# Finiteness problems on Nash manifolds and Nash sets

### José F. Fernando

Universidad Complutense de Madrid, Spain### José Manuel Gamboa

Universidad Complutense de Madrid, Spain### Jesús M. Ruiz

Universidad Complutense de Madrid, Spain

## 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 $(u_1,\dots,u_m):U\to\mathbb R^m$ such that $U\cap X=\{u_1\cdots u_r=0\}$, (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 $N\cap U=\{u_1\ge0,\dots u_r\ge0\}$ for a Nash diffeomorphism $(u_1,\dots,u_m):U\to\mathbb R^m$.

## Cite this article

José F. Fernando, José Manuel Gamboa, Jesús M. Ruiz, Finiteness problems on Nash manifolds and Nash sets. J. Eur. Math. Soc. 16 (2014), no. 3, pp. 537–570

DOI 10.4171/JEMS/439