On the Gotzmann threshold of monomials

Abstract

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 $t_0=\tau_n(u_0)$ such that $u_0x_n^t$ is a Gotzmann monomial in $R_n$ if and only if $t \ge t_0$. Currently, the function $\tau_n$ is exactly known for $n \le 4$ only. We present here an efficient procedure to determine $\tau_n(u_0)$ 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 $\tau_5(x_2^d)$ for all $d$ and we conjecture that for $n \ge 6$, $\tau_n(x_2^d)$ is a polynomial in $d$ of degree $2^{n-2}$ and dominant term equal to that of the $(n-2)$-iterated binomial coefficient $$ \binom {\binom {\binom d2}2}{\stackrel{\cdots}2}. $$