On the $h$-polynomials of cyclotomic standard graded commutative algebras
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912701466279936 |
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| author | Higashitani, Akihiro Ueyama, Kenta |
| author_facet | Higashitani, Akihiro Ueyama, Kenta |
| contents | We call a standard graded commutative $\Bbbk$-algebra cyclotomic if its $h$-polynomial has all its roots on the unit circle in the complex plane. Complete intersections provide typical examples of cyclotomic algebras, since the $h$-polynomial of any standard graded complete intersection is a product of polynomials of the form $1 + t + \cdots + t^{m-1}$. We refer to such polynomials as being of type CI. A natural question is whether there exists a cyclotomic standard graded $\Bbbk$-algebra whose $h$-polynomial is not of type CI. In this paper, we give a partial answer to this question. We show that the $h$-polynomial $h_R(t)$ of a cyclotomic standard graded $\Bbbk$-algebra $R$ is of type CI whenever $h_R(1) \in \{1, 4, 6\}$ or $h_R(1)$ is prime. On the other hand, if $n \ge 8$ and $n$ is not prime, then there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1) = n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07937 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the $h$-polynomials of cyclotomic standard graded commutative algebras Higashitani, Akihiro Ueyama, Kenta Commutative Algebra Combinatorics Primary: 13A02, Secondary: 05E40, 13F55, 13H10 We call a standard graded commutative $\Bbbk$-algebra cyclotomic if its $h$-polynomial has all its roots on the unit circle in the complex plane. Complete intersections provide typical examples of cyclotomic algebras, since the $h$-polynomial of any standard graded complete intersection is a product of polynomials of the form $1 + t + \cdots + t^{m-1}$. We refer to such polynomials as being of type CI. A natural question is whether there exists a cyclotomic standard graded $\Bbbk$-algebra whose $h$-polynomial is not of type CI. In this paper, we give a partial answer to this question. We show that the $h$-polynomial $h_R(t)$ of a cyclotomic standard graded $\Bbbk$-algebra $R$ is of type CI whenever $h_R(1) \in \{1, 4, 6\}$ or $h_R(1)$ is prime. On the other hand, if $n \ge 8$ and $n$ is not prime, then there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1) = n$. |
| title | On the $h$-polynomials of cyclotomic standard graded commutative algebras |
| topic | Commutative Algebra Combinatorics Primary: 13A02, Secondary: 05E40, 13F55, 13H10 |
| url | https://arxiv.org/abs/2511.07937 |