Binomial expansion and the $\mathrm{v}$-number
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911910688980992 |
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| author | Saha, Kamalesh |
| author_facet | Saha, Kamalesh |
| contents | Let $I\subset A$ and $J\subset B$ be two monomial ideals, where $A$ and $B$ are two polynomial rings with disjoint variables. Considering a general set-up of monomial filtrations, we study the behaviour of the $\mathrm{v}$-function under binomial expansion. As an application, we get an explicit formula of $\mathrm{v}((I+J)^{(k)})$ in terms of $\mathrm{v}(I^{(i)})$ and $\mathrm{v}(J^{(j)})$, where $L^{(k)}$ denote the symbolic power of an ideal $L$. Furthermore, an analogous formula is extended for the $\mathrm{v}$-function of integral closure of $(I+J)^k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05567 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Binomial expansion and the $\mathrm{v}$-number Saha, Kamalesh Commutative Algebra 13F20, 13F55, 13B22 Let $I\subset A$ and $J\subset B$ be two monomial ideals, where $A$ and $B$ are two polynomial rings with disjoint variables. Considering a general set-up of monomial filtrations, we study the behaviour of the $\mathrm{v}$-function under binomial expansion. As an application, we get an explicit formula of $\mathrm{v}((I+J)^{(k)})$ in terms of $\mathrm{v}(I^{(i)})$ and $\mathrm{v}(J^{(j)})$, where $L^{(k)}$ denote the symbolic power of an ideal $L$. Furthermore, an analogous formula is extended for the $\mathrm{v}$-function of integral closure of $(I+J)^k$. |
| title | Binomial expansion and the $\mathrm{v}$-number |
| topic | Commutative Algebra 13F20, 13F55, 13B22 |
| url | https://arxiv.org/abs/2406.05567 |