Full characterization of the bifurcation set B(F) for semialgebraic maps

Determine a complete characterization of the bifurcation set B(F) for a semialgebraic map F: R^n → R^m, where B(F) denotes the set of points t ∈ R^m at which F fails to be locally trivial; that is, t ∈ B(F) if and only if there does not exist a neighborhood U of t and a diffeomorphism h: F^{-1}(t) × U → F^{-1}(U) satisfying F ∘ h = pr_2.

Background

For a semialgebraic map F: R^n → R^m, local triviality at a point t means there is a neighborhood U of t and a diffeomorphism h: F^{-1}(t) × U → F^{-1}(U) such that F ∘ h equals the projection to U. The bifurcation set B(F) collects those points where such local triviality fails. It is known that B(F) consists of singular values together with so-called bifurcation values at infinity.

Several frameworks provide upper bounds for B(F). Rabier’s asymptotic critical values K∞(F) satisfy B(F) ⊂ F(Sing F) ∪ K∞(F). Other constructions, such as S(F), yield B(F) ⊂ F(Sing F) ∪ S(F). Stratified variants for restrictions f = F|_X to semialgebraic sets X, including K(f, W) and S(f, W), similarly provide inclusions B(f) ⊂ K(f, W) or B(f) ⊂ S(f, W). Despite these bounds, an exact, necessary-and-sufficient description of B(F) remains unresolved.