# Desingularization of semialgebraic sets. Applications

Antonio Carbone  Dipartimento di Matematica, Via Sommarive, 14, Università di
Trento, 38123 Povo (ITALY<EMAIL_ADDRESS>and  José F. Fernando
Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias
Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040
MADRID (SPAIN<EMAIL_ADDRESS>
(Date: 13/07/2023)

###### Abstract.

Hironaka’s resolution of singularities of algebraic varieties (over a field of
characteristic $0$) is a widespread celebrated discipline that has many
applications in many areas of Mathematics. In this work we approach the
desingularization of semialgebraic sets
${\mathcal{S}}\subset{\mathbb{R}}^{n}$. To that end, we prove the counterpart
to ‘Hironaka’s desingularization of real algebraic sets’ in the semialgebraic
setting. In particular, it involves Nash manifolds with corners instead of
non-singular real algebraic sets. The obtained results are optimal when
${\mathcal{S}}$ is a closed semialgebraic set. The main tools involved are:
Hironaka’s desingularization of real algebraic sets, Hironaka’s embedded
desingularization of real algebraic subsets of non-singular real algebraic
sets and drilling blow-ups of Nash manifolds with centers closed Nash
submanifolds (which were developed in a former work of the second author).

We also show how to ‘build’ a Nash manifold with corners
${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ from a suitable Nash manifold
$M\subset{\mathbb{R}}^{n}$ (of its same dimension), which contains
${\mathcal{Q}}$ as a closed subset, by folding $M$ along the irreducible
components of a normal-crossings divisor of $M$, which is the smallest Nash
subset of $M$ that contains the boundary $\partial{\mathcal{Q}}$ of
${\mathcal{Q}}$. In addition, one can choose as the Nash manifold $M$ the Nash
‘double’ $D({\mathcal{Q}})$ of ${\mathcal{Q}}$, which is analogous to the Nash
double of a Nash manifold with (smooth) boundary, but $D({\mathcal{Q}})$ takes
into account the peculiarities of the boundary of a Nash manifold with
corners.

We include several applications of the previous two results: (1) Weak
desingularization of closed semialgebraic sets using Nash manifolds with
(smooth) boundary, (2) Representation of compact semialgebraic sets connected
by analytic paths as images under Nash maps of closed unit balls, (3) Explicit
construction of Nash models for compact orientable smooth surfaces of genus
$g\geq 0$, and (4) Nash approximation of continuous semialgebraic maps whose
target spaces are Nash manifolds with corners.

###### Key words and phrases:

Desingularization, semialgebraic sets, Nash manifolds with corners, folding
Nash manifolds, Nash manifolds with boundary, Nash maps and images,
semialgebraic sets connected by analytic paths, approximation.

###### 2010 Mathematics Subject Classification:

Primary: 14P10, 14P20, 14E15; Secondary: 14B05, 57R12, 57Q55.

The first author is supported by GNSAGA of INDAM

The second author is supported by Spanish STRANO PID2021-122752NB-I00 and
Grupos UCM 910444. This article has been developed during several one month
research stays of the second author in the Dipartimento di Matematica of the
Università di Trento. The second author would like to thank the department for
the invitations and the very pleasant working conditions.

##  1\. Introduction

Hironaka’s resolution of singularities [Hi1] on an algebraic variety (over a
field of characteristic $0$) is a widespread celebrated discipline that has
many applications in many areas of Mathematics [Ko, L]. It has also been
developed by many other authors in the analytic and subanalytic cases (we
refer to Abhyankar [A1, A2, A3], Bierstone-Milman [BM1, BM3, BM2], Villamayor
[Vi], Zariski [Z1, Z2, Z3] as an example). For simplicity we restrict to the
case when the ground field is ${\mathbb{R}}$. The general approach consists of
the following: Given a real algebraic set $X\subset{\mathbb{R}}^{n}$, one
finds a non-singular real algebraic set $X^{\prime}\subset{\mathbb{R}}^{m}$
together with a proper polynomial map $f:X^{\prime}\to X$ that is a biregular
diffeomorphism outside the set of singular points of $X$. We next recall the
precise statement:

######  Theorem 1.1 (Hironaka’s desingularization).

Let $X\subset{\mathbb{R}}^{n}$ be a real algebraic set. Then there exist a
non-singular real algebraic set $X^{\prime}\subset{\mathbb{R}}^{m}$ and a
proper polynomial map $f:X^{\prime}\to X$ such that the restriction
$f|_{X^{\prime}\setminus f^{-1}(\operatorname{Sing}(X))}:X^{\prime}\setminus
f^{-1}(\operatorname{Sing}(X))\to X\setminus\operatorname{Sing}(X)$ is a
diffeomorphism whose inverse map is a regular map.

######  Remark 1.2.

If $X$ is pure dimensional, $X\setminus\operatorname{Sing}(X)$ is dense in $X$
and, as $f$ is proper, it is surjective.

A related important construction (which will also be very useful for our
purposes) is Hironaka’s embedded resolution of singularities [Hi1], which
involves the concept of normal-crossings divisor. Let $X\subset
Y\subset{\mathbb{R}}^{n}$ be real algebraic sets such that $Y$ is non-singular
and has dimension $d$. Recall that $X$ is a normal-crossings divisor of $Y$ if
the irreducible components of $X$ are non-singular, have codimension $1$ in
$Y$ and constitute a transversal family. More precisely, for each point $x\in
X$ there exists a regular system of parameters ${\tt x}_{1},\ldots,{\tt
x}_{d}$ for $Y$ at $x$ such that $X$ is given on an open Zariski neighborhood
of $x$ in $Y$ by the equation ${\tt x}_{1}\cdots{\tt x}_{k}=0$ for some $k\leq
d$.

######  Theorem 1.3 (Hironaka’s embedded desingularization).

Let $X\subset Y\subset{\mathbb{R}}^{n}$ be real algebraic sets such that $Y$
is non-singular. Then there exists a non-singular real algebraic set
$Y^{\prime}\subset{\mathbb{R}}^{m}$ and a proper surjective polynomial map
$g:Y^{\prime}\to Y$ such that $g^{-1}(X)$ is a normal-crossings divisor of
$Y^{\prime}$ and the restriction $g|_{Y^{\prime}\setminus
g^{-1}(X)}:Y^{\prime}\setminus g^{-1}(X)\to Y\setminus X$ is a diffeomorphism
whose inverse map is a regular map.

###  1.1. Semialgebraic setting

In Real Algebraic Geometry not only algebraic equalities, but also algebraic
inequalities appear, and it seems reasonable to explore until what extent it
is possible to prove similar results to Hironaka’s desingularization (Theorem
1.1) in the context of semialgebraic sets. Recall that a subset
${\mathcal{S}}\subset{\mathbb{R}}^{n}$ is semialgebraic when it admits a
description in terms of a finite boolean combination of polynomial equalities
and inequalities, which we will call a semialgebraic description. The category
of semialgebraic sets is closed under basic boolean operations but also under
usual topological operations: taking closures (denoted by
$\operatorname{Cl}(\cdot)$), interiors (denoted by
$\operatorname{Int}(\cdot)$), connected components, etc. If
${\mathcal{S}}\subset{\mathbb{R}}^{m}$ and
${\mathcal{T}}\subset{\mathbb{R}}^{n}$ are semialgebraic sets, a map
$f:{\mathcal{S}}\to{\mathcal{T}}$ is semialgebraic if its graph is a
semialgebraic set. A Nash map on an open semialgebraic set
$U\subset{\mathbb{R}}^{n}$ is a semialgebraic smooth map
$f:U\to{\mathbb{R}}^{m}$. Given a semialgebraic set
${\mathcal{S}}\subset{\mathbb{R}}^{n}$, a Nash map on ${\mathcal{S}}$ is the
restriction to ${\mathcal{S}}$ of a Nash map $F:U\to{\mathbb{R}}^{m}$ on an
open semialgebraic neighborhood $U\subset{\mathbb{R}}^{n}$ of ${\mathcal{S}}$.
A Nash subset $X\subset M$ of a Nash manifold $M\subset{\mathbb{R}}^{n}$ (that
is, a semialgebraic set that is a smooth submanifold of ${\mathbb{R}}^{n}$) is
the zero set of a Nash function $f:M\to{\mathbb{R}}$, whereas the Nash closure
in $M$ of a semialgebraic set ${\mathcal{S}}\subset M$ is the smallest Nash
subset $X$ of $M$ that contains ${\mathcal{S}}$. In this setting, a Nash
normal-crossings divisor of a Nash manifold $M$ is a Nash set $X\subset M$
whose irreducible components are Nash manifolds of codimension $1$ in $M$ and
constitute a transversal family.

###  1.2. Desingularization of semialgebraic sets by Nash manifolds with
corners.

A first step to approach desingularization of semialgebraic sets concerns the
choice of a suitable class of ‘non-singular semialgebraic domains’ to
represent any semialgebraic set as the image under a proper polynomial map of
one of such domains. As semialgebraic sets
${\mathcal{S}}\subset{\mathbb{R}}^{n}$ may have boundaries, which are
semialgebraic sets of smaller dimension, we must be careful with the choice of
the class of ‘non-singular semialgebraic domains’. A lighter approach of the
desingularization of semialgebraic sets can be found in [BP] (see Remark 1.5
for a more detailed explanation).

As we would like that the involved maps are polynomial and proper, we find
additional rigidity. To lighten the initial approach to the problem, we begin
with the resolution of closed semialgebraic sets and we use as ‘non-singular
semialgebraic domains’ Nash manifolds with corners, that is, semialgebraic
sets that are smooth submanifolds with corners of ${\mathbb{R}}^{n}$. This
choice takes into account not only the fact that a semialgebraic set may have
boundary, but also that the Zariski closure of such boundary may have
different irreducible components.

