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Generators of the initial ideal of simplicial toric ideals

Ryotaro Hanyu

Abstract

We describe a generating set for the initial ideal of simplicial toric ideals with respect to the graded reverse lexicographic order, using representations of elements of affine monoids as sums of irreducible elements. Although the resulting generating set is not necessarily minimal, we demonstrate, through an example, how one can obtain the reduced Gröbner basis from it. Moreover, we compare the maximal degree of the Gröbner basis and the Castelnuvo-Mumford regularity.

Generators of the initial ideal of simplicial toric ideals

Abstract

We describe a generating set for the initial ideal of simplicial toric ideals with respect to the graded reverse lexicographic order, using representations of elements of affine monoids as sums of irreducible elements. Although the resulting generating set is not necessarily minimal, we demonstrate, through an example, how one can obtain the reduced Gröbner basis from it. Moreover, we compare the maximal degree of the Gröbner basis and the Castelnuvo-Mumford regularity.
Paper Structure (9 sections, 16 theorems, 92 equations)

This paper contains 9 sections, 16 theorems, 92 equations.

Table of Contents

  1. Intoroduction
  2. Background
  3. Generation of pointed cones
  4. Simplicial affine semigroup rings
  5. Decomposition of simplicial affine semigroup rings
  6. Initial ideal of simplicial toric ideals
  7. Monomials in the initial ideal of simplicial toric ideals
  8. Example
  9. Degree bounds

Key Result

Theorem 1.2

Let $\ker\pi\subset S$ be a simplicial toric ideal. Thus, there exist $n_1,\ldots, n_r\in\mathscr{N}_1\cup\mathscr{N}_2$ such that $\mathop{\mathrm{in}}\nolimits_{\prec}(\ker\pi)=(n_1,\ldots, n_r)$.

Theorems & Definitions (50)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Corollary 2.4
  • proof
  • Remark 2.5
  • ...and 40 more