On the Hilbert depth of the quotient ring of the edge ideal of a star graph
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| Format: | Preprint |
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2025
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| _version_ | 1866929689447104512 |
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| author | Balanescu, Silviu Cimpoeas, Mircea Cipu, Mihai |
| author_facet | Balanescu, Silviu Cimpoeas, Mircea Cipu, Mihai |
| contents | Let $S_n=K[x_1,\ldots,x_n,y]$ and $I_n=(x_1y,x_2y,\ldots,x_ny)\subset S_n$ be the edge ideal of star graph. We prove that $\operatorname{hdepth}(S_n/I_n)\geq \left\lceil \frac{n}{2} \right\rceil + \left\lfloor \sqrt{n} \right\rfloor - 2$. Also, we show that for any $\varepsilon>0$, there exists some integer $A=A(\varepsilon)\geq 0$ such that $\operatorname{hdepth}(S_n/I_n)\leq \left\lceil \frac{n}{2} \right\rceil + \left\lfloor \varepsilon n \right\rfloor + A - 2$. We deduce that $\lim\limits_{n\to\infty} \frac{1}{n}\operatorname{hdepth}(S_n/I_n) = \frac{1}{2}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_16742 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Hilbert depth of the quotient ring of the edge ideal of a star graph Balanescu, Silviu Cimpoeas, Mircea Cipu, Mihai Commutative Algebra Combinatorics 05A18, 06A07, 13C15, 13P10, 13F20 Let $S_n=K[x_1,\ldots,x_n,y]$ and $I_n=(x_1y,x_2y,\ldots,x_ny)\subset S_n$ be the edge ideal of star graph. We prove that $\operatorname{hdepth}(S_n/I_n)\geq \left\lceil \frac{n}{2} \right\rceil + \left\lfloor \sqrt{n} \right\rfloor - 2$. Also, we show that for any $\varepsilon>0$, there exists some integer $A=A(\varepsilon)\geq 0$ such that $\operatorname{hdepth}(S_n/I_n)\leq \left\lceil \frac{n}{2} \right\rceil + \left\lfloor \varepsilon n \right\rfloor + A - 2$. We deduce that $\lim\limits_{n\to\infty} \frac{1}{n}\operatorname{hdepth}(S_n/I_n) = \frac{1}{2}$. |
| title | On the Hilbert depth of the quotient ring of the edge ideal of a star graph |
| topic | Commutative Algebra Combinatorics 05A18, 06A07, 13C15, 13P10, 13F20 |
| url | https://arxiv.org/abs/2501.16742 |