Recall that a Nash manifold with corners
${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ is contained as a closed
(semialgebraic) subset of a Nash manifold $M\subset{\mathbb{R}}^{n}$ of its
same dimension [FGR, Prop.1.2]. In fact, we restrict our scope to Nash
manifolds ${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ with corners such that the
Nash closure in $M$ of the boundary $\partial{\mathcal{Q}}$ is a (Nash)
normal-crossings divisor of $M$ (maybe after shrinking $M$). Connected Nash
manifolds with corners are connected by analytic paths [Fe2, Main Thm.1.4,
Lem.8.2, Lem.8.3], so their images under Nash maps (in particular polynomial
or regular maps) are connected by analytic paths as a consequence of [Fe2,
Main Thm.1.4]. As in Hironaka’s results (Theorems 1.1 and 1.3), we would like
that each connected component of the ‘non-singular semialgebraic domains’
covers the largest possible part of the (closed) semialgebraic set
${\mathcal{S}}\subset{\mathbb{R}}^{n}$ we want to desingularize. Using the
analytic path-connected components of a semialgebraic set
${\mathcal{S}}\subset{\mathbb{R}}^{n}$ (Theorem 2.2), we reduce the problem of
desingularization to the case of semialgebraic sets connected by analytic
paths (Corollary 2.3).

######  Theorem 1.4 (Desingularization of closed semialgebraic sets).

Let ${\mathcal{S}}\subset{\mathbb{R}}^{m}$ be a $d$-dimensional closed
semialgebraic set connected by analytic paths and let
$\overline{{\mathcal{S}}}^{\operatorname{zar}}$ be its Zariski closure. Then
there exist:

  * (i)

A $d$-dimensional irreducible non-singular real algebraic set
$X\subset{\mathbb{R}}^{n}$ and a normal-crossings divisor $Y\subset X$.

  * (ii)

A connected Nash manifold with corners ${\mathcal{Q}}\subset X$ (which is a
closed subset of $X$) whose boundary $\partial{\mathcal{Q}}$ has $Y$ as its
Zariski closure.

  * (iii)

A polynomial map $f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{m}$ such that the
restrictions $f|_{X}:X\to\overline{{\mathcal{S}}}^{\operatorname{zar}}$ and
$f|_{{\mathcal{Q}}}:{\mathcal{Q}}\to{\mathcal{S}}$ are proper and
$f({\mathcal{Q}})={\mathcal{S}}$.

  * (iv)

A closed semialgebraic set ${\mathcal{R}}\subset{\mathcal{S}}$ of dimension
strictly smaller than $d$ such that ${\mathcal{S}}\setminus{\mathcal{R}}$ and
${\mathcal{Q}}\setminus f^{-1}({\mathcal{R}})$ are Nash manifolds and the
polynomial map $f|_{{\mathcal{Q}}\setminus
f^{-1}({\mathcal{R}})}:{\mathcal{Q}}\setminus
f^{-1}({\mathcal{R}})\to{\mathcal{S}}\setminus{\mathcal{R}}$ is a Nash
diffeomorphism.

In particular, if $\overline{{\mathcal{S}}}^{\operatorname{zar}}$ is compact,
also $X$ is compact.

######  Remark 1.5.

Theorem 1.4 is related to a previous result of Bierstone-Parusiński [BP,
Rmk.2.6] concerning desingularization of semialgebraic sets (and also of
subanalytic sets). However, there are two main differences between their
result and ours:

  * (1)

The ‘non-singular (closed) semialgebraic domain’
$\operatorname{Cl}(f^{-1}({\mathcal{S}}\setminus{\mathcal{R}}))\subset X$ that
maps onto the semialgebraic set ${\mathcal{S}}\subset{\mathbb{R}}^{m}$ under
the polynomial map $f$ does not need to be a Nash manifold with corners in
[BP, Rmk.2.6], but only the closure of a union of connected components of
$X\setminus Y$.

  * (2)

If ${\mathcal{S}}$ is connected by analytic paths,
$\operatorname{Cl}(f^{-1}({\mathcal{S}}\setminus{\mathcal{R}}))\subset X$ does
not need to be connected in [BP, Rmk.2.6].

As Nash manifolds with corners are locally compact, their images under proper
maps are also locally compact, so Theorem 1.4 cannot be extended directly to
general semialgebraic sets. In order to solve this, we will use Nash quasi-
manifolds with corners, which are essentially Nash manifolds with corners with
some ‘faces’ erased. More precisely, we construct a natural semialgebraic
partition ${\mathfrak{S}}({\mathcal{Q}})$ of any Nash manifold with corners
${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ that takes into account the structure
of its boundary (Definition 4.11). A semialgebraic set
${\mathcal{T}}\subset{\mathbb{R}}^{n}$ is a Nash quasi-manifold with corners
if ${\mathcal{Q}}:=\operatorname{Cl}({\mathcal{T}})$ is a Nash manifold with
corners and ${\mathcal{Q}}\setminus{\mathcal{T}}$ is a union of elements of
${\mathfrak{S}}({\mathcal{Q}})$. In this case, it is necessary to substitute
(proper) polynomial maps by (proper) Nash maps. We explain in Example 4.14 why
this additional change is mandatory.

######  Theorem 1.6 (Desingularization of semialgebraic sets).

Let ${\mathcal{S}}\subset{\mathbb{R}}^{m}$ be a $d$-dimensional semialgebraic
set connected by analytic paths. Then there exist:

  * (i)

A $d$-dimensional connected compact (irreducible) non-singular real algebraic
set $X\subset{\mathbb{R}}^{n}$ and a normal-crossings divisor $Y\subset X$.

  * (ii)

A connected Nash quasi-manifold with corners ${\mathcal{S}}^{\bullet}\subset
X$ whose closure in $X$ is a compact connected Nash manifold with corners
${\mathcal{Q}}^{\bullet}\subset X$ and whose boundary
$\partial{\mathcal{Q}}^{\bullet}$ has $Y$ as its Zariski closure.

  * (iii)

A Nash map $f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{m}$ such that the restriction
$f|_{{\mathcal{S}}^{\bullet}}:{\mathcal{S}}^{\bullet}\to{\mathcal{S}}$ is
proper and $f({\mathcal{S}}^{\bullet})={\mathcal{S}}$.

  * (iv)

A closed semialgebraic set ${\mathcal{R}}\subset{\mathcal{S}}$ of dimension
strictly smaller than $d$ such that ${\mathcal{S}}\setminus{\mathcal{R}}$ and
${\mathcal{S}}^{\bullet}\setminus f^{-1}({\mathcal{R}})$ are Nash manifolds
and the Nash map $f|_{{\mathcal{S}}^{\bullet}\setminus
f^{-1}({\mathcal{R}})}:{\mathcal{S}}^{\bullet}\setminus
f^{-1}({\mathcal{R}})\to{\mathcal{S}}\setminus{\mathcal{R}}$ is a Nash
diffeomorphism.

####  1.2.1. Drilling blow-up

Apart from Hironaka’s desingularization (Theorems 1.1 and 1.3), a main tool to
prove Theorems 1.4 and 1.6 is the drilling blow-up of a Nash manifold $M$ with
center a closed Nash submanifold $N$, see [Fe2]. We refer the reader to [S,
Hi2] for the oriented blow-up of a real analytic space with center a closed
subspace, which is the counterpart of the construction in [Fe2] for the real
analytic setting. In [HPV, §5] appears a presentation of the oriented blow-up
in the analytic case closer to the drilling blow-up described in [Fe2]. There
the authors consider the case of the oriented blow-up of a real analytic
manifold $M$ with center a closed real analytic submanifold $N$ whose
vanishing ideal inside $M$ is finitely generated (this happens for instance if
$N$ is compact). In [Fe1, §3] a similar construction is presented in the
semialgebraic setting, which is used to ‘appropriately embed’ semialgebraic
sets in affine spaces. In [FFQU, §3] oriented blow-ups appear to compare
regoluous images of ${\mathbb{R}}^{2}$ with regular images of
${\mathbb{R}}^{2}$. In Section 3 we recall the main properties of drilling
blow-up developed in [Fe2] and we improve some of them in order to prove
Theorems 1.4 and 1.6.

###  1.3. Folding Nash manifolds to build Nash manifolds with corners

Recall that a Nash manifold ${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ with
corners is contained as a closed (semialgebraic) subset of a Nash manifold
$M\subset{\mathbb{R}}^{n}$ of its same dimension and the Nash closure of its
boundary $\partial{\mathcal{Q}}$ is a Nash normal-crossings divisor of $M$.
Our next result shows that the Nash manifold $M$ can be ‘folded’ to
reconstruct the Nash manifold with corners ${\mathcal{Q}}$. Namely,

######  Theorem 1.7 (Folding Nash manifolds).

Let ${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ be a $d$-dimensional Nash manifold
with corners. Then there exist

  * (i)

A $d$-dimensional Nash manifold $M\subset{\mathbb{R}}^{n}$ that contains
${\mathcal{Q}}$ as a closed subset.

  * (ii)

A Nash normal-crossings divisor $Y\subset M$ that is the Nash closure of
$\partial{\mathcal{Q}}$ in $M$ and satisfies ${\mathcal{Q}}\cap
Y=\partial{\mathcal{Q}}$.

  * (iii)

A proper Nash map $f:\operatorname{Cl}(M)\to{\mathcal{Q}}$ such that
$f|_{\mathcal{Q}}:{\mathcal{Q}}\to{\mathcal{Q}}$ is a semialgebraic
homeomorphism close to the identity map and
$f|_{\operatorname{Int}({\mathcal{Q}})}:\operatorname{Int}({\mathcal{Q}})\to\operatorname{Int}({\mathcal{Q}})$
is a Nash diffeomorphism.

