Nash trivial simultaneous resolution for a family of zero-sets of Nash mappings
Let \(F(x;t)\) \((t\in J)\) be a family of Nash mappings defined over a compact Nash manifold or a family of Nash map-germs, where the parameter space \(J\) is a semialgebraic set in some Euclidean space. We prove the following finiteness theorems for the family of zero-sets \(F^{-1}(0)\): (i) There is a finite partition of \(J\) into Nash manifolds such that the family of zero-sets \(F^{-1}(0)\) admits a Nash trivial simultaneous resolution over each Nash manifold. (ii) Suppose that the dimension of the source space is at least 3. Then there is a finite partition of \(J\) into Nash manifolds such that the family of zero-sets \(F^{-1}(0)\) admits a Blow-semialgebraic trivialization along each Nash manifold.
scientific article; zbMATH DE number 1488559