Minimal generating sets of large powers of bivariate monomial ideals

Abstract

It is known that for a monomial ideal I, the number of minimal generators, $\mu \left(I^n\right)$, eventually follows a polynomial pattern for increasing n. In general, little is known about the power at which this pattern emerges. Even less is known about the exact form of the minimal generators after this power. Let $s\ge \mu \left(I\right)(d^2-1)+1$, where d is a constant bounded above by the maximal x- or y-degree appearing in the set $G\left(I\right)$ of minimal generators of I. We show that every higher power $I^{s+\ell }$ for any $\ell \ge 0$ can be constructed from certain subideals of $I^s$. This provides an explicit description of $G\left(I^{s+\ell }\right)$ in terms of $G\left(I^s\right)$. Given $G\left(I^s\right)$, this construction significantly reduces computational complexity in determining larger powers of I. This further enables us to explicitly compute $\mu \left(I^n\right)$ for all $n\ge s$ in terms of a linear polynomial in n. We include runtime measurements for the attached implementation in SageMath.