Nash uniformization of chessboard sets by Nash manifolds with corners

Bierstone and Parusi\'nski studied the desingularization of $d$-dimensional closed subanalytic sets and in particular of $d$-dimensional closed semialgebraic sets. Their main tools are Hironaka's desingularization of real algebraic sets (to `uniform' the Zariski closure of the closed semialgebraic set) and Hironaka's embedded desingularization of real algebraic subsets of non-singular real algebraic sets (to uniform afterwards the Zariski closure of the boundary of the uniformed closed semialgebraic set). The obtained models in the desingularization process, that we call in the following closed chessboard sets, are the closures of (finite) unions of connected components of the complements of normal-crossings divisors of non-singular real algebraic sets. The local models for $d$-dimensional chessboard sets are unions of (standard) closed orthants of ${\mathbb R}^d$, that is, $\bigcup_{(\varepsilon_1,\ldots,\varepsilon_d)\in{\mathfrak F}}\{\varepsilon_1{\tt x}_1\geq0,\ldots,\varepsilon_d{\tt x}_d\geq0\}\subset{\mathbb R}^d$ for some set ${\mathfrak F}\subset\{-1,1\}^d$. We study the Nash uniformization of $d$-dimensional closed chessboard sets ${\mathcal S}$ using Nash manifolds with corners ${\mathcal Q}$ with the same number of connected components as ${\mathcal S}$ (or equivalently the same number of irreducible components). Nash manifolds with corners are closed chessboard set whose local models are either ${\mathbb R}^d$ or semialgebraic sets of the type $\{{\tt x}_1\geq0,\ldots,{\tt x}_k\geq0\}$ for some $1\leq k\leq d$. More generally, a chessboard set is a semialgebraic set in between a finite union of connected components of the complement of a normal-crossings divisor of non-singular real algebraic set and its closure. We also provide a Nash uniformization result for general chessboard sets ${\mathcal S}$.