Integral points on symmetric affine cubic surfaces

TL;DR

This paper investigates integral solutions on symmetric affine cubic surfaces defined by sums of a monic cubic polynomial, showing that for almost all integers, these surfaces lack an integral Brauer-Manin obstruction to the Hasse principle.

Contribution

It proves that for all but finitely many integers, the symmetric affine cubic surfaces have no integral Brauer-Manin obstruction, advancing understanding of integral points on these surfaces.

Findings

Most such surfaces have no integral Brauer-Manin obstruction.

Finiteness of exceptions where obstructions may occur.

Supports the Hasse principle for almost all cases.

Abstract

We show that if $f(u)\in \mathbb{Z}[u]$ is a monic cubic polynomial, then for all but finitely many $n\in \mathbb{Z}$ the affine cubic surface $f(u_{1})+f(u_{2})+f(u_{3})=n \subset \mathbb{A}^{3}_{\mathbb{Z}}$ has no integral Brauer-Manin obstruction to the Hasse principle.