LAUSR

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 exist only finitely many l such that l!{l!} is represented by NA⁢(x){N_{A}(x)}, where NA{N_{A}} is a norm form constructed from the field norm of a field extension K/????{K/\mathbf{Q}}. We also deal with the equation NA⁢(x)=l!S{N_{A}(x)=l!_{S}}, where l!S{l!_{S}} is the Bhargava factorial. In this paper, we also show that the Oesterlé–Masser conjecture implies that for any infinite subset S of ????{\mathbf{Z}} and for any polynomial P⁢(x)∈????⁢[x]{P(x)\in\mathbf{Z}[x]} of degree 2 or more the equation P⁢(x)=l!S{P(x)=l!_{S}} has only finitely many solutions (x,l){(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{P(x)=l!_{S}} unconditionally.