Finiteness problems on Nash manifolds and Nash sets

@article{Fernando2014,
abstract = {We study here several finiteness problems concerning affine Nash manifolds $M$ and Nash subsets $X$. Three main results are: (i) A Nash function on a semialgebraic subset $Z$ of $M$ has a Nash extension to an open semialgebraic neighborhood of $Z$ in $M$, (ii) A Nash set $X$ that has only normal crossings in $M$ can be covered by finitely many open semialgebraic sets $U$ equipped with Nash diffeomorphisms $(u_1,\dots ,u_m):U\rightarrow \mathbb \{R\}^m$ such that $U\cap X=\lbrace u_1\cdots u_r=0\rbrace $, (iii) Every affine Nash manifold with corners $N$ is a closed subset of an affine Nash manifold $M$ where the Nash closure of the boundary $\partial N$ of $N$ has only normal crossings and $N$ can be covered with finitely many open semialgebraic sets $U$ such that each intersection $N\cap U=\lbrace u_1\ge 0,\dots u_r\ge 0\rbrace $ for a Nash diffeomorphism $(u_1,\dots ,u_m):U\rightarrow \mathbb \{R\}^m$.},
author = {Fernando, José F., Gamboa, José Manuel, Ruiz, Jesús M.},
journal = {Journal of the European Mathematical Society},
keywords = {finiteness; Nash functions and Nash sets; semialgebraic sets; Nash manifolds with corners; extension; normal crossings at a point; normal crossings divisor; Finiteness; Nash functions and Nash sets; semialgebraic sets; Nash manifolds with corners; extension; normal crossings at a point; normal crossing divisor.},
language = {eng},
number = {3},
pages = {537-570},
publisher = {European Mathematical Society Publishing House},
title = {Finiteness problems on Nash manifolds and Nash sets},
url = {http://eudml.org/doc/277463},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Fernando, José F.
AU - Gamboa, José Manuel
AU - Ruiz, Jesús M.
TI - Finiteness problems on Nash manifolds and Nash sets
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 3
SP - 537
EP - 570
AB - We study here several finiteness problems concerning affine Nash manifolds $M$ and Nash subsets $X$. Three main results are: (i) A Nash function on a semialgebraic subset $Z$ of $M$ has a Nash extension to an open semialgebraic neighborhood of $Z$ in $M$, (ii) A Nash set $X$ that has only normal crossings in $M$ can be covered by finitely many open semialgebraic sets $U$ equipped with Nash diffeomorphisms $(u_1,\dots ,u_m):U\rightarrow \mathbb {R}^m$ such that $U\cap X=\lbrace u_1\cdots u_r=0\rbrace $, (iii) Every affine Nash manifold with corners $N$ is a closed subset of an affine Nash manifold $M$ where the Nash closure of the boundary $\partial N$ of $N$ has only normal crossings and $N$ can be covered with finitely many open semialgebraic sets $U$ such that each intersection $N\cap U=\lbrace u_1\ge 0,\dots u_r\ge 0\rbrace $ for a Nash diffeomorphism $(u_1,\dots ,u_m):U\rightarrow \mathbb {R}^m$.
LA - eng
KW - finiteness; Nash functions and Nash sets; semialgebraic sets; Nash manifolds with corners; extension; normal crossings at a point; normal crossings divisor; Finiteness; Nash functions and Nash sets; semialgebraic sets; Nash manifolds with corners; extension; normal crossings at a point; normal crossing divisor.
UR - http://eudml.org/doc/277463
ER -