Minimal generating sets of large powers of bivariate monomial ideals
Jutta Rath, Roswitha Rissner
TL;DR
The authors address the problem of understanding minimal generating sets for large powers of bivariate monomial ideals. By introducing persistent generators and the link operation, they show that beyond an explicit bound s, powers I^{s+ℓ} can be assembled from stable components derived from I^s, yielding explicit descriptions of G(I^{s+ℓ}) and a linear growth of μ(I^n). They develop the ABH decomposition for regular staircase factors and prove that large powers admit an O(ℓ) update mechanism, enabling efficient computation via SageMath. The approach is complemented by practical runtime benchmarks demonstrating significant speedups over standard exponentiation in Macaulay2. Overall, the work provides both theoretical insight into the structure of high powers and practical algorithms for fast computation of minimal generators and their counts.
Abstract
It is known that for a monomial ideal $I$, the number of minimal generators, $μ(I^n)$, 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 $s\ge μ(I)(d^2-1)+1$, where $d$ is a constant bounded above by the maximal $x$- or $y$-degree appearing in the set $\mathsf{G}(I)$ 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~$\mathsf{G}(I^{s+\ell})$ in terms of $\mathsf{G}(I^s)$. Given $\mathsf{G}(I^s)$, this construction significantly reduces computational complexity in determining larger powers of~$I$. This further enables us to explicitly compute $μ(I^n)$ for all $n\ge s$ in terms of a linear polynomial in $n$. We include runtime measurements for the attached implementation in SageMath.