Real points in a homogeneous space of a real algebraic group

TL;DR

This paper presents a method, based on nonabelian Galois cohomology, for determining the existence of real points in homogeneous spaces of real algebraic groups, suitable for computational implementation.

Contribution

It introduces a novel approach using second nonabelian Galois cohomology to find or prove the absence of real points in homogeneous spaces.

Findings

Method effectively identifies real points or proves their absence.

Suitable for computer-assisted calculations.

Advances understanding of rational points in algebraic groups.

Abstract

Let G be a linear algebraic group over the field of real numbers R, and let Y be a right homogeneous space of G. We wish to find a real point of Y or to prove that Y has no real points. We describe a method to do that, implicitly using second nonabelian Galois cohomology. Our method is suitable for computer-assisted calculations.