Certain monomial ideals whose numbers of generators of powers descend

TL;DR

This paper constructs a class of monomial ideals in polynomial rings where the number of minimal generators of their powers decreases in a specific pattern, revealing unexpected behaviors and examples of descending Cohen-Macaulay type.

Contribution

It introduces a novel class of monomial ideals with decreasing numbers of generators across powers, showcasing unusual algebraic properties not previously documented.

Findings

Constructed monomial ideals with $\mu(I)>\mu(I^2)>\cdots >\mu(I^n)$

Provided examples where Cohen-Macaulay type of $R/I^n$ decreases

Demonstrated unexpected behavior of the generator count function $\mu(I^k)$

Abstract

This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal $I$ in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound $\mu (I^2)\ge 9$ for the number of minimal generators of $I^2$ with $\mu(I)\geq 6$. Recently, Gasanova constructed monomial ideals such that $\mu(I)>\mu(I^n)$ for any positive integer $n$. In reference to them, we construct a certain class of monomial ideals such that $\mu(I)>\mu(I^2)>\cdots >\mu(I^n)=(n+1)^2$ for any positive integer $n$, which provides one of the most unexpected behaviors of the function $\mu(I^k)$. The monomial ideals also give a peculiar example such that the Cohen-Macaulay type (or the index of irreducibility) of $R/I^n$ descends.