Simultaneous resolution of singularities in the Nash category: finiteness and effectiveness
In this paper we present new proofs using real spectra of the finiteness theorem on Nash trivial simultaneous resolution and the finiteness theorem on Blow-Nash triviality for isolated real algebraic singularities. That is, we prove that a family of Nash sets in a Nash manifold indexed by a semialgebraic set always admits a Nash trivial simultaneous resolution after a partition of the parameter space into finitely many semialgebraic pieces and in the case of isolated singularities it admits a finite Blow-Nash trivialization. We also complement the finiteness results with recursive bounds.
KeywordsNash manifolds Resolution Singularities Real spectrum Blow-Nash
Mathematics Subject Classification14P10 14P20
I would like to thank Michel Coste for his advice during the preparation of this paper. I also would like to thank the anonymous referees for reading carefully the previous version and for their remarkable suggestions.