Finiteness of semialgebraic types of Nash mappings defined on a Nash surface (Q2312817)

We say that two polynomial mappings \(f , g : \mathbb{R}^m \to \mathbb{R}^n\) are topologically equivalent if there are homeomorphisms \(\sigma : \mathbb{R}^m \to \mathbb{R}^m\) and \(\tau : \mathbb{R}^n \to \mathbb{R}^n\) such that \(\tau \circ f = g \circ \sigma.\) We say that two polynomial mappings \(f , g : \mathbb{R}^m \to \mathbb{R}^n\) are semialgebraically equivalent if we can take the \(\sigma\) and \(\tau\) as semialgebraic homeomorphisms. The main two results in the paper under review are as follows. Theorem 1. Let \(\{ f_t : V \to N \ | \ t \in J \}\) be a Nash family of Nash mappings from a Nash surface \(V\) in \(\mathbb{R}^m\) to a nonsingular Nash variety \(N\) in \(\mathbb{R}^n\) with the parameter space \(J\) which is a semialgebraic set in \(\mathbb{R}^a.\) Suppose that \(V\) has isolated singularities. Then there exists a finite partition of \(J\) into Nash open simplices \(J = Q_1 \cup \cdots \cup Q_u\) such that \[ \{ f_t : V \to N \ | \ t \in Q_i \} \] is semialgebraically trivial over each \(Q_i.\) Here a Nash open simplex means a Nash manifold which is Nash diffeomorphic to an open simplex in some Euclidean space. Theorem 2. There exist a real algebraic surface \(X\) with a one-dimensional singular locus and a polynomial family of polynomial mappings from \(X\) to \(\mathbb{R}^2\) such that the topological types appearing in the family have the cardinal number of the continuum.