Generators of the initial ideal of simplicial toric ideals

TL;DR

This paper provides a method to generate the initial ideal of simplicial toric ideals using affine monoid representations, and compares the degrees of Gr"obner bases with Castelnuovo-Mumford regularity.

Contribution

It introduces a new approach to generate initial ideals for simplicial toric ideals and discusses how to derive reduced Gr"obner bases from this set.

Findings

Generated initial ideal using affine monoid sums.

Demonstrated how to obtain reduced Gr"obner basis from the generating set.

Compared degrees of Gr"obner basis with Castelnuovo-Mumford regularity.

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 Gr\"obner basis from it. Moreover, we compare the maximal degree of the Gr\"obner basis and the Castelnuvo-Mumford regularity.