The Strong Factorial Conjecture

Let $m\in\mathbb N_+$ and ${\mathbb C}^{[m]}={\mathbb C}[X_1,\ldots,X_m]$. Let ${\cal L}:{\mathbb C}^{[m]}\to{\mathbb C}$ be the factorial map, defined by

$${\cal L}(X_1^{l_1}\cdots X_m^{l_m})=l_1!\cdots l_m!.$$

For $n\in\mathbb N_+$, define

$$F_n^{[m]}=\{f\in{\mathbb C}^{[m]}\setminus\{0\};\ (\exists k\in\{n,\ldots,n+{\cal N}(f)-1\})\,{\cal L}(f^k)\ne0\}\cup\{0\},$$

where ${\cal N}(f)$ is the number of nonzero monomials in $f$, and set

$$F_\cap^{[m]}=\bigcap_{n\in\mathbb N_+}F_n^{[m]}.$$

Strong Factorial Conjecture. $F_\cap^{[m]}={\mathbb C}^{[m]}$; equivalently, every polynomial belongs to the strong factorial set. This conjecture is explicitly stated as stronger than the Factorial Conjecture. The paper proves it in some special cases, but leaves the general assertion open.