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 Gotzmann monomial if its associated Borel-stable monomial ideal 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 Gotzmann threshold is the unique non-negative 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 \( \left( {\begin{array}{c}\left( {\begin{array}{c}\left( {\begin{array}{c}d\\ 2\end{array}}\right) \\ 2\end{array}}\right) \\ {\mathop {2}\limits ^{\cdots }}\end{array}}\right) . \)