In addition, for each $x\in\partial{\mathcal{Q}}$ there exist open
semialgebraic neighborhoods $U,V\subset M$ of $x$ equipped with Nash
diffeomorphisms $\varphi:U\to{\mathbb{R}}^{d}$ and $\psi:V\to{\mathbb{R}}^{d}$
and $1\leq s\leq d$ such that

$\psi\circ f\circ\varphi^{-1}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d},\
(x_{1},\ldots,x_{d})\mapsto(x_{1}^{2},\ldots,x_{s}^{2},x_{s+1},\ldots,x_{d}).$

In addition, we construct for Nash manifolds with corners
${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ the counterpart of the Nash double of a
Nash manifold with (smooth) boundary and such Nash double $D({\mathcal{Q}})$
is Nash diffeomorphic around ${\mathcal{Q}}$ to the Nash manifold $M$ provided
by Theorem 1.7 (Theorem 5.13). As a straightforward consequence of Theorems
1.4 and 5.13 and Lemma 2.8, the reader can prove the following:

######  Corollary 1.8 (Nash ramified covering map).

Let ${\mathcal{S}}\subset{\mathbb{R}}^{m}$ be a $d$-dimensional closed
semialgebraic set connected by analytic paths. Then there exist:

  * (i)

A $d$-dimensional irreducible non-singular real algebraic set
$X\subset{\mathbb{R}}^{n}$.

  * (ii)

A polynomial map $f:{\mathbb{R}}^{n}\to{\mathbb{R}}^{m}$ such that the
restriction $f|_{C}:C\to{\mathcal{S}}$ is proper and $f(C)={\mathcal{S}}$ for
each connected component $C$ of $X$.

  * (iii)

A closed semialgebraic set ${\mathcal{R}}\subset{\mathcal{S}}$ of dimension
strictly smaller than $d$ such that ${\mathcal{S}}\setminus{\mathcal{R}}$ and
$X\setminus f^{-1}({\mathcal{R}})$ are Nash manifolds and the polynomial map
$f|_{X\setminus f^{-1}({\mathcal{R}})}:X\setminus
f^{-1}({\mathcal{R}})\to{\mathcal{S}}\setminus{\mathcal{R}}$ is a Nash
covering map whose fibers have constant cardinality.

###  1.4. Some further applications

We next propose some further applications of both techniques proposed above
(desingularization of closed semialgebraic sets (connected by analytic paths)
and folding of Nash manifolds to construct Nash manifolds with corners).

####  1.4.1. Weak desingularization of closed semialgebraic sets by Nash
manifolds with boundary.

A similar result to Theorem 1.4 changing the Nash manifold with corners
${\mathcal{Q}}$ by a Nash manifold with boundary ${\mathcal{H}}$ (that is, a
semialgebraic set that is a smooth submanifold with (smooth) boundary of
${\mathbb{R}}^{n}$) seems difficult to be obtained (even in the compact case)
if we want to keep that the map $f$ is polynomial and that the semialgebraic
set ${\mathcal{R}}$ has dimension strictly smaller than the dimension of
${\mathcal{S}}$. We propose the following statement, whose proof is based on
the folding of Nash manifolds (Theorem 1.7):

######  Theorem 1.9 (Weak desingularization of closed semialgebraic sets).

Let ${\mathcal{S}}\subset{\mathbb{R}}^{n}$ be a $d$-dimensional closed
semialgebraic set connected by analytic paths and let $\varepsilon>0$ be a
strictly positive continuous semialgebraic function on ${\mathcal{S}}$. Then
there exist:

  * (i)

A $d$-dimensional irreducible non-singular real algebraic set
$X\subset{\mathbb{R}}^{m}$.

  * (ii)

A Nash manifold with boundary
${\mathcal{H}}_{\varepsilon}\subset{\mathbb{R}}^{m}$ such that the Zariski
closure $Z_{\varepsilon}$ of $\partial{\mathcal{H}}_{\varepsilon}$ is a non-
singular real algebraic set $Z_{\varepsilon}\subset X$ of dimension $d-1$ and
$\operatorname{Int}({\mathcal{H}}_{\varepsilon})$ is a connected component of
$X\setminus Z_{\varepsilon}$.

  * (iii)

A proper Nash map $f:{\mathcal{H}}_{\varepsilon}\to{\mathcal{S}}$ such that
$f({\mathcal{H}}_{\varepsilon})={\mathcal{S}}$.

  * (iv)

The restriction $f|_{{\mathcal{H}}_{\varepsilon}\setminus
f^{-1}({\mathcal{T}}_{\varepsilon})}:{\mathcal{H}}_{\varepsilon}\setminus
f^{-1}({\mathcal{T}}_{\varepsilon})\to{\mathcal{S}}\setminus{\mathcal{T}}_{\varepsilon}$
is a Nash diffeomorphism, where
${\mathcal{T}}_{\varepsilon}:=\\{x\in{\mathcal{S}}:\ {\rm
dist}(x,{\mathcal{R}})\leq\varepsilon\\}$ for a certain closed semialgebraic
set ${\mathcal{R}}\subset{\mathcal{S}}$ of dimension strictly smaller than
$d$.

If ${\mathcal{S}}$ is compact, we may assume in addition: $\varepsilon$ is
constant, $X$ and ${\mathcal{H}}_{\varepsilon}$ are also compact and $X$ is
connected.

Related to the previous result in [Fe2, Proof of Thm.1.4, §8.C.12] it is shown
that if ${\mathcal{S}}\subset{\mathbb{R}}^{n}$ is a semialgebraic set
connected by analytic paths, then there exists a Nash manifold ${\mathcal{H}}$
with smooth boundary and a surjective Nash map
$f:{\mathcal{H}}\to{\mathcal{S}}$. However, the Nash map provided there is far
from being proper even if ${\mathcal{S}}$ is a closed semialgebraic set.

####  1.4.2. Nash images of the closed unit ball

In Section 6.2 we provide an alternative proof of the following result
proposed in [CF].

######  Theorem 1.10 (Compact Nash images).

Let ${\mathcal{S}}\subset{\mathbb{R}}^{n}$ be a $d$-dimensional compact
semialgebraic set. The following assertions are equivalent:

  * (i)

There exists a Nash map $f:{\mathbb{R}}^{d}\to{\mathbb{R}}^{n}$ such that
$f(\overline{{\mathcal{B}}}_{d})={\mathcal{S}}$.

  * (ii)

${\mathcal{S}}$ is connected by analytic paths.

A related (and much more previous in time) question is to determine until what
extent we can represent semialgebraic sets connected by analytic paths using
polynomial maps in the vein of Gamboa’s problem proposed in the 1990
Oberwolfach reelle algebraische Geometrie week [G] (see also [E, §3.IV, p.69]
and [FGU] for a survey concerning this problem):

######  Problem 1.11 (Gamboa’s problem).

To characterize the (semialgebraic) subsets of ${\mathbb{R}}^{n}$ that are
either polynomial or regular images of ${\mathbb{R}}^{m}$.

The reader can confront Theorem 1.10 with partial results [FU, Thm.1.4 &
Thm.1.6] concerning polynomial images of the closed unit balls. Polynomial
images of models connected by polynomial paths (affine spaces, closed balls
etc.) are connected by polynomial paths, which is a stronger condition than
being connected by analytic paths. In general, semialgebraic sets do not
contain rational paths and consequently not even polynomial paths. By [C, V] a
generic complex hypersurface $Z$ of ${\mathbb{C}}{\mathbb{P}}^{m}$ of degree
$d\geq 2m-2$ for $m\geq 4$ and of degree $d\geq 2m-1$ for $m=2,3$ does not
contain rational curves. If ${\mathcal{S}}\subset{\mathbb{R}}^{m}$ is a
semialgebraic set whose Zariski closure in ${\mathbb{R}}{\mathbb{P}}^{m}$ is a
generic hypersurface of high enough degree, then its Zariski closure $Z$ in
${\mathbb{C}}{\mathbb{P}}^{m}$ does not contain rational curves, so
${\mathcal{S}}$ cannot contain rational paths. This means in particular that
general semialgebraic sets (even if they are connected by analytic paths) do
not contain polynomial paths, so they are not polynomial images of either
affine spaces or closed balls. In view of Theorem 1.4 and Corollary 1.8 the
obstruction to solve Problem 1.11 mainly concentrates on non-singular real
algebraic sets, which in general do not contain polynomial paths.

####  1.4.3. Explicit Nash models for orientable compact smooth surfaces

Theorem 1.7 provides an explicit procedure to construct Nash models for
orientable compact smooth surfaces, see §6.3.

####  1.4.4. Approximation of continuous semialgebraic maps by Nash maps

Approximation of maps of certain type by maps of a better subtype is an
important tool in Mathematics. A celebrated example is Whitney’s approximation
theorem [Wh]: a continuous map (defined on a locally compact subset of an
affine space) whose target space is a $\mathcal{C}^{r}$ submanifold $M$ of
${\mathbb{R}}^{n}$ for $r\in{\mathbb{N}}\cup\\{\infty,\omega\\}$ can be
approximated by a $\mathcal{C}^{r}$ map whose target space is $M$. Whitney
approximation theorem has been extended in many directions, like the case of
manifolds with boundary (using partitions of unit and collars, [ORR, Ch. III,
Thm. 6.1]). New results on approximation in Whitney’s style have been proved
in [FGh2] when the target space has ‘singularities’ and admits ‘nice’
triangulations. As a consequence: every continuous map between a locally
compact subset $X\subset{\mathbb{R}}^{m}$ and a smooth manifold with corners
${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ can be approximated by a smooth map
with respecto to the ${\mathcal{C}}^{0}$ topology.

There are also relevant results concerning approximation in the semialgebraic
setting. Efroymson [Ef] showed that every continuous semialgebraic function on
a Nash manifold can be approximated by Nash functions. Shiota [Sh] improved
this result in many directions. He proved relative and absolute versions with
(strong) control on finitely many derivatives of the approximation map.
Approximation results also appear when the target space has singularities. In
[BFR, Thm.1.7] the target space is a Nash set with monomial singularities and
approximation of continuous semialgebraic maps by Nash maps is proved under
certain restrictions. In [FGh1] it is presented differentiable approximation
of continuous semialgebraic maps when the target space admits ‘nice’
triangulations. As a consequence: every continuous semialgebraic map between a
compact semialgebraic set $X\subset{\mathbb{R}}^{m}$ and a Nash manifold with
corners ${\mathcal{Q}}\subset{\mathbb{R}}^{n}$, can be approximated by a
$\mathcal{C}^{r}$ semialgebraic map with respect to the ${\mathcal{C}}^{0}$
semialgebraic topology.

