Computational methods for t-spread monomial ideals

TL;DR

This paper introduces optimized algorithms and a Macaulay2 package for efficiently computing and managing t-spread monomial ideals and related algebraic invariants, advancing computational tools in this area.

Contribution

It presents new algorithms for computing minimal t-spread sets and a Macaulay2 package to facilitate their manipulation and analysis.

Findings

Algorithms for smallest t-spread lexicographic and strongly stable sets

Methods to compute cardinality without set construction

Implementation of a Macaulay2 package for t-spread ideals

Abstract

Let $K$ be a field and $S=K[x_1,\ldots,x_n]$ a standard polynomial ring over $K$. In this paper, some new optimized algorithms to compute the smallest $t$-spread lexicographic set and the smallest $t$-spread strongly stable set containing a given set of $t$-spread monomials of $S$ are presented. Some technical tools allowing to compute the cardinality of $t$-spread strongly stable sets avoiding their construction are given. Then, a \emph{Macaulay2} package, \texttt{TSpreadIdeals}, providing methods to easily manage $t$-spread monomials and $t$-spread ideals is implemented. Some functions to ease the calculation of well known results about algebraic invariants for $t$-spread ideals are also provided.