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

Abstract

Let $\mathbb{Q}$ be the field of rational numbers and let $X$ be a subset of $\mathbb{R}^n$. We say that $X$ is $\mathbb{Q}$-algebraic if it is the common zero set in $\mathbb{R}^n$ of a family of polynomials in $\mathbb{Q}[\mathtt{x}_1,\ldots,\mathtt{x}_n]$. If $X$ is $\mathbb{Q}$-algebraic and of dimension $d$, then we say that $X$ is $\mathbb{Q}$-nonsingular if, for all $a\in X$, there exist a neighborhood $U$ of $a$ in $\mathbb{R}^n$ and $f_1,\ldots,f_{n-d}\in\mathbb{Q}[\mathtt{x}_1,\ldots,\mathtt{x}_n]$ such that $\nabla f_1(a),\ldots,\nabla f_{n-d}(a)$ are linearly independent and $X\cap U=\{x\in U:f_1(x)=0,\cdots,f_{n-d}(x)=0\}$. The celebrated Nash-Tognoli theorem asserts the following: if $M$ is a compact smooth manifold of dimension $d$ and $\psi:M\to\mathbb{R}^{2d+1}$ is a smooth embedding, then $\psi$ can be approximated by an arbitrarily close smooth embedding $\phi:M\to\mathbb{R}^{2d+1}$ whose image $\phi(M)$ is a nonsingular algebraic subset of $\mathbb{R}^{2d+1}$. In this article, we prove that $\phi$ can be chosen in such a way that $\phi(M)$ is a $\mathbb{Q}$-nonsingular $\mathbb{Q}$-algebraic subset of $\mathbb{R}^{2d+1}$. This guarantees for the first time that, up to smooth diffeomorphisms, every compact smooth manifold $M$ 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 $\mathbb{Q}[\mathtt{x}_1,\ldots,\mathtt{x}_{2d+1}]$ vanishing on $\phi(M)$. We extend our result to the singular setting by proving that every real algebraic set with finitely many singularities is semialgebraically homeomorphic to a $\mathbb{Q}$-algebraic set with the same number of singularities.