The techniques developed in [FGh1] make an essential use of $\mathcal{C}^{r}$
semialgebraic partitions of unity, so the proofs proposed there do not extend
to Nash approximation. However, as an additional application of Theorem 1.7,
we prove approximation of continuous semialgebraic maps by Nash maps when the
target space is a Nash manifold with corners.

######  Theorem 1.12 (Nash approximation).

Let ${\mathcal{S}}\subset{\mathbb{R}}^{m}$ be a locally compact semialgebraic
set, ${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ a Nash manifold with corners and
$h:{\mathcal{S}}\to{\mathcal{Q}}$ a continuous semialgebraic map. Then there
exist Nash maps $g:{\mathcal{S}}\to{\mathcal{Q}}$ arbitrarily close to $h$
with respect to the ${\mathcal{C}}^{0}$ semialgebraic topology.

###  1.5. Structure of the article

The article is organized as follows. In Section 2 we recall the concepts and
main properties of regular and smooth points of a semialgebraic set. We also
analyze the main properties of Nash manifolds with corners and we refer the
reader to [FGR] for further details. In this section also the concept of
analytic path-connected components of a semialgebraic set appears and we
recall the existence and uniqueness of such components. In addition, we show
that each connected Nash manifold can be understood as a connected component
of an irreducible non-singular algebraic set whose connected components are
pairwise Nash diffeomorphic. We also prove in this section Theorems 1.4 and
1.6 for the $1$-dimensional case. In Section 3 we recall the main properties
of the drilling blow-up of a Nash manifold along a closed Nash submanifold
proposed in [Fe2, §3]. We also present some additional new properties of the
drilling blow-up that we need to prove Theorems 1.4 and 1.6. In Section 4 we
prove Theorem 1.4 and 1.6 (for semialgebraic sets of dimension $d\geq 2$),
whereas in Section 5 we prove Theorem 1.7 and we construct the Nash double of
a Nash manifold with corners (Theorem 5.13). Section 6 is devoted to prove
Theorems 1.9, 1.10 and 1.12. We also present an explicit procedure to
construct Nash models for orientable compact smooth surfaces.

###  1.6. Acknowledgements

The authors are very grateful to E. Bierstone for useful comments and to S.
Schramm for a careful reading of the final version and for the suggestions to
refine its redaction.

##  2\. Preliminary results

In this section we collect some preliminary concepts and results that will be
used freely along this article. We include them for the sake of completeness
and to ease the reading of the article. We refer the reader to [Fe2, FGR] for
further details.

###  2.1. Regular points versus smooth points.

The set $\operatorname{Reg}({\mathcal{S}})$ of regular points of a
semialgebraic set ${\mathcal{S}}\subset{\mathbb{R}}^{n}$ is defined as
follows. Let $X$ be the Zariski closure of ${\mathcal{S}}$ in
${\mathbb{R}}^{n}$ and $\widetilde{X}$ the complexification of $X$, that is,
the smallest complex algebraic subset of ${\mathbb{C}}^{n}$ that contains $X$.
Define $\operatorname{Reg}(X):=X\setminus\operatorname{Sing}(\widetilde{X})$
and let $\operatorname{Reg}({\mathcal{S}})$ be the interior of
${\mathcal{S}}\setminus\operatorname{Sing}(\widetilde{X})$ in
$\operatorname{Reg}(X)$. Observe that $\operatorname{Reg}({\mathcal{S}})$ is a
finite union of disjoint Nash manifolds maybe of different dimensions. We
refer the reader to [Fe2, §2.A] for further details concerning the set of
regular points of a semialgebraic set.

A point $x\in{\mathcal{S}}$ is smooth if there exists an open neighborhood
$U\subset{\mathbb{R}}^{n}$ of $x$ such that $U\cap{\mathcal{S}}$ is a Nash
manifold. It holds that each regular point is a smooth point, but the converse
is not always true even if ${\mathcal{S}}=X$ is a real algebraic set [Fe2,
Ex.2.1]. The set $\operatorname{Sth}({\mathcal{S}})$ of smooth points of a
semialgebraic set ${\mathcal{S}}\subset{\mathbb{R}}^{n}$ is by [St] a
semialgebraic subset of ${\mathbb{R}}^{n}$ (and consequently a union of Nash
submanifolds of ${\mathbb{R}}^{n}$ possibly of different dimension), which
contains $\operatorname{Reg}({\mathcal{S}})$ (maybe as a proper subset as it
happens in [Fe2, Ex.2.1]), and it is open in ${\mathcal{S}}$. The set of
points of dimension $k$ of $\operatorname{Sth}({\mathcal{S}})$ is either the
empty-set or a Nash manifold of dimension $k$ for each $k=0,1,\ldots,d$. In
particular, if ${\mathcal{S}}$ is pure dimensional,
$\operatorname{Sth}({\mathcal{S}})$ is a Nash submanifold of
${\mathbb{R}}^{n}$. If $X$ is a real algebraic set, $\operatorname{Sing}(X)$
is always an algebraic subset of $X$ whereas the set
$X\setminus\operatorname{Sth}(X)$ of non-smooth points is in general only a
semialgebraic subset of $X$ (see [Fe2, Ex.2.1]).

If $X$ is a non-singular real algebraic set of dimension $d$, then
$\operatorname{Reg}(X)=\operatorname{Sth}(X)$. Thus, if ${\mathcal{S}}\subset
X$ is a pure dimensional semialgebraic set of dimension $d$, we have
$\operatorname{Reg}({\mathcal{S}})=\operatorname{Sth}({\mathcal{S}})$, because
the Zariski closure $\overline{{\mathcal{S}}}^{\operatorname{zar}}$ of
${\mathcal{S}}$ is a union of irreducible components of $X$, so it is a non-
singular real algebraic set of dimension $d$.

###  2.2. Analytic path-connected components of a semialgebraic set

We recall next the concept of analytic path-connected components of a
semialgebraic set introduced in [Fe2, §9].

######  Definition 2.1.

A semialgebraic set ${\mathcal{S}}\subset{\mathbb{R}}^{n}$ admits a
decomposition into analytic path-connected components if there exist
semialgebraic sets
${\mathcal{S}}_{1},\ldots,{\mathcal{S}}_{r}\subset{\mathcal{S}}$ such that:

  * (i)

Each ${\mathcal{S}}_{i}$ is connected by analytic paths.

  * (ii)

If ${\mathcal{T}}\subset{\mathcal{S}}$ is a semialgebraic set connected by
analytic paths that contains ${\mathcal{S}}_{i}$, then
${\mathcal{S}}_{i}={\mathcal{T}}$.

  * (iii)

${\mathcal{S}}_{i}\not\subset\bigcup_{j\neq i}S_{j}$.

  * (iv)

${\mathcal{S}}=\bigcup_{i=1}^{r}{\mathcal{S}}_{i}$.

In [Fe2, Thm.9.2] the existence and uniqueness of the analytic path-connected
components of a semialgebraic set is shown.

######  Theorem 2.2 ([Fe2, Thm.9.2]).

Let ${\mathcal{S}}\subset{\mathbb{R}}^{n}$ be a semialgebraic set. Then
${\mathcal{S}}$ admits a decomposition into analytic path-connected components
and this decomposition is unique. In addition, the analytic path-connected
components of a semialgebraic set are closed in ${\mathcal{S}}$.

######  Corollary 2.3.

Let ${\mathcal{S}}\subset{\mathbb{R}}^{n}$ be a semialgebraic set and let
$\\{{\mathcal{S}}_{i}\\}_{i=1}^{r}$ be the analytic path-connected components
of ${\mathcal{S}}$. Consider the semialgebraic set
${\mathcal{S}}^{*}:=\bigcup_{i=1}^{r}{\mathcal{S}}_{i}\times\\{i\\}\subset{\mathbb{R}}^{n+1}$
and the projection
$\pi:{\mathbb{R}}^{n+1}\to{\mathbb{R}}^{n}\,(x,x_{n})\mapsto x$. Then the
restriction $\pi|_{{\mathcal{S}}^{*}}:{\mathcal{S}}^{*}\to{\mathcal{S}}$ is a
surjective proper projection such that
$\pi({\mathcal{S}}_{i}\times\\{i\\})={\mathcal{S}}_{i}$ for $i=1,\ldots,r$.

###  2.3. Nash manifolds with corners.

Let ${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ be a Nash manifold with corners.
The set of internal points of a Nash manifold with corners ${\mathcal{Q}}$ is
$\operatorname{Int}({\mathcal{Q}}):=\operatorname{Sth}({\mathcal{Q}})$. The
boundary $\partial{\mathcal{Q}}$ of ${\mathcal{Q}}$ is
$\partial{\mathcal{Q}}:={\mathcal{Q}}\setminus\operatorname{Int}({\mathcal{Q}})={\mathcal{Q}}\setminus\operatorname{Sth}({\mathcal{Q}})$.
If $\overline{{\mathcal{Q}}}^{\operatorname{zar}}$ is a non-singular real
algebraic set, then
$\operatorname{Reg}({\mathcal{Q}})=\operatorname{Sth}({\mathcal{Q}})=\operatorname{Int}({\mathcal{Q}})$
and
$\partial{\mathcal{Q}}={\mathcal{Q}}\setminus\operatorname{Reg}({\mathcal{Q}})$.
Otherwise,
$\partial{\mathcal{Q}}\subset{\mathcal{Q}}\setminus\operatorname{Reg}({\mathcal{Q}})$
and the inclusion could be strict.

