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 and Nash subsets . Three main results are: (i) A Nash function on a semialgebraic subset of has a Nash extension to an open semialgebraic neighborhood of in , (ii) A Nash set that has only normal crossings in can be covered by finitely many open semialgebraic sets equipped with Nash diffeomorphisms such that , (iii) Every affine Nash manifold with corners is a closed subset of an affine Nash manifold where the Nash closure of the boundary of has only normal crossings and can be covered with finitely many open semialgebraic sets such that each intersection for a Nash diffeomorphism .

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