Searcharxiv⌕ Search

arXiv subjects

José F. Fernando

Publications and source records attributed to José F. Fernando.

30 records · Page 2Linked to original sources
arXiv2017

Normalization of complex analytic spaces from a global viewpoint

In this work we study some algebraic and topological properties of the ring ${\mathcal O}(X^ν)$ of global analytic functions of the normalization $(X^ν,{\mathcal O}_{X^ν})$ of a reduced complex analytic space $(X,{\mathcal O}_X)$. If $(X,{\mathcal O}_X)$ is a Stein space, we characterize ${\mathcal O}(X^ν)$ in terms of the (topological) completion of the integral closure $\overline{{\mathcal O}(X)}^ν$ of the ring ${\mathcal O}(X)$ of global holomorphic functions on $X$ (inside its total ring of fractions) with respect to the usual Fréchet topology of $\overline{{\mathcal O}(X)}^ν$. This shows that not only the Stein space $(X,{\mathcal O}_X)$ but also its normalization is completely determined by the ring ${\mathcal O}(X)$ of global analytic functions on $X$. This result was already proved in 1988 by Hayes-Pourcin when $(X,{\mathcal O}_X)$ is an irreducible Stein space whereas in this paper we afford the general case. We also analyze the real underlying structures $(X^{\mathbb R},{\mathcal O}_X^{\mathbb R})$ and $(X^{ν\,{\mathbb R}},{\mathcal O}_{X^ν}^{\mathbb R})$ of a reduced complex analytic space $(X,{\mathcal O}_X)$ and its normalization $(X^ν,{\mathcal O}_{X^ν})$. We prove that the complexification of $(X^{ν\,{\mathbb R}},{\mathcal O}_{X^ν}^{\mathbb R})$ provides the normalization of the complexification of $(X^{\mathbb R},{\mathcal O}_X^{\mathbb R})$ if and only if $(X^{\mathbb R},{\mathcal O}_X^{\mathbb R})$ is a coherent real analytic space. Roughly speaking, coherence of the real underlying structure is equivalent to the equality of the following two combined operations: (1) normalization + real underlying structure + complexification, and (2) real underlying structure + complexification + normalization.

math.AG↗
arXiv2015

On globally defined semianalytic sets

In this work we present the concept of $C$-semianalytic subset of a real analytic manifold and more generally of a real analytic space. $C$-semianalytic sets can be understood as the natural generalization to the semianalytic setting of global analytic sets introduced by Cartan ($C$-analytic sets for short). More precisely $S$ is a $C$-semianalytic subset of a real analytic space $(X,{\mathcal O}_X)$ if each point of $X$ has a neighborhood $U$ such that $S\cap U$ is a finite boolean combinations of global analytic equalities and strict inequalities on $X$. By means of paracompactness $C$-semianalytic sets are the locally finite unions of finite boolean combinations of global analytic equalities and strict inequalities on $X$. The family of $C$-semianalytic sets is closed under the same operations as the family of semianalytic sets: locally finite unions and intersections, complement, closure, interior, connected components, inverse images under analytic maps, sets of points of dimension $k$, etc. although they are defined involving only global analytic functions. In addition, we characterize subanalytic sets as the images under proper analytic maps of $C$-semianalytic sets. We prove also that the image of a $C$-semianalytic set $S$ under a proper holomorphic map between Stein spaces is again a $C$-semianalytic set. The previous result allows us to understand better the structure of the set $N(X)$ of points of non-coherence of a $C$-analytic subset $X$ of a real analytic manifold $M$. We provide a global geometric-topological description of $N(X)$ inspired by the corresponding local one for analytic sets due to Tancredi-Tognoli (1980), which requires complex analytic normalization. As a consequence it holds that $N(X)$ is a $C$-semianalytic set of dimension $\leq\dim(X)-2$.

math.AG↗
arXiv2015

On the irreducible components of globally defined semianalytic sets

In this work we present the concept of amenable $C$-semianalytic subset of a real analytic manifold $M$ and study the main properties of this type of sets. Amenable $C$-semianalytic sets can be understood as globally defined semianalytic sets with a neat behavior with respect to Zariski closure. This fact allows us to develop a natural definition of irreducibility and the corresponding theory of irreducible components for amenable $C$-semianalytic sets. These concepts generalize the parallel ones for: complex algebraic and analytic sets, $C$-analytic sets, Nash sets and semialgebraic sets.

math.AG↗
arXiv2015

On complements of convex polyhedra as polynomial images of ${\mathbb R}^n$

In this work we prove constructively that the complement ${\mathbb R}^n\setminus{\mathcal K}$ of an $n$-dimensional unbounded convex polyhedron ${\mathcal K}\subset{\mathbb R}^n$ and the complement ${\mathbb R}^n\setminus{\rm Int}({\mathcal K})$ of its interior are polynomial images of ${\mathbb R}^n$ whenever ${\mathcal K}$ does not disconnect ${\mathbb R}^n$. The compact case and the case of convex polyhedra of small dimension were approached by the authors in previous works. Consequently, the results of this article provide a full answer to the representation as polynomial images of Euclidean spaces of complements of convex polyhedra and its interiors. The techniques here are more sophisticated than those corresponding to the compact case and require a rational separation result for certain type of (non-compact) semialgebraic sets, that has interest by its own.

math.AG↗
arXiv2015

On the remainder of the semialgebraic Stone-Cěch compactification of a semialgebraic set

