On the Betti numbers of monomial ideals and their powers

TL;DR

This paper investigates the Betti numbers of monomial ideals and their powers, focusing on the minimal generators and specific Betti numbers for certain families of ideals in polynomial rings.

Contribution

It provides new results on the minimal number of generators of powers of monomial ideals and computes specific Betti numbers for particular cases.

Findings

Derived formulas for minimal generators of ideal powers

Computed Betti numbers for selected families of monomial ideals

Identified patterns in Betti numbers for special variable counts

Abstract

Let $S=\mathbb{K}[x_1,\ldots,x_n]$ the polynomial ring over a field $\mathbb{K}$. In this paper for some families of monomial ideals $I \subset S$ we study the minimal number of generators of $I^k$. We use this results to find some other Betti numbers of these families of ideals for special choices of $n$, the number of variables.