The Nash-Tognoli theorem over the rationals and its version for isolated singularities

Riccardo Ghiloni; Enrico Savi

Paper

The Nash-Tognoli theorem over the rationals and its version for isolated singularities

Abstract

Let

be the field of rational numbers and let

be a subset of

. We say that

-algebraic if it is the common zero set in

of a family of polynomials in

. If

-algebraic and of dimension

, then we say that

-nonsingular if, for all

, there exist a neighborhood

and

such that

are linearly independent and

. The celebrated Nash-Tognoli theorem asserts the following: if

is a compact smooth manifold of dimension

and

is a smooth embedding, then

can be approximated by an arbitrarily close smooth embedding

whose image

is a nonsingular algebraic subset of

. In this article, we prove that

can be chosen in such a way that

is a

-nonsingular

-algebraic subset of

. This guarantees for the first time that, up to smooth diffeomorphisms, every compact smooth manifold

can be described both globally and locally by means of finitely many exact data, such as a finite system of generators of the ideal of polynomials in

vanishing on

. We extend our result to the singular setting by proving that every real algebraic set with finitely many singularities is semialgebraically homeomorphic to a

-algebraic set with the same number of singularities.