Minimal generating sets of large powers of bivariate monomial ideals

For a monomial ideal $I$, it is known that for increasing $n$ the number of minimal generators $μ(I^n)$ eventually follows a polynomial pattern. 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. We show that for sufficiently large $s\in\mathbb{N}$, every higher power $I^{s+\ell}$ can be constructed from certain subideals of $I^s$. We further show that such an $s$ can be chosen to satisfy $s\leμ(I)(d^2-1)+1$, 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 $I$. This provides an explicit description of $\mathsf{G}(I^n)$ in terms of $\mathsf{G}(I^s)$, significantlyreducing computational complexity in determining high powers of bivariate monomial ideals. 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.