Computational methods for $t$-spread monomial ideals

Abstract

Let $K$ be a field and $S=K[x_1,\ldots,x_n]$ a standard polynomial ring over $K$. In this paper, we give new combinatorial 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 $S$. Some technical tools allowing to compute the cardinality of $t$-spread strongly stable sets avoiding their construction are also presented. Such functions are also implemented in a \emph{Macaulay2} package, \texttt{TSpreadIdeals}, to ease the computation of well-known results about algebraic invariants for $t$-spread ideals.