# Finiteness problems on Nash manifolds and Nash sets

José F. Fernando; José Manuel Gamboa; Jesús M. Ruiz

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 3, page 537-570
- ISSN: 1435-9855

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topFernando, 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 -

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