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In this paper we present new proofs using real spectra of the finitenesstheorem on Nash trivial simultaneous resolution and the finiteness theorem onBlow-Nash triviality for isolated real algebraic singularities. That is, weprove that a family of Nash sets in a Nash manifold indexed by a semialgebraicset always admits a Nash trivial simultaneous resolution after a partition ofthe parameter space into finitely many semialgebraic pieces and in the case ofisolated singularities it admits a finite Blow-Nash trivialization. We alsocomplement the finiteness results with recursive bounds.

Simultaneous resolution of singularities in the Nash category: finiteness and effectiveness