In this work we analyze some topological properties of the remainder $\partial M:=β_s^* M\setminus M$ of the semialgebraic Stone-Cěch compactification $β_s^* M$ of a semialgebraic set $M\subset{\mathbb R}^m$ in order to `distinguish' its points from those of $M$. To that end we prove that the set of points of $β_s^* M$ that admit a metrizable neighborhood in $β_s^* M$ equals $M_{\rm lc}\cup( {\rm Cl}_{β_s^* M}(\overline{M}_{\leq1})\setminus\overline{M}_{\leq1})$ where $M_{\rm lc}$ is the largest locally compact dense subset of $M$ and $\overline{M}_{\leq1}$ is the closure in $M$ of the set of $1$-dimensional points of $M$. In addition, we analyze the properties of the sets $\widehat{\partial}M$ and $\widetilde{\partial}M$ of free maximal ideals associated with formal and semialgebraic paths. We prove that both are dense subsets of the remainder $\partial M$ and that the differences $\partial M\setminus\widehat{\partial}M$ and $\widehat{\partial} M\setminus\widetilde{\partial}M$ are also dense subsets of $\partial M$. It holds moreover that all the points of $\widehat{\partial}M$ have countable systems of neighborhoods in $β_s^* M$.

math.AG↗
arXiv2014

On the set of points at infinity of a polynomial image of ${\mathbb R}^n$

In this work we prove that the set of points at infinity $S_\infty:={\rm Cl}_{{\mathbb R}{\mathbb P}^m}(S)\cap\mathsf{H}_\infty$ of a semialgebraic set $S\subset{\mathbb R}^m$ which is the image of a polynomial map $f:{\mathbb R}^n\to{\mathbb R}^m$ is connected. This result is no further true in general if $f$ is a regular map, although it still works for a large family of regular maps that we call quasi-polynomial maps.

math.AG↗
arXiv2014

On the complements of 3-dimensional convex polyhedra as polynomial images of ${\mathbb R}^3$

We prove that the complement ${\mathcal S}:={\mathbb R}^3\setminus{\mathcal K}$ of a 3-dimensional convex polyhedron ${\mathcal K}\subset{\mathbb R}^3$ and its closure $\overline{\mathcal S}$ are polynomial images of ${\mathbb R}^3$. The former techniques cannot be extended in general to represent such semialgebraic sets ${\mathcal S}$ and $\overline{\mathcal S}$ as polynomial images of ${\mathbb R}^n$ if $n\geq4$.

math.AG↗
arXiv2013

On spectral types of semialgebraic sets

In this work we prove that a semialgebraic set $M\subset{\mathbb R}^m$ is determined (up to a semialgebraic homeomorphism) by its ring ${\mathcal S}(M)$ of (continuous) semialgebraic functions while its ring ${\mathcal S}^*(M)$ of (continuous) bounded semialgebraic functions only determines $M$ besides a distinguished finite subset $η(M)\subset M$. In addition it holds that the rings ${\mathcal S}(M)$ and ${\mathcal S}^*(M)$ are isomorphic if and only if $M$ is compact. On the other hand, their respective maximal spectra $β_s M$ and $β_s^* M$ endowed with the Zariski topology are always homeomorphic and topologically classify a `large piece' of $M$. The proof of this fact requires a careful analysis of the points of the remainder $\partial M:=β_s^* M\setminus M$ associated with formal paths.

math.AG↗
arXiv2013

Approximation on Nash sets with monomial singularities

This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to functions defined on Nash subsets X of M whose singularities are monomial. To that end we discuss first "finiteness" and "weak normality" for such sets X. Namely, we prove that (i) X is the union of finitely many open subsets, each Nash diffeomorphic to a finite union of coordinate linear varieties of an affine space and (ii) every function on X which is Nash on every irreducible component of X extends to a Nash function on M. Then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on Nash sets with monomial singularities. As a nice consequence we show that m-dimensional affine Nash manifolds with divisorial corners which are class k semialgebraically diffeomorphic, for k>m^2, are also Nash diffeomorphic.

math.AG↗
arXiv2013

On the Krull dimension of rings of semialgebraic functions

Let $R$ be a real closed field and let ${\mathcal S}(M)$ be the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset R^n$ and let ${\mathcal S}^*(M)$ be its subring of bounded semialgebraic functions. In this work we introduce the concept of \em semialgebraic depth \em of a prime ideal $\gtp$ of ${\mathcal S}(M)$ in order to provide an elementary proof of the finiteness of the Krull dimension of the rings ${\mathcal S}(M)$ and ${\mathcal S}^*(M)$, inspired in the classical way of doing to compute the dimension of a ring of polynomials on a complex algebraic set and without involving the sophisticated machinery of real spectra. We also show that $\dim{\mathcal S}(M)=\dim{\mathcal S}^*(M)=\dim M$ and we prove that in both cases the height of a maximal ideal corresponding to a point $p\in M$ coincides with the local dimension of $M$ at $p$. In case $\gtp$ is a prime \em $z$-ideal \em of ${\mathcal S}(M)$, its semialgebraic depth coincides with the transcendence degree over $R$ of the real closed field $\qf({\mathcal S}(M)/\gtp)$.

math.AG↗
arXiv2012

On complements of convex polyhedra as polynomial and regular images of $\R^n$

In this work we prove constructively that the complement $\R^n\setminus\pol$ of a convex polyhedron $\pol\subset\R^n$ and the complement $\R^n\setminus\Int(\pol)$ of its interior are regular images of $\R^n$. If $\pol$ is moreover bounded, we can assure that $\R^n\setminus\pol$ and $\R^n\setminus\Int(\pol)$ are also polynomial images of $\R^n$. The construction of such regular and polynomial maps is done by double induction on the number of \em facets \em (faces of maximal dimension) and the dimension of $\pol$; the careful placing (\em first \em and \em second trimming positions\em) of the involved convex polyhedra which appear in each inductive step has interest by its own and it is the crucial part of our technique.

math.AG↗