JournalsjemsVol. 16, No. 3pp. 537–570

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
Finiteness problems on Nash manifolds and Nash sets cover
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Abstract

We study here several finiteness problems concerning affine Nash manifolds MM and Nash subsets XX. Three main results are: (i) A Nash function on a semialgebraic subset ZZ of MM has a Nash extension to an open semialgebraic neighborhood of ZZ in MM, (ii) A Nash set XX that has only normal crossings in MM can be covered by finitely many open semialgebraic sets UU equipped with Nash diffeomorphisms (u1,,um):URm(u_1,\dots,u_m):U\to\mathbb R^m such that UX={u1ur=0}U\cap X=\{u_1\cdots u_r=0\}, (iii) Every affine Nash manifold with corners NN is a closed subset of an affine Nash manifold MM where the Nash closure of the boundary N\partial N of NN has only normal crossings and NN can be covered with finitely many open semialgebraic sets UU such that each intersection NU={u10,ur0}N\cap U=\{u_1\ge0,\dots u_r\ge0\} for a Nash diffeomorphism (u1,,um):URm(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