Methods for finding and verifying linear quotients of ideals

This repository contains the following Macaulay2 files: findLinearOrderings.m2, isLinear.m2, and getQuotients.m2. Collectively, these files consist of 3 methods for working with ideals of polynomial rings. The usage and behavior of each method are detailed below.

isLinear ( ordering, n )

Use this method to test if the given order of a list of generators yields linear quotients.

Parameters:

• ordering: a list of generators in Q = QQ[x_1 .. x_n]
• n: the dimension of Q, the ideal's polynomial ring

Returns:

• true: if ordering is a linear quotients ordering
• false: otherwise

getQuotients ( ordering )

Use this method to return the quotients of a given list of ideal generators.

Parameters:

• ordering: a list of generators in Q = QQ[x_1 .. x_n]

Returns:

• {quots}: a list of the linear quotients

findLinearOrderings ( edgeIdeal, n )

This method is used to find the "most linear" ordering of an ideal's generators; if an ordering exists that yields all linear quotients, it will be returned. Otherwise, an ordering will be returned that has the most linear quotients.

Parameters:

• edgeIdeal: an edge ideal in Q = QQ[x_1 .. x_n]
• n: the dimension of Q, the ideal's polynomial ring

Returns:

• {order}: a list of the most linear ordering of the given ideal