Failures of the Integral Hasse Principle for Affine Quadric Surfaces

TL;DR

This paper investigates how often integral solutions fail the Hasse principle on affine quadric surfaces, despite the principle holding for rational points, revealing the rarity and distribution of counter-examples.

Contribution

It provides a systematic analysis of the frequency and distribution of counter-examples to the integral Hasse principle on affine quadric surfaces.

Findings

Counter-examples are rare but systematically occur in certain families.

The distribution of failures can be characterized statistically.

The work extends understanding of local-global principles for integral points.

Abstract

Quadric hypersurfaces are well-known to satisfy the Hasse principle. However, this is no longer true in the case of the Hasse principle for integral points, where counter-examples are known to exist in dimension 1 and 2. This work explores the frequency that such counter-examples arise in a family of affine quadric surfaces defined over the integers.