Minimal generating sets of large powers of bivariate monomial ideals

It is known that for a monomial ideal $I$, the number of minimal generators, $\mu(I^n)$, 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 $s\ge \mu(I)(d^2-1)+1$, 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 $\mu(I^n)$ for all $n\ge s$ in terms of a linear polynomial in $n$. We include runtime measurements for the attached implementation in SageMath.