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| Định dạng: | Preprint |
| Được phát hành: |
2026
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| Những chủ đề: | |
| Truy cập trực tuyến: | https://arxiv.org/abs/2602.09607 |
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| _version_ | 1866911437511720960 |
|---|---|
| author | Bordianu, Andreea I. Cimpoeas, Mircea |
| author_facet | Bordianu, Andreea I. Cimpoeas, Mircea |
| contents | Let $n\geq m$ be two positive integers, $S_{n,m}=K[x_1,\ldots,x_n,y_1,\ldots,y_m]$ and $I_{n,m}=(x_iy_j\;:\;1\leq i\leq n,1\leq j\leq m)\subset S_{n,m}$ the edge ideal of a complete bipartite graph. Denote $h(n,m)=\operatorname{hdepth}(S_{n,m}/I_{n,m})$. We prove that $h(n,m)\geq \left\lceil \frac{n}{2} \right\rceil$ and the equality holds if $m$ belong to a certain interval centered in $\left\lceil \frac{n} {2} \right\rceil$. Also, we find some tight bounds for $h(n,n)$ and we prove several inequalities between $h(n,m)$ and $h(n,m')$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09607 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Hilbert depth of the quotient ring of the edge ideal of a complete bipartite graph Bordianu, Andreea I. Cimpoeas, Mircea Commutative Algebra 05A18, 06A07, 13C15, 13P10, 13F20 Let $n\geq m$ be two positive integers, $S_{n,m}=K[x_1,\ldots,x_n,y_1,\ldots,y_m]$ and $I_{n,m}=(x_iy_j\;:\;1\leq i\leq n,1\leq j\leq m)\subset S_{n,m}$ the edge ideal of a complete bipartite graph. Denote $h(n,m)=\operatorname{hdepth}(S_{n,m}/I_{n,m})$. We prove that $h(n,m)\geq \left\lceil \frac{n}{2} \right\rceil$ and the equality holds if $m$ belong to a certain interval centered in $\left\lceil \frac{n} {2} \right\rceil$. Also, we find some tight bounds for $h(n,n)$ and we prove several inequalities between $h(n,m)$ and $h(n,m')$. |
| title | On the Hilbert depth of the quotient ring of the edge ideal of a complete bipartite graph |
| topic | Commutative Algebra 05A18, 06A07, 13C15, 13P10, 13F20 |
| url | https://arxiv.org/abs/2602.09607 |