Algebraic varieties are homeomorphic to varieties defined over number fields
Adam Parusiński
Université de Nice Sophia Antipolis, FranceGuillaume Rond
Université d’Aix-Marseille, Marseille, France
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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