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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.09497 |
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| _version_ | 1866910367259557888 |
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| author | Bonanzinga, Vittoria Eliahou, Shalom |
| author_facet | Bonanzinga, Vittoria Eliahou, Shalom |
| contents | Let $R_n=K[x_1,\dots,x_n]$ be the $n$-variable polynomial ring over a field $K$. Let $S_n$ denote the set of monomials in $R_n$. A monomial $u \in S_n$ is a \textit{Gotzmann monomial} if the Borel-stable monomial ideal $\langle u \rangle$ it generates in $R_n$ is a Gotzmann ideal. A longstanding open problem is to determine all Gotzmann monomials in $R_n$. Given $u_0 \in S_{n-1}$, its \textit{Gotzmann threshold} is the unique nonnegative integer $t_0=τ_n(u_0)$ such that $u_0x_n^t$ is a Gotzmann monomial in $R_n$ if and only if $t \ge t_0$. Currently, the function $τ_n$ is exactly known for $n \le 4$ only. We present here an efficient procedure to determine $τ_n(u_0)$ for all $n$ and all $u_0 \in S_{n-1}$. As an application, in the critical case $u_0=x_2^d$, we determine $τ_5(x_2^d)$ for all $d$ and we conjecture that for $n \ge 6$, $τ_n(x_2^d)$ is a polynomial in $d$ of degree $2^{n-2}$ and dominant term equal to that of the $(n-2)$-iterated binomial coefficient $$ \binom {\binom {\binom d2}2}{\stackrel{\cdots}2}. $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09497 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Gotzmann threshold of monomials Bonanzinga, Vittoria Eliahou, Shalom Commutative Algebra Combinatorics 13F20, 13D40, 05E40 Let $R_n=K[x_1,\dots,x_n]$ be the $n$-variable polynomial ring over a field $K$. Let $S_n$ denote the set of monomials in $R_n$. A monomial $u \in S_n$ is a \textit{Gotzmann monomial} if the Borel-stable monomial ideal $\langle u \rangle$ it generates in $R_n$ is a Gotzmann ideal. A longstanding open problem is to determine all Gotzmann monomials in $R_n$. Given $u_0 \in S_{n-1}$, its \textit{Gotzmann threshold} is the unique nonnegative integer $t_0=τ_n(u_0)$ such that $u_0x_n^t$ is a Gotzmann monomial in $R_n$ if and only if $t \ge t_0$. Currently, the function $τ_n$ is exactly known for $n \le 4$ only. We present here an efficient procedure to determine $τ_n(u_0)$ for all $n$ and all $u_0 \in S_{n-1}$. As an application, in the critical case $u_0=x_2^d$, we determine $τ_5(x_2^d)$ for all $d$ and we conjecture that for $n \ge 6$, $τ_n(x_2^d)$ is a polynomial in $d$ of degree $2^{n-2}$ and dominant term equal to that of the $(n-2)$-iterated binomial coefficient $$ \binom {\binom {\binom d2}2}{\stackrel{\cdots}2}. $$ |
| title | On the Gotzmann threshold of monomials |
| topic | Commutative Algebra Combinatorics 13F20, 13D40, 05E40 |
| url | https://arxiv.org/abs/2403.09497 |