Monomial ideals with tiny squares

TL;DR

This paper investigates the minimal number of generators of the square of a monomial ideal in two variables, challenging previous assumptions and providing new insights into their algebraic structure.

Contribution

It disproves the expectation that the number of generators of the square always exceeds that of the original ideal for monomial ideals in two variables.

Findings

Found counterexamples where ^2 has fewer generators than .

Established that ^2 can be smaller than , contrary to prior beliefs.

Provided a new understanding of the behavior of monomial ideals with small squares.

Abstract

Let $I \subset K[x,y]$ be a monomial ideal. How small can $\mu(I^2)$ be in terms of $\mu(I)$? It has been expected that the inequality $\mu(I^2) > \mu(I)$ should hold whenever $\mu(I) \ge 2$. Here we disprove this expectation and provide a somewhat surprising answer to the above question.