Minimal generating sets of large powers of bivariate monomial ideals
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911577628737536 |
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| author | Rath, Jutta Rissner, Roswitha |
| author_facet | Rath, Jutta Rissner, Roswitha |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21466 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimal generating sets of large powers of bivariate monomial ideals Rath, Jutta Rissner, Roswitha Commutative Algebra Rings and Algebras 13C99, 13E15, 68W30, 13F20, 13A15, 13B22 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. |
| title | Minimal generating sets of large powers of bivariate monomial ideals |
| topic | Commutative Algebra Rings and Algebras 13C99, 13E15, 68W30, 13F20, 13A15, 13B22 |
| url | https://arxiv.org/abs/2503.21466 |