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

TL;DR

This paper investigates integral points outside the branch locus of projections from hypersurfaces, providing effective bounds and constructive methods with applications to solving certain Diophantine equations.

Contribution

It generalizes previous approaches by Zannier, offering sharper bounds and effective constructions for integral points on complements of branch loci in projective space.

Findings

Provides effective bounds for integral points

Offers constructive methods to find all integral points

Applies results to solving specific Diophantine equations

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 $D$.