# On the finiteness of solutions for polynomial-factorial Diophantine equations

## TL;DR

This paper proves the finiteness of solutions for certain polynomial and factorial Diophantine equations, linking number theory conjectures and special factorial functions, with some results unconditional and others conditional on conjectures.

## Contribution

It establishes finiteness results for polynomial-factorial Diophantine equations, including cases involving norm forms and Bhargava factorials, and connects these to the Oesterlé-Masser conjecture.

## Key findings

- Finiteness of solutions for equations involving norm forms and factorials.
- Conditional finiteness results assuming the Oesterlé-Masser conjecture.
- Unconditional finiteness for specific infinite subsets of integers.

## 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.