Algebraic varieties are homeomorphic to varieties defined over number fields

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 carefully choosing a small deformation of the coefficients of the original equations. This deformation preserves all polynomial relations over $\mathbb{Q}$ satisfied by these coefficients and is equisingular in the sense of Zariski.

Moreover we construct an algorithm, that, given a system of equations defining a variety $V$, produces a system of equations with coefficients in $\bar{\mathbb{Q}}$ of a variety homeomorphic to $V$.