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

### Andrea Ciappi

Scuola Normale Superiore, Pisa, Italy

## Abstract

We study the integral points on $P_{n}∖D$, where $D$ is the branch locus of a projection from a hypersurface in $P_{n+1}$ to a hyperplane $H≃P_{n}$. We extend to the general case a result by Zannier (whose approach we follow) and we also obtain a sharper bound that yields, in some cases, the finiteness of integral points. The results presented are effective and the proofs provide a way to actually construct a set containing all the integral points in question. Thus, there are concrete applications to the study of Diophantine equations, more precisely to the problem of finding integral solutions to equations $F(x_{0},…,x_{n})=c$, where $c$ is a given nonzero value and $F$ is a homogeneous form defining the branch locus $D$, i.e. a discriminant.

## Cite this article

Andrea Ciappi, Integral points on the complement of the branch locus of projections from hypersurfaces. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 28 (2017), no. 2, pp. 277–291

DOI 10.4171/RLM/762