Strict $\mathcal{C}^p$-triangulations – a new approach to desingularization

Abstract

Let R be any real closed field expanded by some o-minimal structure. Let f: A\to R^d be a definable and continuous mapping defined on a definable, closed, bounded subset A of R^n . Let \mathcal{E} be a finite family of definable subsets of R^n contained in A . Let p be any positive integer. We prove that then there exists a finite simplicial complex \mathcal{T} in R^n and a definable homeomorphism h: |\mathcal{T} |\to A , where |\mathcal{T}|:= \bigcup \mathcal{T} , such that for each simplex \varDelta\in \mathcal{T} , the restriction of h to its relative interior \mathring{\varDelta} is a \mathcal{C}^p -embedding of \mathring{\varDelta} into R^n and moreover both h and f\circ h are of class \mathcal{C}^p in the sense that they have definable \mathcal{C}^p -extensions defined on an open definable neighborhood of | \mathcal{T}| in R^n . We then call a pair ( \mathcal{T}, h) a strict \mathcal{C}^p -triangulation of A . In addition, this triangulation can be made compatible with \mathcal{E} in the sense that for each E\in \mathcal{E} , h^{-1}(E) is a union of some \mathring{\varDelta} , where \varDelta\in \mathcal{T} . We also give an application to approximation theory.