Related papers: The Nash-Tognoli theorem over the rationals and it…

We prove a relative version over $\mathbb{Q}$ of Nash-Tognoli theorem, that is: Let $M$ be a compact smooth manifold with closed smooth submanifolds $M_1,\dots,M_\ell$ in general position, then there exists a nonsingular real algebraic set…

We prove that every smooth compact submanifold of $\R^n$ can be approximated up to a small isotopy by the real locus of a nonsingular complex algebraic subset of $\C^n$ defined over $\R$. This settles a version of a conjecture posed in 1952…

We present new real algebraic maps of non-positive codimensions with prescribed images whose boundaries consist of explicit non-singular real algebraic hypersurfaces satisfying so-called "transversality" as follows. Explicit information on…

In this work we characterize the subsets of ${\mathbb R}^n$ that are images of Nash maps $f:{\mathbb R}^m\to{\mathbb R}^n$. We prove Shiota's conjecture and show that a subset ${\mathcal S}\subset{\mathbb R}^n$ is the image of a Nash map…

In this monograph, we lay the foundations for a new theory that generalizes real algebraic geometry. Let $R|K$ be a field extension, where $R$ is a real closed field and $K$ is an ordered subfield of $R$. The main objective is to study…

Let $\mathcal{M}\subset \mathbb{R}^n$ be a compact and sufficiently smooth manifold of dimension $d$. Suppose $\mathcal{M}$ is nowhere completely flat. Let $N_{\mathcal{M}}(\delta,Q)$ denote the number of rational vectors $\mathbf{a}/q$…

We show that a uniformly Euclidean metric with isolated singularity on $M^n = T^n \# M_0$, where $4\leq n\leq 7$ or $n\geq 4$, $M_0$ spin, and nonnegative scalar curvature on the smooth part is Ricci flat and extends smoothly over the…

We show that a finite connected quiver Q with no oriented cycles is tame if and only if for each dimension vector $\mathbf{d}$ and each integral weight $\theta$ of Q, the moduli space $\mathcal{M}(Q,\mathbf{d})^{ss}_{\theta}$ of…

If $X$ is a projective, geometrically irreducible variety defined over a finite field $\F_q$, such that it is smooth and its Chow group of 0-cycles fulfills base change, i.e. $CH_0(X\times_{\F_q}\bar{\F_q(X)})=\Q$, then the second author's…

The renowned Theorem of Nobile, proved by Nobile in 1975, states that a pure dimensional complex analytic set $X$ is analytically smooth if and only if its Nash transformation $\eta: \mathcal{N}(X) \to X$ is an analytic isomorphism. While…

Nash and Tognoli show that smooth closed manifolds can be the zero sets of some real polynomial maps and non-singular. The canonical projections of spheres naturally embedded in the $1$-dimensional higher Euclidean spaces and some natural…

The celebrated Nash Embedding Theorem asserts that every closed Riemannian manifold can be isometrically embedded into a sufficiently high-dimensional Euclidean space. In this paper, we prove an analogous result in the conformally compact…

The classical Theorem of Mumford states that a topologically regular complex algebraic surface in $\mathbb{C}^3$ with an isolated singular point is smooth. We proof that any Lipschitz regular complex algebraic set is smooth. No restriction…

We prove that if $\phi:(X,0)\to (X,0)$ is a finite endomorphism of an isolated singularity such that $\operatorname{deg}(\phi)\geq 2$ and $\phi$ is \'etale in codimension 1, then $X$ is $\mathbb{Q}$-Gorenstein and log canonical.

Let $T\subset{\mathbb R}^n$ be a semialgebraic set and let $\mu\ge0$ be a non-negative integer. We say that $T$ is a {\em Nash $\mu$-approximation target space} (or a $({\mathcal N},\mu)$-${\tt ats}$ for short) if it has the following…

Let $\varphi: \mathbb{P}^{n}_{F} \to \mathbb{P}^{n}_{F}$ where $F$ is a complete valued field. If $x$ is a fixed point, such that the action of $\varphi$ on $T_{x}$ has eigenvalues $\lambda_{1}, \ldots, \lambda_{n}$, with $\lambda_{1},…

Let $E$ be a number field and $X$ a smooth geometrically connected variety defined over a characteristic $p$ finite field. Given an $n$-dimensional pure $E$-compatible system of semisimple $\lambda$-adic representations of the \'etale…

Let M be a singular irreducible complex manifold of dimension n. There are Q divisors D[-1], D[0], D[1],...,D[n+1] on Nash's manifold U -> M such that D[n+1] is relatively ample on bounded sets, D[n] is relatively eventually basepoint free…

In the present paper, we study an extended theory of statistical manifolds in application to affine differential geometry. Any smooth hypersurface $M \subset \mathbb{R}^{n+1}$ with a transverse vector field $\xi$ naturally admits a…

We show that if $\phi : X \to X$ is an automorphism of a smooth projective variety and $D \subset X$ is an irreducible divisor for which the set of $d$ in $D$ with $\phi^n(d)$ in $D$ for some nonzero $n$ is not Zariski dense, then $(X,…