A Nash manifold with corners ${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ is locally
closed. Consequently, ${\mathcal{Q}}$ is a closed Nash submanifold with
corners of the Nash manifold
${\mathbb{R}}^{n}\setminus(\operatorname{Cl}({\mathcal{Q}})\setminus{\mathcal{Q}})$.
In [FGR, Thm.1.11] it is shown that ${\mathcal{Q}}$ is a closed subset of an
affine Nash manifold of its same dimension. Recall that a Nash subset $Y$ of a
Nash manifold $M\subset{\mathbb{R}}^{n}$ has only Nash normal-crossings in $M$
if for each point $y\in Y$ there exists an open semialgebraic neighborhood
$U\subset M$ such that $Y\cap U$ is a Nash normal-crossings divisor of $U$.

######  Theorem 2.4 ([FGR, Thm.1.11]).

Let ${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ be a $d$-dimensional Nash manifold
with corners. There exists a $d$-dimensional Nash manifold
$M\subset{\mathbb{R}}^{n}$ that contains ${\mathcal{Q}}$ as a closed subset
and satisfies:

  * (i)

The Nash closure $Y$ of $\partial{\mathcal{Q}}$ in $M$ has only Nash normal-
crossings in $M$ and ${\mathcal{Q}}\cap Y=\partial{\mathcal{Q}}$.

  * (ii)

For every $x\in\partial{\mathcal{Q}}$ the smallest analytic germ that contains
the germ $\partial{\mathcal{Q}}_{x}$ is $Y_{x}$.

  * (iii)

$M$ can be covered by finitely many open semialgebraic subsets $U_{i}$ (for
$i=1,\ldots,r$) equipped with Nash diffeomorphisms
$u_{i}:=(u_{i1},\dots,u_{id}):U_{i}\to{\mathbb{R}}^{d}$ such that:

$\begin{cases}\text{$U_{i}\subset\operatorname{Int}({\mathcal{Q}})$ or
$U_{i}\cap{\mathcal{Q}}=\varnothing$},&\text{\\!if $U_{i}$ does not meet
$\partial{\mathcal{Q}}$,}\\\ U_{i}\cap{\mathcal{Q}}=\\{u_{i1}\geq
0,\dots,u_{ik_{i}}\geq 0\\},&\text{\\!if $U_{i}$ meets $\partial{\mathcal{Q}}$
(for a suitable $k_{i}\geq 1$).}\end{cases}$

The Nash manifold $M$ is called a Nash envelope of ${\mathcal{Q}}$. In
general, it is not guaranteed that the Nash closure $Y$ of
$\partial{\mathcal{Q}}$ in $M$ is a Nash normal-crossings divisor of $M$ as we
show next.

Figure 2.1. The teardrop.

######  Example 2.5.

The teardrop ${\mathcal{Q}}:=\\{{\tt x}\geq 0,{\tt y}^{2}\leq{\tt x}^{2}-{\tt
x}^{4}\\}\subset{\mathbb{R}}^{2}$ is a Nash manifold with corners (Figure
2.1). Given any open semialgebraic neighborhood $M$ of ${\mathcal{Q}}$ in
${\mathbb{R}}^{2}$ the Nash closure of $\partial{\mathcal{Q}}$ in $M$ is not a
Nash normal-crossings divisor.

We define now Nash manifolds with divisorial corners.

######  Definition 2.6.

A Nash manifold with corners ${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ is a Nash
manifold with divisorial corners if there exists a Nash envelope
$M\subset{\mathbb{R}}^{n}$ such that the Nash closure of
$\partial{\mathcal{Q}}$ in $M$ is a Nash normal-crossings divisor.

A facet of a Nash manifold with corners ${\mathcal{Q}}\subset{\mathbb{R}}^{n}$
is the (topological) closure in ${\mathcal{Q}}$ of a connected component of
$\operatorname{Sth}(\partial{\mathcal{Q}})$. As
$\partial{\mathcal{Q}}={\mathcal{Q}}\setminus\operatorname{Sth}({\mathcal{Q}})$
is semialgebraic, the facets are semialgebraic and finitely many. The non-
empty intersections of facets of ${\mathcal{Q}}$ are the faces of
${\mathcal{Q}}$. In [FGR] the following characterization for Nash manifolds
with divisorial corners is shown:

######  Theorem 2.7 ([FGR, Thm.1.12, Cor.6.5]).

Let ${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ be a $d$-dimensional Nash manifold
with corners. The following assertions are equivalent:

  * (i)

There exists a Nash envelope $M\subset{\mathbb{R}}^{n}$ where the Nash closure
of $\partial{\mathcal{Q}}$ is a Nash normal-crossings divisor.

  * (ii)

Every facet ${\mathcal{F}}$ of ${\mathcal{Q}}$ is contained in a Nash manifold
$X\subset{\mathbb{R}}^{n}$ of dimension $d-1$.

  * (iii)

The number of facets of ${\mathcal{Q}}$ that contain every given point
$x\in\partial{\mathcal{Q}}$ coincides with the number of connected components
of the germ $\operatorname{Sth}(\partial{\mathcal{Q}})_{x}$.

  * (iv)

All the facets of ${\mathcal{Q}}$ are Nash manifold with divisorial corners.

If that is the case, the Nash manifold $M$ in (i) can be chosen such that the
Nash closure in $M$ of every facet ${\mathcal{F}}$ of ${\mathcal{Q}}$ meets
${\mathcal{Q}}$ exactly along ${\mathcal{F}}$.

Note that properties (iii) and (iv) are intrinsic properties of
${\mathcal{Q}}$ and do not depend on the Nash envelope $M$. The faces of a
Nash manifold with divisorial corners are again Nash manifolds with divisorial
corners. If a Nash envelope $M\subset{\mathbb{R}}^{n}$ of ${\mathcal{Q}}$
satisfies (one of the equivalent) conditions of Theorem 2.7, every open
semialgebraic neighborhood $M^{\prime}\subset M$ of ${\mathcal{Q}}$ satisfies
such conditions. For the rest of this article: A Nash manifold with corners
means a Nash manifold with divisorial corners.

###  2.4. Connected components of a non-singular real algebraic set

The following result allows the reader to understand connected Nash manifolds
as connected components of irreducible non-singular real algebraic sets whose
connected components are pairwise diffeomorphic.

######  Lemma 2.8.

Let $M\subset{\mathbb{R}}^{n}$ be a $d$-dimensional connected Nash manifold.
Then there exists a non-singular irreducible real algebraic set
$X\subset{\mathbb{R}}^{n+k}$ whose connected components are Nash diffeomorphic
to $M$. In addition,

  * (i)

Each connected component of $X$ projects onto $M$.

  * (ii)

Given any two of the connected components of $X$, there exists an automorphism
of $X$ that swaps them.

  * (iii)

If $M$ is compact, we may assume that $X$ has at most two connected
components.

###### Proof.

Consider the Nash map $f:M\hookrightarrow{\mathbb{R}}^{n},\ x\mapsto x$. By
Artin-Mazur’s description [BCR, Thm.8.4.4] of Nash maps there exist $s\geq 1$
and a non-singular irreducible real algebraic set
$Z\subset{\mathbb{R}}^{2n+s}$ of dimension $d$, a connected component $N$ of
$Z$ and a Nash diffeomorphism $g:M\to N$ such that the following diagram is
commutative:

$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}\times{\mathbb{R}}^{s}\equiv{\mathbb{R}}^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\pi_{1}}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{g}$$\scriptstyle{f}$$\textstyle{{\mathbb{R}}^{n}}$

We denote the projection of
${\mathbb{R}}^{n}\times{\mathbb{R}}^{n}\times{\mathbb{R}}^{s}$ onto the first
space ${\mathbb{R}}^{n}$ with $\pi_{1}$ and the projection of
${\mathbb{R}}^{n}\times{\mathbb{R}}^{n}\times{\mathbb{R}}^{s}$ onto the second
space ${\mathbb{R}}^{n}$ with $\pi_{2}$. Write $m:=2n+s$. By [M] applied to
$N$ and the union of the remaining connected components of $Z$ there exist
finitely many polynomials
$P_{1},\ldots,P_{\ell},Q_{1},\ldots,Q_{\ell}\in{\mathbb{R}}[{\tt
x}]:={\mathbb{R}}[{\tt x}_{1},\ldots,{\tt x}_{m}]$ such that each $Q_{j}$ is
strictly positive on ${\mathbb{R}}^{m}$ and

$N=Z\cap\Big{\\{}\sum_{j=1}^{\ell}P_{j}\sqrt{Q_{j}}>0\Big{\\}}.$

If $M$ is compact, also $N$ is compact and by [CF, Lem.6.1] there exists a
polynomial $P\in{\mathbb{R}}[{\tt x}]$ such that $N=Z\cap\\{P>0\\}$.

A general $N$ is the projection of the real algebraic set

$Y:=\Big{\\{}(z,y,t)\in Z\times{\mathbb{R}}^{\ell}\times{\mathbb{R}}:\
\Big{(}\sum_{j=1}^{\ell}P_{j}y_{j}^{2}\Big{)}t^{2}-1=0,\ y_{j}^{4}-Q_{j}=0\
\text{for $j=1,\ldots,\ell$}\Big{\\}}$

under
$\pi:{\mathbb{R}}^{m}\times{\mathbb{R}}^{\ell}\times{\mathbb{R}}\to{\mathbb{R}}^{m},\
(z,y,t)\mapsto z$. If $M$ is compact, $N$ is the projection of the real
algebraic set $Y:=\\{(z,t)\in Z\times{\mathbb{R}}:\ P(z)t^{2}-1=0\\}$ under
$\pi:{\mathbb{R}}^{m}\times{\mathbb{R}}\to{\mathbb{R}}^{m},\ (z,t)\mapsto z$.
Denote

$h:=\begin{cases}P&\text{if $M$ is compact,}\\\
\sqrt{\sum_{j=1}^{\ell}P_{j}\sqrt{Q_{j}}}&\text{if $M$ is non-
compact.}\end{cases}$

Fix
$\epsilon:=(\epsilon_{1},\ldots,\epsilon_{\ell},\epsilon_{\ell+1})\in\\{-1,1\\}^{\ell+1}$
(where $\ell=0$ if $N$ is compact) and let
$N_{\epsilon}:=Y\cap\\{\epsilon_{1}y_{1}>0,\ldots,\epsilon_{\ell}y_{\ell}>0,\epsilon_{\ell+1}t>0\\}$.
Consider the Nash diffeomorphism

$\varphi_{\epsilon}:N\to N_{\epsilon},\
z\mapsto\Big{(}z,\epsilon_{1}\sqrt[4]{Q_{1}(z)},\ldots,\epsilon_{\ell}\sqrt[4]{Q_{\ell}(z)},\epsilon_{\ell+1}\frac{1}{h(z)}\Big{)}$

whose inverse map is the restriction of the projection $\pi$ to
$N_{\epsilon}$.

