Integral points on the complement of the branch locus of projections from hypersurfaces

Abstract

We study the integral points on ${\mathbb{P}}_n\setminus D$, where $D$ is the branch locus of a projection from an hypersurface in ${\mathbb{P}}_{n+1}$ to a hyperplane $H\simeq {\mathbb{P}}_n$. In doing that we follow the approach proposed in a paper by Zannier but we prove a more general result that also gives a sharper bound that may lead to prove the finiteness of integral points and has more applications. The proofs we present in this paper are effective and they provide a way to actually construct a set containing all the integral points in question. Our results find a concrete application to Diophantine equations, more specifically to the problem of finding integral solutions to equations $F(x_0,\dots ,x_n)=c$, where $c$ is a given nonzero value and $F$ is a homogeneous form defining the branch locus $D$.