A note on some polynomial-factorial Diophantine equations

In 1876 Brocard, and independently in 1913 Ramanujan, asked to find all integer solutions for the equation \(n!=x^2-1\). It is conjectured that this equation has only three solutions, but up to now this is an open problem. Overholt observed that a weak form of Szpiro's-conjecture implies that Brocard's equation has finitely many integer solutions. More generally, assuming the ABC-conjecture, Luca showed that equations of the form \(n!=P(x)\) where \(P(x)\in\mathbb{Z}[x]\) of degree \(d\geq 2\) have only finitely many integer solutions with \(n\gt 0\). And if \(P(x)\) is irreducible, Berend and Harmse proved unconditionally that \(P(x)=n!\) has only finitely many integer solutions. In this note we study Diophantine equations of the form \(g(x_1,\ldots,x_r)=P(x)\) where \(P(x)\in\mathbb{Z}[x]\) of degree \(d\geq 2\) and \(g(x_1,\ldots,x_r)\in \mathbb{Z}[x_1,\ldots,x_r]\) where for the \(x_i\) one may also plug in \(A^{n}\) or the Bhargava factorial \(n!_S\). We want to understand when there are finitely many or infinitely many integer solutions. Moreover, we study Diophantine equations of the form \(g(x_1,\ldots,x_r)=f(x,y)\) where \(f(x,y)\in\mathbb{Z}[x,y]\) is a homogeneous polynomial of degree \(\geq2\).