Gr\"obner bases of contraction ideals

TL;DR

This paper studies Gr"obner bases of contraction ideals under monomial homomorphisms, generalizing previous results and enabling the identification of many toric ideals with square-free or quadratic initial ideals.

Contribution

It introduces new theoretical results on Gr"obner bases of contraction ideals, extending prior work and broadening the class of toric ideals with desirable initial ideal properties.

Findings

Generalized previous results on contraction ideals

Provided examples of toric ideals with square-free initial ideals

Identified conditions under which contraction ideals admit quadratic initial ideals

Abstract

We investigate Gr\"obner bases of contraction ideals under some monomial homomorphisms. As an application of our theorem, we generalize the result of Aoki--Hibi--Ohsugi--Takemura and Hibi-Ohsugi. Using our results, one can provide many examples of toric ideals that admit square-free or quadratic initial ideals.