Algebraic varieties are homeomorphic to varieties defined over number fields

TL;DR

This paper proves that any real or complex algebraic variety can be topologically transformed into a variety defined over algebraic numbers, using small deformations and Zariski equisingular families, and provides an algorithm for this process.

Contribution

It introduces a method to construct homeomorphisms from varieties over real or complex fields to those over algebraic numbers, including an explicit algorithm for the transformation.

Findings

Every real or complex algebraic variety is homeomorphic to one over algebraic numbers.

A deformation method based on Zariski equisingular families is used.

An algorithm for transforming defining equations is provided.

Abstract

We show that every affine or projective algebraic variety defined over the field of real or complex numbers is homeomorphic to a variety defined over the field of algebraic numbers. We construct such a homeomorphism by choosing a small deformation of the coefficients of the original equations. This method is based on the properties of Zariski equisingular families of varieties. Moreover we construct an algorithm, that, given a system of equations defining a variety $V$, produces a system of equations with algebraic coefficients of a variety homeomorphic to $V$