Observe that $\\{N_{\epsilon}\\}_{\epsilon\in\\{-1,1\\}^{\ell+1}}$ is the
collection of the connected components of $Y$. As $\pi(N_{\epsilon})=N$ and
using the diagram above, we deduce

$(\pi_{2}\circ\pi)(N_{\epsilon})=\pi_{2}(N)=(f\circ\pi_{1})(N)=f(M)=M.$

In addition, each $N_{\epsilon}$ is Nash diffeomorphic to $M$ and for
$\epsilon\neq\epsilon^{\prime}$ the polynomial map

$\phi_{\epsilon,\epsilon^{\prime}}:{\mathbb{R}}^{m}\times{\mathbb{R}}^{\ell}\times{\mathbb{R}}\to{\mathbb{R}}^{m}\times{\mathbb{R}}^{\ell}\times{\mathbb{R}},\
(x,y,t)\mapsto(x,(\epsilon\cdot\epsilon^{\prime})\cdot(y,t))$

induces an involution of $Y$ such that
$\phi_{\epsilon,\epsilon^{\prime}}(N_{\epsilon})=N_{\epsilon^{\prime}}$. We
have denoted

$(\epsilon\cdot\epsilon^{\prime})\cdot(y,t):=(\epsilon_{1}\epsilon_{1}^{\prime}y_{1},\ldots,\epsilon_{\ell}\epsilon_{\ell}^{\prime}y_{\ell},\epsilon_{\ell+1}\epsilon_{\ell+1}^{\prime}t).$

As $Z$ is non-singular, also $Y$ is non-singular. Let $X$ be the irreducible
component of $Y$ that contains $N_{(1,\ldots,1)}$, which has at most two
connected components if $M$ is compact. Then $k:=m+\ell+1-n$ and the non-
singular real algebraic set $X$ satisfies the requirements in the statement. ∎

###  2.5. The $1$-dimensional case

Let us prove here Theorems 1.4 and 1.6 when
${\mathcal{S}}\subset{\mathbb{R}}^{n}$ is a $1$-dimensional semialgebraic set
connected by analytic paths. As $1$-dimensional semialgebraic sets are locally
compact, we can merge both results in one by choosing the hypothesis of
Theorem 1.6 and obtaining the thesis of Theorem 1.4.

###### Proof of Theorems 1.4 and 1.6 for the $1$-dimensional case.

Following the proof of [Fe2, Prop.1.6] we find a $1$-dimensional non-singular
real algebraic set $X\subset{\mathbb{R}}^{m}$, a proper polynomial map
$f:X\to\overline{{\mathcal{S}}}^{\operatorname{zar}}$, and a connected
$1$-dimensional semialgebraic subset ${\mathcal{T}}\subset X$ such that:

  * (i)

The restriction $f|_{X\setminus
f^{-1}(\operatorname{Sing}(\overline{{\mathcal{S}}}^{\operatorname{zar}}))}:X\setminus
f^{-1}(\operatorname{Sing}(\overline{{\mathcal{S}}}^{\operatorname{zar}}))\to\overline{{\mathcal{S}}}^{\operatorname{zar}}\setminus\operatorname{Sing}(\overline{{\mathcal{S}}}^{\operatorname{zar}})$
is a biregular diffeomorphism.

  * (ii)

The restriction $f|_{{\mathcal{T}}}:{\mathcal{T}}\to{\mathcal{S}}$ is proper
and surjective.

We have proved in [Fe2, Prop.1.6] that the connected components of $X$ are
Nash diffeomorphic either to ${\mathbb{R}}$ or to the sphere
${\mathbb{S}}^{1}$. Thus, ${\mathcal{T}}$ is Nash diffeomorphic to either
$(0,1)$, $[0,1)$, $[0,1]$ or ${\mathbb{S}}^{1}$. In the latter two cases
($[0,1]$ and ${\mathbb{S}}^{1}$), the properties stated in Theorem 1.4 follow
readily. In the first two cases, we consider
${\mathcal{A}}:=\operatorname{Cl}({\mathcal{T}})\setminus{\mathcal{T}}$, which
is a finite set with at most two elements. Observe that ${\mathcal{T}}$ is
closed in $X\setminus{\mathcal{A}}$, which is a Zariski open subset of $X$.
Thus, the $1$-dimensional non-singular real algebraic set
$X^{\prime}:=\\{(x,y)\in X\times{\mathbb{R}}:\ yh(x)=1\\}$, where
$h\in{\mathbb{R}}[{\tt x}]$ is any polynomial whose zero set is
${\mathcal{A}}$, together with the projection $\pi:X\times{\mathbb{R}}\to X$
(onto the first factor) satisfy that $\pi|_{X^{\prime}}:X^{\prime}\to
X\setminus{\mathcal{A}}$ is a biregular diffeomorphism. Substituting $f$ by
$f^{\prime}:=f\circ\pi|_{X^{\prime}}$ the properties stated in Theorem 1.4 are
proved straightforwardly, as required. ∎

In Section 4 we prove Theorems 1.4 and 1.6 for semialgebraic sets of dimension
$d\geq 2$.

##  3\. Drilling blow-up

In this section we recall the main properties of the drilling blow-up of a
Nash manifold with center a closed Nash submanifold [Fe2, §5]. We prove some
additional results (Remark 3.4, Fact 3.5, Remark 3.10 an §3.4) that we need in
the subsequent sections.

###  3.1. Local structure of the drilling blow-up.

Let $M\subset{\mathbb{R}}^{m}$ be a $d$-dimensional Nash manifold and
$N\subset M$ a closed $e$-dimensional Nash submanifold. As we are interested
in the local structure, assume that there exists a Nash diffeomorphism
$u:=(u_{1},\ldots,u_{d}):M\to{\mathbb{R}}^{d}$ such that
$N=\\{u_{e+1}=0,\ldots,u_{d}=0\\}$. Denote
$\psi:=u^{-1}:{\mathbb{R}}^{d}\equiv{\mathbb{R}}^{e}\times{\mathbb{R}}^{d-e}\to
M$. Let $\zeta_{e+1},\ldots,\zeta_{d}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{k}$ be
Nash maps such that the vectors $\zeta_{e+1}(y,0),\ldots,\zeta_{d}(y,0)$ are
linearly independent for each $y\in{\mathbb{R}}^{e}$. Write
$z\in{\mathbb{R}}^{d-e}$ as $z:=(z_{e+1},\ldots,z_{d})$. Consider the Nash
maps

$\displaystyle\varphi:{\mathbb{R}}^{d}\equiv{\mathbb{R}}^{e}\times{\mathbb{R}}^{d-e}\to{\mathbb{R}}^{k},\
(y,z)\mapsto\zeta_{e+1}(y,z)z_{e+1}+\cdots+\zeta_{d}(y,z)z_{d},$
$\displaystyle\phi:{\mathbb{R}}^{e}\times{\mathbb{R}}\times{\mathbb{S}}^{d-e-1}\to{\mathbb{R}}^{k},\
(y,\rho,w)\mapsto\zeta_{e+1}(y,\rho w)w_{e+1}+\cdots+\zeta_{d}(y,\rho w)w_{d}$

and assume that $\varphi(y,z)=0$ if and only if $z=0$. Consider the (well-
defined) Nash map:

$\Phi:{\mathbb{R}}^{e}\times{\mathbb{R}}\times{\mathbb{S}}^{d-e-1}\to
M\times{\mathbb{S}}^{k-1},\ (y,\rho,w)\mapsto\Big{(}\psi(y,\rho
w),\frac{\phi(y,\rho,w)}{\|\phi(y,\rho,w)\|}\Big{)}.$

######  Fact 3.1.

Fix $\epsilon=\pm$ and denote

$I_{\epsilon}:=\begin{cases}[0,+\infty)&\text{ if $\epsilon=+$,}\\\
(-\infty,0]&\text{ if $\epsilon=-$.}\end{cases}$

The closure $\widetilde{M}_{\epsilon}$ in $M\times{\mathbb{S}}^{k-1}$ of the
set

$\Gamma_{\epsilon}:=\Big{\\{}\Big{(}\psi(y,z),\epsilon\frac{\varphi(y,z)}{\|\varphi(y,z)\|}\Big{)}\in
M\times{\mathbb{S}}^{k-1}:\,z\neq 0\Big{\\}}$

is a Nash manifold with boundary such that:

  * (i)

$\widetilde{M}_{\epsilon}\subset\operatorname{im}(\Phi)$.

  * (ii)

The restriction of $\Phi$ to ${\mathbb{R}}^{e}\times
I_{\epsilon}\times{\mathbb{S}}^{d-e-1}$ induces a Nash diffeomorphism between
${\mathbb{R}}^{e}\times I_{\epsilon}\times{\mathbb{S}}^{d-e-1}$ and
$\widetilde{M}_{\epsilon}$. Consequently,
$\partial\widetilde{M_{\epsilon}}=\Phi({\mathbb{R}}^{e}\times\\{0\\}\times{\mathbb{S}}^{d-e-1})$
and
$\Gamma_{\epsilon}=\operatorname{Int}(\widetilde{M_{\epsilon}})=\Phi({\mathbb{R}}^{e}\times(I_{\epsilon}\setminus\\{0\\})\times{\mathbb{S}}^{d-e-1})$.

