Finding all monomials in a polynomial ideal

TL;DR

This paper presents an elementary algorithm to find the largest A-graded ideal within any ideal in a polynomial ring, generalizing the process of finding the largest monomial ideal in a given ideal.

Contribution

The paper introduces a simple, elementary algorithm for computing the largest A-graded ideal contained in any ideal, extending to the case of finding the largest monomial ideal.

Findings

Algorithm efficiently finds the largest A-graded ideal

Special case recovers the largest monomial ideal in a given ideal

Provides a new elementary approach to ideal grading and monomial ideal extraction

Abstract

Given a $d \times n$ integer matrix $A$, the main result is an elementary, simple-to-state algorithm that finds the largest $A$-graded ideal contained in any ideal $I$ in a polynomial ring $\Bbbk[x_1,\ldots,x_n]$. The special case where $A$ is an identity matrix yields that $(t.I) \cap \Bbbk[x_1,\ldots,x_n]$ is the largest monomial ideal in $I$, where the generators of $t.I$ are those of $I$ but with each variable $x_i$ replaced by $t_i x_i$ for an invertible variable $t_i$.