What Kind of Morphisms Induces Covering Maps over a Real Closed Field?

TL;DR

This paper explores how certain morphisms over real closed fields induce covering maps on rational points, providing new triviality results and interpretations of cylindrical algebraic decomposition.

Contribution

It demonstrates that flat morphisms with locally constant fibers induce covering maps over real closed fields and offers new triviality results and algebraic interpretations.

Findings

Flat morphisms induce covering maps over real closed fields

New triviality results different from Hardt's

Cylindrical algebraic decomposition interpreted algebraically

Abstract

In this article, we show that a flat morphism of $k$-varieties ($\mathop{\mathrm{char}} k=0$) with locally constant geometric fibers becomes finite \'etale after reduction. When $k$ is a real closed field, we prove that such a morphism induces a covering map on the rational points. We further give a triviality result different from Hardt's and a new interpretation of the construction of cylindrical algebraic decomposition as applications.