Finiteness problems on Nash manifolds and Nash sets

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 $N\cap U=\{u_1\ge 0,\dots u_r\ge 0\}$ for a Nash diffeomorphism $(u_1,\dots ,u_m):U\to {\mathbb{R}}^m$.