Weighted Borel Generators

TL;DR

This paper introduces weighted Borel generators, a refined subset of generators for a special class of monomial ideals called strongly stable ideals, enabling more efficient algebraic computations.

Contribution

It defines weighted Borel generators for w-stable ideals and provides a Macaulay2 package for their computation, extending prior formulas for Hilbert series and Betti numbers.

Findings

Defined weighted Borel generators for w-stable ideals

Developed a Macaulay2 package for computations

Extended formulas for Hilbert series and Betti numbers

Abstract

Strongly stable ideals are a class of monomial ideals which correspond to generic initial ideals in characteristic zero and can be described completely by their Borel generators, a subset of the minimal monomial generators of the ideal. Francisco, Mermin, and Schweig developed formulas for the Hilbert series and Betti numbers of strongly stable ideals in terms of their Borel generators. In this work, a specialization of strongly stable ideals is presented which further restricts the subset of relevant generators. A choice of weight vector $w\in\mathbb{N}_{> 0}^n$ restricts the set of strongly stable ideals to a subset designated as $w$-stable ideals. This restriction further compresses the Borel generators to a subset termed the weighted Borel generators of the ideal. A new Macaulay2 package wStableIdeals.m2 has been developed alongside this paper and segments of code support computations within.