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 $\Delta \in \mathcal{T}$, the restriction of $h$ to its relative interior $\mathring{\Delta }$ is a ${\mathcal{C}}^p$-embedding of $\mathring{\Delta }$ 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 $\mathring{\Delta }$, where $\Delta \in \mathcal{T}$. We also give an application to approximation theory.