On the finiteness of solutions for polynomial-factorial Diophantine equations

Abstract

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 $K/\mathbf Q$. 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.