On the finiteness of solutions for polynomial-factorial Diophantine equations

We study the Diophantine equations obtained by equating a polynomial and the factorial function, and prove the finiteness of integer solutions under certain conditions. For example, we show that there exists only finitely many $l$ such that $l!$ is represented {by} $N_A(x)$, where $N_A$ is a norm form constructed from the field norm of a field extension $K\mathrm{/}\mathbf{Q}$. We also deal with the equation $N_A(x)=l{!}_S$, where $l{!}_S$ is the Bhargava factorial. In this paper, we also show that the Oesterl'e-Masser conjecture implies that for any infinite subset $S$ of $\mathbf{Z}$ and for any polynomial $P(x)\in \mathbf{Z}[x]$ of degree $2$ or more the equation $P(x)=l{!}_S$ has only finitely many solutions $(x,l)$. For some special infinite subsets $S$ of $\mathbf{Z}$, we can show the finiteness of solutions for the equation $P(x)=l{!}_S$ unconditionally.