######  Fact 3.2.

Denote ${\mathcal{R}}:=\partial\widetilde{M}_{+}=\partial\widetilde{M}_{-}$
and
$\widehat{M}:=\widetilde{M}_{+}\cup\widetilde{M}_{-}=\Gamma_{+}\sqcup{\mathcal{R}}\sqcup\Gamma_{-}$.
Then $\Phi$ induces a Nash diffeomorphism between
${\mathbb{R}}^{e}\times{\mathbb{R}}\times{\mathbb{S}}^{d-e-1}$ and
$\widehat{M}$, which is the Nash closure of $\widetilde{M}_{+}$ and
$\widetilde{M}_{-}$ in $M\times{\mathbb{S}}^{k-1}$. In addition, the Nash map
$\sigma:M\times{\mathbb{S}}^{k-1}\to M\times{\mathbb{S}}^{k-1},(a,b)\to(a,-b)$
induces a Nash involution on $\widehat{M}$ without fixed points such that
$\sigma(\widetilde{M}_{+})=\widetilde{M}_{-}$ and
$\Phi(y,-\rho,-w)=(\sigma\circ\Phi)(y,\rho,w)$ for each
$(y,\rho,w)\in{\mathbb{R}}^{e}\times{\mathbb{R}}\times{\mathbb{S}}^{d-e-1}$.

######  Fact 3.3.

Consider the projection $\pi:M\times{\mathbb{S}}^{k-1}\to M$ onto the first
factor and denote $\pi_{\epsilon}:=\pi|_{\widetilde{M_{\epsilon}}}$ for
$\epsilon=\pm$. Then

  * (i)

$\pi_{\epsilon}$ is proper, $\pi_{\epsilon}(\widetilde{M_{\epsilon}})=M$ and
${\mathcal{R}}=\pi^{-1}_{\epsilon}(N)$.

  * (ii)

The restriction $\pi_{\epsilon}|_{\Gamma_{\epsilon}}:\Gamma_{\epsilon}\to
M\setminus N$ is a Nash diffeomorphism.

  * (iii)

For each $q\in N$ it holds
$\pi_{\epsilon}^{-1}(q)=\\{q\\}\times{\mathbb{S}}_{q}^{d-e-1}$ where
${\mathbb{S}}_{q}^{d-e-1}$ is the sphere of dimension $d-e-1$ obtained when
intersecting the sphere ${\mathbb{S}}^{k-1}$ with the linear subspace $L_{q}$
generated by $(\zeta_{e+1}\circ u)(q),\ldots,(\zeta_{d}\circ u)(q)$.

$\Phi^{-1}(\widehat{\pi}^{-1}(N))$$\bullet$$u(N)$$(y,\rho,w)$$(y,\rho
w)$${\mathbb{R}}^{d}$${\mathbb{R}}^{e}\times[0,+\infty)\times{\mathbb{S}}^{d-e-1}$$u\circ\pi_{+}\circ\Phi$$\bullet$
Figure 3.1. Local structure of the drilling blow-up $\widetilde{M}_{+}$ of $M$
of center $N$ (figure borrowed from [Fe2, Fig.3]).

Denote $\widehat{\pi}:=\pi|_{\widehat{M}}$ and consider the commutative
diagram.

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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(3.1)

As a consequence, we have: The Nash maps $\pi_{\epsilon}$ and $\widehat{\pi}$
have local representations

$(x_{1},\ldots,x_{d})\mapsto(x_{1},\ldots,x_{e},x_{e+1},x_{e+1}x_{e+2},\ldots,x_{e+1}x_{d})$

in an open neighborhood of each point $p\in{\mathcal{R}}$. In addition,
$d\pi_{p}(T_{p}\widehat{M})\not\subset T_{\pi(p)}N$.

######  Remarks 3.4.

Denote $g:=u\circ\widehat{\pi}\circ\Phi$ and
$g_{+}:=g|_{{\mathbb{R}}^{e}\times[0,+\infty)\times{\mathbb{S}}^{d-e-1}}=u\circ\pi_{+}\circ\Phi|_{{\mathbb{R}}^{e}\times[0,+\infty)\times{\mathbb{S}}^{d-e-1}}$.
Consider the Nash normal-crossings divisor $Z:=\\{{\tt y}_{e+1}\cdots{\tt
y}_{d}=0\\}\subset{\mathbb{R}}^{d}$. Consider coordinates
$(w_{e+1},\ldots,w_{d})$ in ${\mathbb{R}}^{d-e}$ and the sphere
${\mathbb{S}}^{d-e-1}=\\{{\tt w}_{e+1}^{2}+\cdots+{\tt w}_{d}^{2}=1\\}$.

(i) Write $Z_{k}:=\\{{\tt y}_{k}=0\\}$ for $k=e+1,\ldots,d$ and observe that

$\displaystyle g^{-1}(Z_{k})$
$\displaystyle=({\mathbb{R}}^{e}\times\\{0\\}\times{\mathbb{S}}^{d-e-1})\cup({\mathbb{R}}^{e}\times{\mathbb{R}}\times({\mathbb{S}}^{d-e-1}\cap\\{{\tt
w}_{k}=0\\})),$ $\displaystyle g_{+}^{-1}(Z_{k})$
$\displaystyle=({\mathbb{R}}^{e}\times\\{0\\}\times{\mathbb{S}}^{d-e-1})\cup({\mathbb{R}}^{e}\times[0,+\infty)\times({\mathbb{S}}^{d-e-1}\cap\\{{\tt
w}_{k}=0\\}))$

for $k=e+1,\ldots,d$. Thus,

$g^{-1}(Z)=({\mathbb{R}}^{e}\times\\{0\\}\times{\mathbb{S}}^{d-e-1})\cup\bigcup_{k=e+1}^{d}({\mathbb{R}}^{e}\times{\mathbb{R}}\times({\mathbb{S}}^{d-e-1}\cap\\{{\tt
w}_{k}=0\\}))$

is a Nash normal-crossings divisor and

$g_{+}^{-1}(Z)=({\mathbb{R}}^{e}\times\\{0\\}\times{\mathbb{S}}^{d-e-1})\cup\bigcup_{k=e+1}^{d}({\mathbb{R}}^{e}\times[0,+\infty)\times({\mathbb{S}}^{d-e-1}\cap\\{{\tt
w}_{k}=0\\})).$

The Nash closure of $g_{+}^{-1}(Z)$ is $g^{-1}(Z)$.

(ii) Let $\epsilon:=(\epsilon_{e+1},\ldots,\epsilon_{d})$ where
$\epsilon_{k}=\pm 1$ and denote
${\mathcal{Q}}_{\epsilon}:=\\{\epsilon_{e+1}{\tt y}_{e+1}\geq
0,\ldots,\epsilon_{d}{\tt y}_{d}\geq 0\\}$. Write
$-\epsilon:=(-\epsilon_{e+1},\ldots,-\epsilon_{d})$. We have:

$\operatorname{Cl}(g_{+}^{-1}({\mathcal{Q}}_{\epsilon}\setminus
Z))={\mathbb{R}}^{e}\times[0,+\infty)\times({\mathbb{S}}^{d-e-1}\cap\\{\epsilon_{e+1}{\tt
w}_{e+1}\geq 0,\ldots,\epsilon_{d}{\tt w}_{d}\geq 0\\}).$

Consequently, $\operatorname{Cl}(g_{+}^{-1}({\mathcal{Q}}_{\epsilon}\setminus
Z))\cap\operatorname{Cl}(g_{+}^{-1}({\mathcal{Q}}_{-\epsilon}\setminus
Z))=\varnothing$. More generally, if

$\epsilon:=(\epsilon_{e+1},\ldots,\epsilon_{m},\epsilon_{m+1},\ldots,\epsilon_{d})\
\text{and}\
\epsilon^{\prime}:=(\epsilon_{e+1},\ldots,\epsilon_{m},-\epsilon_{m+1},\ldots,-\epsilon_{d})$

where $e<m<d$, then

${\mathcal{Q}}_{\epsilon}\cap{\mathcal{Q}}_{\epsilon^{\prime}}=\\{\epsilon_{e+1}{\tt
y}_{e+1}\geq 0,\ldots,\epsilon_{m}{\tt y}_{m}\geq 0,{\tt
y}_{m+1}=0,\ldots,{\tt y}_{d}=0\\},$

which has dimension $e+(d-e)-(d-m)=m$. In addition,

$\operatorname{Cl}(g_{+}^{-1}({\mathcal{Q}}_{\epsilon}\setminus
Z))\cap\operatorname{Cl}(g_{+}^{-1}({\mathcal{Q}}_{\epsilon^{\prime}}\setminus
Z))\\\
={\mathbb{R}}^{e}\times[0,+\infty)\times({\mathbb{S}}^{d-e-1}\cap\\{\epsilon_{e+1}{\tt
w}_{e+1}\geq 0,\ldots,\epsilon_{m}{\tt w}_{m}\geq 0,{\tt
w}_{m+1}=0,\ldots,{\tt w}_{d}=0\\}),$

which has dimension $e+1+(d-e-1-(d-m))=m$.

