Finiteness problems on Nash manifolds and Nash sets

Fernando, José F., Gamboa, José Manuel, and Ruiz, Jesús M.. "Finiteness problems on Nash manifolds and Nash sets." Journal of the European Mathematical Society 016.3 (2014): 537-570. <http://eudml.org/doc/277463>.

@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 -