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

DOI10.4171/RLM/762zbMATH Open1416.11051arXiv1411.2282OpenAlexW2963103159MaRDI QIDQ2360050

Andrea Ciappi

Publication date: 23 June 2017

Published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni (Search for Journal in Brave)

Abstract: We study the integral points on $mathbb{P}_{n}setminusD$, where $D$ is the branch locus of a projection from an hypersurface in $mathbb{P}_{n+1}$ to a hyperplane $Hsimeqmathbb{P}_n$. 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 $D$.

Full work available at URL: https://arxiv.org/abs/1411.2282

zbMATH Keywords

integral pointsDiophantine equationsDiophantine geometry

Mathematics Subject Classification ID

Varieties over global fields (11G35) Diophantine equations in many variables (11D72)

Related Items (2)

On the integral points on certain surfaces ⋮ ON THE INTEGRAL POINTS ON THE COMPLEMENT OF RAMIFICATION-DIVISORS

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