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A combinatorial simplicial cone decomposition

Guoce Xin, Xinyu Xu, Zihao Zhang

TL;DR

This work develops an algebraic combinatorial framework that connects constant-term extraction to shifted simplicial cone decompositions, providing a flexible, strategy-driven method (SimpCone[S]) to decompose polytopes and solve inhomogeneous linear Diophantine systems. By embedding lattice-point counting in a field of iterated Laurent series and leveraging the CT and Z_y operators, the authors translate analytic generator manipulations into geometric cone decompositions, enabling efficient volume computations and unimodular cone decompositions via DecDenu. The approach offers a departure from traditional geometric triangulations, supporting parametric polyhedra and linking to Barvinok’s scheme while offering practical variants (S0–S2) and applications to knapsack/denumerant cones. The framework yields new proofs of Stanley reciprocity and demonstrates competitive performance through examples like magic-square cones and random tests, with broad potential for future research in parametric, unimodular, and complexity-aware cone decompositions.

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 \( Aα= \mathbf{b} \), 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.

A combinatorial simplicial cone decomposition

TL;DR

This work develops an algebraic combinatorial framework that connects constant-term extraction to shifted simplicial cone decompositions, providing a flexible, strategy-driven method (SimpCone[S]) to decompose polytopes and solve inhomogeneous linear Diophantine systems. By embedding lattice-point counting in a field of iterated Laurent series and leveraging the CT and Z_y operators, the authors translate analytic generator manipulations into geometric cone decompositions, enabling efficient volume computations and unimodular cone decompositions via DecDenu. The approach offers a departure from traditional geometric triangulations, supporting parametric polyhedra and linking to Barvinok’s scheme while offering practical variants (S0–S2) and applications to knapsack/denumerant cones. The framework yields new proofs of Stanley reciprocity and demonstrates competitive performance through examples like magic-square cones and random tests, with broad potential for future research in parametric, unimodular, and complexity-aware cone decompositions.

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 , where is an integral matrix and 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 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.
Paper Structure (16 sections, 17 theorems, 104 equations, 1 table)

This paper contains 16 sections, 17 theorems, 104 equations, 1 table.

Table of Contents

  1. Introduction
  2. Concepts and Tools
  3. Cones
  4. Brief introduction to the XinPF Algorithm
  5. Cone Decomposition from Constant Term Extraction
  6. Matrix Forms
  7. Constant term extraction over matrix forms
  8. The SimpCone algorithm according to a strategy
  9. The Inhomogeneous System
  10. The homogeneous System
  11. On Unimodular Cone Decomposition
  12. Existing algorithms
  13. On simplicial cone of the form $K^{\raisebox{1mm}{$\centerdot$}}(M_r\langle J\rangle)$
  14. On general full-dimensional simplicial cone
  15. Examples
  16. ...and 1 more sections

Key Result

Proposition 1

2015A Suppose the partial fraction decomposition of $E$ is given by where the $u_i$'s are independent of $\lambda$, $P(\lambda)$, $p(\lambda)$, and the $A_i(\lambda)$'s are all polynomials, $\deg p(\lambda)<k$, and $\deg A_i(\lambda)<a_i$ for all $i$. Then we have Moreover, if $E$ is proper in $\lambda$, i.e., $\deg_\lambda E<0$, then If $E|_{\lambda=0}=\lim_{\lambda\rightarrow 0} E$ exists, i.

Theorems & Definitions (48)

  • Proposition 1
  • Definition 2
  • Lemma 3
  • proof
  • Definition 4
  • Definition 5
  • Example 6
  • Proposition 7
  • proof
  • Corollary 8
  • ...and 38 more