Monomial ideals with arbitrarily high tiny powers in any number of variables

TL;DR

This paper constructs monomial ideals in any number of variables with arbitrarily high powers having fewer minimal generators than the original ideal, challenging previous expectations about their growth.

Contribution

It demonstrates the existence of monomial ideals with decreasing minimal generators in their powers, countering the common belief of growth in generator count.

Findings

Existence of monomial ideals with fewer generators in higher powers

Counterexample to the growth expectation of minimal generators

Construction applicable in any number of variables and powers

Abstract

Powers of (monomial) ideals is a subject that still calls attraction in various ways. Let $I\subset \mathbb K[x_1,\ldots,x_n]$ be a monomial ideal and let $G(I)$ denote the (unique) minimal monomial generating set of $I$. How small can $|G(I^i)|$ be in terms of $|G(I)|$? We expect that the inequality $|G(I^2)|>|G(I)|$ should hold and that $|G(I^i)|$, $i\ge 2$, grows further whenever $|G(I)|\ge 2$. In this paper we will disprove this expectation and show that for any $n$ and $d$ there is an $\mathfrak m$-primary monomial ideal $I\subset \mathbb K[x_1,\ldots,x_n]$ such that $|G(I)|>|G(I^i)|$ for all $i\le d$.