Algebraic varieties are homeomorphic to varieties defined over number fields

  • Adam Parusiński

    Université de Nice Sophia Antipolis, France
  • Guillaume Rond

    Université d’Aix-Marseille, Marseille, France
Algebraic varieties are homeomorphic to varieties defined over number fields cover
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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 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 , produces a system of equations with coefficients in of a variety homeomorphic to .

Cite this article

Adam Parusiński, Guillaume Rond, Algebraic varieties are homeomorphic to varieties defined over number fields. Comment. Math. Helv. 95 (2020), no. 2, pp. 339–359

DOI 10.4171/CMH/490