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

Rizeng Chen

TL;DR

This work addresses when morphisms of real varieties induce coverings on $R$-rational points. It introduces the q-étale notion, proving that over characteristic zero a q-étale morphism becomes finite étale after reduction and that its real points yield a covering map in the Euclidean topology. The paper then derives a semi-algebraic triviality result and interprets cylindrical algebraic decomposition through this algebro-geometric lens, while providing explicit constructions via separating elements and subresultants. These results offer a concrete mechanism for understanding parametric polynomial systems over real closed fields and their topological solution sets, with potential impact on real algebraic geometry and computational real algebraic geometry.

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 étale 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.

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

TL;DR

This work addresses when morphisms of real varieties induce coverings on -rational points. It introduces the q-étale notion, proving that over characteristic zero a q-étale morphism becomes finite étale after reduction and that its real points yield a covering map in the Euclidean topology. The paper then derives a semi-algebraic triviality result and interprets cylindrical algebraic decomposition through this algebro-geometric lens, while providing explicit constructions via separating elements and subresultants. These results offer a concrete mechanism for understanding parametric polynomial systems over real closed fields and their topological solution sets, with potential impact on real algebraic geometry and computational real algebraic geometry.

Abstract

In this article, we show that a flat morphism of -varieties () with locally constant geometric fibers becomes finite étale after reduction. When 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.

Paper Structure

This paper contains 13 sections, 14 theorems, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Our Contribution
  4. Related Works
  5. Preliminary
  6. Notations and Conventions
  7. Ingredients from Computational Algebraic Geometry
  8. Subresultants
  9. Separating Elements
  10. Rouillier's Lemma
  11. Reduction of q-étale Morphism
  12. Semi-algebraic Covering Map
  13. Applications

Key Result

Lemma 2.2

Let $\varphi:A\to B$ be a ring homomorphism and $f,g\in A[x]$. Suppose that ${\mathtt{LC}}_x(f),{\mathtt{LC}}_x(g)\notin \ker \varphi$, then and

Figures (3)

  • Figure 1: $X$ as the intersection of zeros of defining equations
  • Figure 2: Examples
  • Figure 3: Non-examples

Theorems & Definitions (29)

  • Definition 1.1
  • Definition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5: Stickelberger
  • Lemma 2.6
  • Lemma 2.7: Rouillier
  • proof
  • Proposition 3.1: Grothendieck
  • ...and 19 more