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 $ψ:M\to\mathbb{R}^{2d+1}$ is a smooth embedding, then $ψ$ can be approximated by an arbitrarily close smooth embedding $ϕ:M\to\mathbb{R}^{2d+1}$ whose image $ϕ(M)$ is a nonsingular algebraic subset of $\mathbb{R}^{2d+1}$. In this article, we prove that $ϕ$ can be chosen in such a way that $ϕ(M)$ 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 $ϕ(M)$. 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.
We consider the question whether one can construct an embedded resolution of singularities of a singular variety $X\subset \textbf{A}^n$ from the data of the irreducible components of the spaces of jets (of $X$) centered at the singular locus of $X.$ We show that the answer is no in general and that it is yes for some birational models of rational triple surface singularities.
Let $X\subset \mathbb{R}^n$ be a compact semialgebraic set and let $f:X\to \mathbb{R}$ be a nonzero Nash function. We give a Solernó and D'Acunto-Kurdyka type estimation of the exponent $\varrho\in[0,1)$ in the Łojasiewicz gradient inequality $|\nabla f(x)|\ge C|f(x)|^\varrho$ for $x\in X$, $|f(x)|<\varepsilon$ for some constants $C,\varepsilon>0$, in terms of the degree of a polynomial $P$ such that $P(x,f(x))=0$, $x\in X$. As a corollary we obtain an estimation of the degree of sufficiency of non-isolated Nash functions singularities
On a Computational Approach to the Nash Blowup Problem
Centro de Investigación en Matemáticas（数学研究中心）；Centro de Ciencias Matematicas, UNAM（墨西哥国立自治大学数学科学中心）；Instituto de Matemáticas, Universidad de Talca（塔尔卡大学数学研究所）
This paper seeks to prove the bijectivity of the "Nash mapping" from the set of irreducible components of the scheme parametrizing analytic arcs on an algebraic surface $X$ whose origin is a singular point, into the set of irreducible components of the exceptional locus of a minimal desingularization $X'$ of $X$ when the base field has positive characteristic. The idea is to view the surface as a specialization of another defined over a field of characteristic zero. A number of related results are proved. Among them, the construction of a scheme of arcs for a suitable one parameter family of s…
Inspired by several works on jet schemes and motivic integration, we consider an extension to singular varieties of the classical definition of discrepancy for morphisms of smooth varieties. The resulting invariant, which we call Jacobian discrepancy, is closely related to the jet schemes and the Nash blow-up of the variety. This notion leads to a framework in which adjunction and inversion of adjunction hold in full generality, and several consequences are drawn from these properties. The main result of the paper is a formula measuring the gap between the dualizing sheaf and the Grauert-Rieme…
In this article we prove two results concerning the motivic Milnor fibres $S^ε(f)$ associated to a map germ $f: (\mathbb{R}^n,0)\to(\mathbb{R},0)$, defined by G. Comte and G. Fichou. Firstly, we prove that if $f,g:(\mathbb{R}^n,0)\to(\mathbb{R},0)$ are arc-analytically equivalent germs of Nash functions then the virtual Poincaré polynomial of the corresponding motivic Milnor fibres are equal. This extends (and provides a new proof) of a result of G. Fichou. Secondly, let $T\subset\mathbb{R}^m$ be a real algebraic set and $f: T\times(\mathbb{R}^n,0)\to\mathbb{R}$ a polynomial function of polyno…
We classify semi-algebraic surfaces in $\mathbb{R}^n$ with isolated singularities up to bi-Lipschitz homeomorphisms with respect to the inner distance. In particular, we obtain complete classifications for the Nash surfaces and the complex algebraic curves. We also address the minimal surfaces with finite total curvature.
