Nash uniformization of chessboard sets by Nash manifolds with corners

TL;DR

The paper develops a unified framework for Nash uniformization of d-dimensional closed chessboard sets by Nash manifolds with corners, leveraging Hironaka-type desingularization and introducing drilling blow-ups to handle non-compact or face-deleted cases. It first handles closed chessboard sets, proving that each can be modeled by a compact Nash manifold with corners Q with a proper map f onto the Zariski closure, such that the target S is the f-image of Q outside a small exceptional set. To extend to general chessboard and semialgebraic sets, the authors introduce Nash quasi-manifolds with corners and the drilling blow-up, allowing iterative reduction of boundary complexity while preserving connected/irreducible component structure. The results unify desingularization and uniformization in a Nash setting, providing compactifications, a robust framework for approximations, and a decomposition of semialgebraic sets into Nash quasi-manifolds with corners, with potential applications to semialgebraic compactification and first-pass contextualization in computational settings. The methods combine explicit local models, strict transforms, and boundary stratifications to achieve global Nash-diffeomorphism-type uniformizations, yielding practical semialgebraic representations of otherwise singular objects and enabling controlled approximation within Nash categories.

Abstract

Bierstone and Parusiński 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 ${\mathcal S}$.

Figures (8)

• Figure 1.1: The closed chessboard sets ${\EuScript S}$ (left) and $f^{-1}({\EuScript S})$ (right).
• Figure 2.1: The teardrop.
• Figure 3.1: Local structure of the drilling blow-up $\widetilde{M}_+$ of $M$ of center $N$ (figure borrowed from fe3).
• Figure 4.1: A general checkerboard set ${\EuScript S}$ (left) and a closed checkerboard set ${\EuScript T}$ (right).
• Figure 4.2: Closed checkerboard set ${\EuScript S}$. Set of points with $e=3$ (cyan), set of points with $e=2$ (red) and set of points with $e=0$ (grey).

Theorems & Definitions (46)

• Theorem 1.1: Hironaka's desingularization
• Remark 1.2
• Theorem 1.3: Hironaka's embedded desingularization
• Theorem 1.4: Desingularization of closed semialgebraic sets, bp
• Theorem 1.6: Nash uniformization of closed chessboard sets
• Theorem 1.8: Nash uniformization of chessboard sets
• Theorem 1.9
• Corollary 1.10: Nash uniformization of semialgebraic sets
• Remark 1.11
• Theorem 2.1: fgr