Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A New Invariant of Lattice polytopes

Winfried Bruns, Takayuki Hibi

Abstract

The maximal degree of monomials belonging to the unique minimal system of monomial generators of the canonical module $ω(K[{\mathcal P}])$ of the toric ring $K[{\mathcal P}]$ defined by a lattice polytope ${\mathcal P}$ will be studied. It is shown that if ${\mathcal P}$ possesses an interior lattice point, then the maximal degree is at most ${\rm dim} {\mathcal P} - 1$, and that this bound is the best possible in general.

A New Invariant of Lattice polytopes

Abstract

The maximal degree of monomials belonging to the unique minimal system of monomial generators of the canonical module of the toric ring defined by a lattice polytope will be studied. It is shown that if possesses an interior lattice point, then the maximal degree is at most , and that this bound is the best possible in general.
Paper Structure (4 sections, 10 theorems, 18 equations)

This paper contains 4 sections, 10 theorems, 18 equations.

Table of Contents

  1. Reduced ${\mathcal{P}}$-degree
  2. Lattice simplices
  3. Lattice polytopes
  4. Examples

Key Result

Lemma 2.1

Let $\sigma \subset {\mathbb R}^m$ be a lattice simplex of dimension $d$. Then

Theorems & Definitions (21)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Lemma 3.3
  • ...and 11 more