Asymptotic behaviour and stability index of v-numbers of graded ideals
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arXiv
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| Natura: | Preprint |
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2024
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| author | Biswas, Prativa Mandal, Mousumi Saha, Kamalesh |
| author_facet | Biswas, Prativa Mandal, Mousumi Saha, Kamalesh |
| contents | Recently, Ficarra and Sgroi initiated the study of v-numbers of powers of graded ideals. They proved that for a graded ideal $I$ in a polynomial ring $S$, $\mathrm{v}(I^k)$ is a linear function in $k$ for $k>>0$. Later, Ficarra conjectured that if $I$ is a monomial ideal with linear powers, then $\mathrm{v}(I^k)=α(I)k-1$ for all $k\geq 1$, where $α(I)$ denotes the initial degree of $I$. In this paper, we generalize this conjecture for graded ideals. We prove this conjecture for several classes of graded ideals: principal ideals, ideals $I$ with $\mathrm{depth}(S/I)=0$, cover ideals of graphs, $t$-path ideals, monomial ideals generated in degree $2$, edge ideals of weighted oriented graphs. We reduce the conjecture for several classes of graded ideals (including square-free monomial ideals) by showing it is enough to prove the conjecture for $k=1$ only. We define the stability index of the $\mathrm{v}$-number for graded ideals and investigate the stability index for edge ideals of graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16583 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic behaviour and stability index of v-numbers of graded ideals Biswas, Prativa Mandal, Mousumi Saha, Kamalesh Commutative Algebra 13A02, 13F20, 13F55, 05E40, 05C38 Recently, Ficarra and Sgroi initiated the study of v-numbers of powers of graded ideals. They proved that for a graded ideal $I$ in a polynomial ring $S$, $\mathrm{v}(I^k)$ is a linear function in $k$ for $k>>0$. Later, Ficarra conjectured that if $I$ is a monomial ideal with linear powers, then $\mathrm{v}(I^k)=α(I)k-1$ for all $k\geq 1$, where $α(I)$ denotes the initial degree of $I$. In this paper, we generalize this conjecture for graded ideals. We prove this conjecture for several classes of graded ideals: principal ideals, ideals $I$ with $\mathrm{depth}(S/I)=0$, cover ideals of graphs, $t$-path ideals, monomial ideals generated in degree $2$, edge ideals of weighted oriented graphs. We reduce the conjecture for several classes of graded ideals (including square-free monomial ideals) by showing it is enough to prove the conjecture for $k=1$ only. We define the stability index of the $\mathrm{v}$-number for graded ideals and investigate the stability index for edge ideals of graphs. |
| title | Asymptotic behaviour and stability index of v-numbers of graded ideals |
| topic | Commutative Algebra 13A02, 13F20, 13F55, 05E40, 05C38 |
| url | https://arxiv.org/abs/2402.16583 |