Year: 2019
Type: article
Abstract: The Macaulay2 package RandomMonomialIdeals.m2 provides users with a set of tools that allow for the systematic generation and study of random monomial ideals.It also introduces new objects, Sample and Model, to allow for streamlined handling of random objects and their statistics in Macaulay2. ERD ŐS-RÉNYI RANDOM MONOMIAL IDEALS.Given their central role in commutative algebra and their inherent combinatorial structure (see, e.g., [Miller and Sturmfels 2005; Stanley 1996]), monomial ideals are a natural class of object to study probabilistically.This study was initiated in [De Loera et al. 2019], where random monomial ideals were produced from random sets of monomial generators in a manner inspired by the Erdős-Rényi model of random graphs.Working within the framework of such a model allows one to ask well-posed questions about the distributions of various algebraic invariants of interest.The [Macaulay2] package RandomMonomialIdeals.m2 implements several basic probabilistic models for monomial ideals based on the work in [De Loera et al. 2019]; it also computes summary statistics of (algebraic properties of ) samples of monomial ideals, and sets up the framework to define new probabilistic models and generate samples from them using two new Macaulay2 types, Sample and Model.The fundamental method in the package is randomMonomialSets, which randomly generates sets of monomials in a fixed number of variables up to a given degree from the Erdős-Rényi-type distribution Ꮾ(n, D, p) defined in [De Loera et al. 2019], as well as various other related distributions.In the following examples, we set the number of variables to n=3 and sample size to N=5.Each command in Table 1 returns a list of five sets of randomly generated monomials.In the case of Ꮾ(n, D, M), if M is larger than the total number of monomials in n variables of degree at most D, all such monomials are returned.
Authors: Sonja Petrović, Despina Stasi, Dane Wilburne
Institutions: Illinois Institute of Technology, York University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2768929134