An Effective Criterion for Covering Maps Between Real Varieties

TL;DR

This paper introduces a new, algorithmically verifiable criterion for when certain morphisms between real algebraic varieties induce covering maps on their real points, enhancing understanding of their topological structure.

Contribution

It establishes a novel criterion linking algebraic properties of morphisms to topological covering maps, with an emphasis on algorithmic verification.

Findings

A quasi-finite, flat morphism with locally constant geometric fibers induces a covering map on real points.

The criterion can be checked algorithmically using developed tools.

Provides a practical method for verifying covering properties in real algebraic geometry.

Abstract

In this paper, we establish a new criterion for covering maps between real algebraic varieties. Specifically, we prove that a quasi-finite, flat morphism with locally constant geometric fibers between varieties over a real closed field induces a covering map on the real points in the Euclidean topology. This result provides an effective method for verifying covering properties, as we demonstrate that the required conditions can be checked algorithmically using the tools developed in this work.