An Effective Criterion for Covering Maps Between Real Varieties

Abstract

We prove that a quasi-finite flat morphism with locally constant geometric-fiber cardinality between varieties over a real closed field induces a covering map on the rational points. The algebraic core of the criterion is a characteristic-zero result showing that every such morphism becomes finite étale after reduction. Since reduction does not change rational points, the induced map is a covering in the Euclidean topology. We extend the covering conclusion to Boolean combinations of closed subschemes and their complements, and construct a finite stratification separating finite covering families from families with positive-dimensional fibers. Finally, we compute the canonical non-finite, non-finite-flat, and non-finite-étale loci, using Gröbner bases. This yields effective tests for the hypotheses of the covering criterion. We conclude with applications illustrating these constructions.

Revised. The proof of Theorem 3.3 is greatly simplified. Expand the section on algorithms. 34 pages, 10 figures. This supersedes arXiv:2502.05834 as the results are largely improved and expanded