithm.

Abstract

This paper introduces an algebraic combinatorial approach to simplicial cone decompositions, a key step in solving inhomogeneous linear Diophantine systems and counting lattice points in polytopes. We use constant term manipulation on the system \( A\alpha = \mathbf{b} \), where \( A \) is an \( r \times n \) integral matrix and \( \mathbf{b} \) is an integral vector. We establish a relationship between special constant terms and shifted simplicial cones. This leads to the \texttt{SimpCone[S]} algorithm, which efficiently decomposes polyhedra into simplicial cones. Unlike traditional geometric triangulation methods, this algorithm is versatile for many choices of the strategy \( \texttt{S} \) and can also be applied to parametric polyhedra. The algorithm is useful for efficient volume computation of polytopes and can be applied to address various new research projects. Additionally, we apply our framework to unimodular cone decompositions. This extends the effectiveness of the newly developed \texttt{DecDenu} algorithm from denumerant cones to general simplicial cones.