Some Results on $\mathrm{v}$-Number of Monomial Ideals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917978741669888 |
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| author | Yang, Liuqing Hu, Kaiwen Chu, Lizhong |
| author_facet | Yang, Liuqing Hu, Kaiwen Chu, Lizhong |
| contents | This paper investigates the v-number of various classes of monomial ideals. First, we considers the relationship between the v-number and the regularity of the mixed product ideal $I$, proving that $\mathrm{v}(I) \leq \mathrm{reg}(S/I)$. Next, we investigate an open conjecture on the v-number: if a monomial ideal $I$ has linear powers, then for all $k \geq 1$, $\mathrm{v}(I^k) = α(I)k - 1.$ We prove that if a monomial ideal $I$ with linear powers is a homogeneous square-free ideal and ($k \geq 1$) has no embedded associated primes, then $\mathrm{v}(I^k) = α(I)k - 1.$ We have also drawn some conclusions about the k-th power of the graph.Additionally, we calculate the v-number of various powers of edge ideals(including ordinary power ,square-free powers, symbolic powers).
Finally, we propose a conjecture that the v-number of ordinary powers of line graph is equal to the v-number of square-free powers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_04478 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some Results on $\mathrm{v}$-Number of Monomial Ideals Yang, Liuqing Hu, Kaiwen Chu, Lizhong Commutative Algebra This paper investigates the v-number of various classes of monomial ideals. First, we considers the relationship between the v-number and the regularity of the mixed product ideal $I$, proving that $\mathrm{v}(I) \leq \mathrm{reg}(S/I)$. Next, we investigate an open conjecture on the v-number: if a monomial ideal $I$ has linear powers, then for all $k \geq 1$, $\mathrm{v}(I^k) = α(I)k - 1.$ We prove that if a monomial ideal $I$ with linear powers is a homogeneous square-free ideal and ($k \geq 1$) has no embedded associated primes, then $\mathrm{v}(I^k) = α(I)k - 1.$ We have also drawn some conclusions about the k-th power of the graph.Additionally, we calculate the v-number of various powers of edge ideals(including ordinary power ,square-free powers, symbolic powers). Finally, we propose a conjecture that the v-number of ordinary powers of line graph is equal to the v-number of square-free powers. |
| title | Some Results on $\mathrm{v}$-Number of Monomial Ideals |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2504.04478 |