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Semigroups of Integer Points in Convex Cones

Grigoriy Blekherman, Jesús A. De Loera, Luze Xu, Shixuan Zhang

Abstract

We study the question whether the affine semigroup of integer points in a convex cone can be finitely generated up to symmetries of the cone. We establish general properties of finite generation up to symmetry, and then concentrate on the case of irrational polyhedral cones.

Semigroups of Integer Points in Convex Cones

Abstract

We study the question whether the affine semigroup of integer points in a convex cone can be finitely generated up to symmetries of the cone. We establish general properties of finite generation up to symmetry, and then concentrate on the case of irrational polyhedral cones.

Paper Structure

This paper contains 14 sections, 24 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. General cones and (R,G)-finite generation
  4. Properties of (R,G)-finite generation
  5. Non-examples of (R,G)-finite generation
  6. Polyhedral cones and (R,G)-finite generation
  7. Properties of polyhedral cones
  8. Sufficient condition for simple polyhedral cones
  9. (R,G)-finitely generated cones in the plane
  10. Pointed cone case
  11. Half-space case
  12. Three-dimensional (R,G)-finitely generated pointed polyhedral cones
  13. Non-finitely generated irrational simple cones
  14. Acknowledgements:

Key Result

Theorem 1

Suppose $C\subset\mathbb{R}^n$ is a full-dimensional nonpolyhedral convex cone such that there are only finitely many extreme rays $[u_1],\dots,[u_m]$ of $C$ with $u_1,\dots,u_m\in\mathbb{Q}^n$ and $\operatorname{span}_\mathbb{R}\{u_1,\dots,u_m\}=\mathbb{R}^n$. Then $C\cap\mathbb{Z}^n$ is not $(R,G)

Figures (1)

  • Figure 1: Two constructions of $P$ in Example \ref{['ex:SOCSlice']}.

Theorems & Definitions (54)

  • Definition 1
  • Theorem 1
  • Example 1: Fermat cones
  • Theorem 2
  • Example 2: weighted $\ell^p$-norm cones
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Conjecture
  • Lemma 6
  • ...and 44 more