Integral Brauer-Manin obstructions for sums of two squares and a power

TL;DR

This paper investigates how Brauer-Manin obstructions account for failures of the integral Hasse principle and strong approximation in equations involving sums of two squares and a power, providing theoretical explanations and an algorithmic approach.

Contribution

It demonstrates that specific Brauer-Manin obstructions fully explain these failures under Schinzel's hypothesis and offers an algorithm to determine the existence of integral solutions.

Findings

Brauer-Manin obstructions explain all failures under certain conditions.

The paper provides an explicit algorithm for solving the equation.

Results depend on Schinzel's hypothesis (H).

Abstract

We use Brauer-Manin obstructions to explain failures of the integral Hasse principle and strong approximation away from infinity for the equation x^2+y^2+z^k=m with fixed integers k>=3 and m. Under Schinzel's hypothesis (H), we prove that Brauer-Manin obstructions corresponding to specific Azumaya algebras explain all failures of strong approximation away from infinity at the variable z. Finally, we present an algorithm that, again under Schinzel's hypothesis (H), finds out whether the equation has any integral solutions.