(iii) Let $Y_{1},Y_{2}$ be intersections of dimension $e+1$ of irreducible
components of $Z$ that contain $N$. We may assume $Y_{1}=\\{{\tt
y}_{e+1}=0,\ldots,{\tt y}_{d-1}=0\\}$ and $Y_{2}=\\{{\tt
y}_{e+1}=0,\ldots,{\tt y}_{d-2}=0,{\tt y}_{d}=0\\}$, so $Y_{1}\cap
Y_{2}=\\{{\tt y}_{e+1}=0,\ldots,{\tt y}_{d}=0\\}=N$. Thus,

$\displaystyle g^{-1}(Y_{1})$
$\displaystyle=({\mathbb{R}}^{e}\times\\{0\\}\times{\mathbb{S}}^{d-e-1})\cup({\mathbb{R}}^{e}\times{\mathbb{R}}\times({\mathbb{S}}^{d-e-1}\cap\\{{\tt
w}_{e+1}=0,\ldots,{\tt w}_{d-1}=0\\})),$ $\displaystyle g_{+}^{-1}(Y_{1})$
$\displaystyle=({\mathbb{R}}^{e}\times\\{0\\}\times{\mathbb{S}}^{d-e-1})\cup({\mathbb{R}}^{e}\times[0,+\infty)\times({\mathbb{S}}^{d-e-1}\cap\\{{\tt
w}_{e+1}=0,\ldots,{\tt w}_{d-1}=0\\})),$ $\displaystyle g^{-1}(Y_{2})$
$\displaystyle=({\mathbb{R}}^{e}\times\\{0\\}\times{\mathbb{S}}^{d-e-1})\cup({\mathbb{R}}^{e}\times{\mathbb{R}}\times({\mathbb{S}}^{d-e-1}\cap\\{{\tt
w}_{e+1}=0,\ldots,{\tt w}_{d-2}=0,{\tt w}_{d}=0\\})),$ $\displaystyle
g_{+}^{-1}(Y_{2})$
$\displaystyle=({\mathbb{R}}^{e}\times\\{0\\}\times{\mathbb{S}}^{d-e-1})\cup({\mathbb{R}}^{e}\times[0,+\infty)\times({\mathbb{S}}^{d-e-1}\cap\\{{\tt
w}_{e+1}=0,\ldots,{\tt w}_{d-2}=0,{\tt w}_{d}=0\\})).$

As the intersection

${\mathbb{S}}^{d-e-1}\cap\\{{\tt w}_{e+1}=0,\ldots,{\tt
w}_{d-1}=0\\}\cap\\{{\tt w}_{e+1}=0,\ldots,{\tt w}_{d-2}=0,{\tt w}_{d}=0\\}$

is empty, we conclude that the intersection

$\operatorname{Cl}(g^{-1}(Y_{1}\setminus
N))\cap\operatorname{Cl}(g^{-1}(Y_{2}\setminus N))\\\
=(g^{-1}(Y_{1})\cap\operatorname{Cl}(g^{-1}(Y_{1}\setminus
N)))\cap(g^{-1}(Y_{2})\cap\operatorname{Cl}(g^{-1}(Y_{2}\setminus N)))$

of the strict transforms of $Y_{1},Y_{2}$ under $g$ is also empty.
Analogously, the intersection

$\operatorname{Cl}(g_{+}^{-1}(Y_{1}\setminus
N))\cap\operatorname{Cl}(g_{+}^{-1}(Y_{2}\setminus N))\\\
=(g_{+}^{-1}(Y_{1})\cap\operatorname{Cl}(g_{+}^{-1}(Y_{1}\setminus
N)))\cap(g_{+}^{-1}(Y_{2})\cap\operatorname{Cl}(g_{+}^{-1}(Y_{2}\setminus
N)))$

of the strict transforms of $Y_{1},Y_{2}$ under $g_{+}$ is empty.∎

Next, we analyze the properties of the local structure of drilling blow-up
when $N$ has dimension $d-1$.

######  Fact 3.5.

Assume $N$ has dimension $e=d-1$. The Nash diffeomorphism
$u:=(u_{1},\ldots,u_{d}):M\to{\mathbb{R}}^{d}$ satisfies $N=\\{u_{d}=0\\}$.
Recall that
$\psi:=u^{-1}:{\mathbb{R}}^{d}\equiv{\mathbb{R}}^{d-1}\times{\mathbb{R}}\to
M$, $\zeta_{d}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{k}$ is a Nash map that does
not take the value $0\in{\mathbb{R}}^{k}$ and define the Nash map

$\Phi:{\mathbb{R}}^{d-1}\times{\mathbb{R}}\times\\{\pm 1\\}\to
M\times{\mathbb{S}}^{k-1},\ (y,\rho,\pm
1)\mapsto\Big{(}\psi(y,\pm\rho),\pm\frac{\zeta_{d}(y,\pm\rho)}{\|\zeta_{d}(y,\pm\rho)\|}\Big{)}.$

Fix $\epsilon=\pm$ and denote

$I_{\epsilon}:=\begin{cases}[0,+\infty)&\text{ if $\epsilon=+$,}\\\
(-\infty,0]&\text{ if $\epsilon=-$.}\end{cases}$

The closure $\widetilde{M}_{\epsilon}$ in $M\times{\mathbb{S}}^{k-1}$ of the
set

$\Gamma_{\epsilon}:=\Big{\\{}\Big{(}\psi(y,z),\epsilon\frac{z}{|z|}\frac{\zeta_{d}(y,z)}{\|\zeta_{d}(y,z)\|}\Big{)}\in
M\times{\mathbb{S}}^{k-1}:\,z\neq 0\Big{\\}}$

is a Nash manifold with boundary such that:

  * (i)

$\widetilde{M}_{\epsilon}\subset\operatorname{im}(\Phi)$.

  * (ii)

The restriction of $\Phi$ to ${\mathbb{R}}^{d-1}\times
I_{\epsilon}\times\\{\pm 1\\}$ induces a Nash diffeomorphism between
${\mathbb{R}}^{d-1}\times I_{\epsilon}\times\\{\pm 1\\}$ and
$\widetilde{M}_{\epsilon}$. Consequently,
$\partial\widetilde{M_{\epsilon}}=\Phi({\mathbb{R}}^{d-1}\times\\{0\\}\times\\{\pm
1\\})$ and
$\Gamma_{\epsilon}=\operatorname{Int}(\widetilde{M_{\epsilon}})=\Phi({\mathbb{R}}^{d-1}\times(I_{\epsilon}\setminus\\{0\\})\times\\{\pm
1\\})$.

Denote ${\mathcal{R}}:=\partial\widetilde{M}_{+}=\partial\widetilde{M}_{-}$
and
$\widehat{M}:=\widetilde{M}_{+}\cup\widetilde{M}_{-}=\Gamma_{+}\sqcup{\mathcal{R}}\sqcup\Gamma_{-}$.
Then $\Phi$ induces a Nash diffeomorphism between
${\mathbb{R}}^{d-1}\times{\mathbb{R}}\times\\{\pm 1\\}$ and $\widehat{M}$,
which is the Nash closure of $\widetilde{M}_{+}$ and $\widetilde{M}_{-}$ in
$M\times{\mathbb{S}}^{k-1}$. In addition, the Nash map
$\sigma:M\times{\mathbb{S}}^{k-1}\to M\times{\mathbb{S}}^{k-1},(a,b)\to(a,-b)$
induces a Nash involution on $\widehat{M}$ without fixed points such that
$\sigma(\widetilde{M}_{+})=\widetilde{M}_{-}$ and $\Phi(y,-\rho,\pm
1)=(\sigma\circ\Phi)(y,\rho,\mp 1)$ for each $(y,\rho,\pm
1)\in{\mathbb{R}}^{d-1}\times{\mathbb{R}}\times\\{\pm 1\\}$.

Consider the projection $\pi:M\times{\mathbb{S}}^{k-1}\to M$ onto the first
factor and denote $\pi_{\epsilon}:=\pi|_{\widetilde{M_{\epsilon}}}$. Then

  * (i)

$\pi_{\epsilon}$ is proper, $\pi_{\epsilon}(\widetilde{M_{\epsilon}})=M$ and
${\mathcal{R}}=\pi^{-1}_{\epsilon}(N)$.

  * (ii)

The restriction $\pi_{\epsilon}|_{\Gamma_{\epsilon}}:\Gamma_{\epsilon}\to
M\setminus N$ is a Nash diffeomorphism.

  * (iii)

For each $q\in N$ it holds
$\pi_{\epsilon}^{-1}(q)=\\{q\\}\times\\{\pm\frac{\\{(\zeta_{d}\circ
u)(q)\\}}{\|\\{(\zeta_{d}\circ u)(q)\\}\|}\\}$, that is, each point $q\in N$
has exactly two preimages under $\pi_{\epsilon}$.

Denote $\widehat{\pi}:=\pi|_{\widehat{M}}$ and consider the commutative
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0.0pt\hbox{$\textstyle{\psi(y,\pm\rho)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
204.72937pt\raise-41.27774pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
170.93079pt\raise-41.27774pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\end{gathered}$
(3.2)

Consequently: The Nash maps $\pi_{\epsilon}$ and $\widehat{\pi}$ have local
representations $(x_{1},\ldots,x_{d})\mapsto(x_{1},\ldots,x_{d})$ in an open
neighborhood of each point $p\in{\mathcal{R}}$. In addition,
$d\pi_{p}(T_{p}\widehat{M})=T_{\pi(p)}M$.

###  3.2. Global definition.

Let $M\subset{\mathbb{R}}^{m}$ be a $d$-dimensional Nash manifold and
$N\subset M$ a closed $e$-dimensional Nash submanifold. Let
$f_{1},\ldots,f_{k}\in{\mathcal{N}}(M)$ be a finite system of generators of
the ideal $\mathcal{I}(N)$ of Nash functions on $M$ vanishing identically on
$N$. Consider the Nash map

$F:M\setminus
N\to{\mathbb{S}}^{k-1},x\mapsto\frac{(f_{1}(x),\ldots,f_{k}(x))}{\|(f_{1}(x),\ldots,f_{k}(x))\|}.$

We have:

######  Fact 3.6.

Fix $\epsilon=\pm$. The closure $\widetilde{M}_{\epsilon}$ in
$M\times{\mathbb{S}}^{k-1}$ of the graph

$\Gamma_{\epsilon}:=\\{(x,\epsilon F(x))\in M\times{\mathbb{S}}^{k-1}:\ x\in
M\setminus N\\}$

is a Nash manifold with boundary. Denote
${\mathcal{R}}:=\partial\widetilde{M}_{+}=\partial\widetilde{M}_{-}$ and
$\widehat{M}:=\widetilde{M}_{+}\cup\widetilde{M}_{-}=\Gamma_{+}\sqcup{\mathcal{R}}\sqcup\Gamma_{-}$,
which is the Nash closure of $\widetilde{M}_{+}$ and $\widetilde{M}_{-}$ in
$M\times{\mathbb{S}}^{k-1}$ if $M$ is connected. In addition, $\widehat{M}$ is
a Nash manifold and the Nash map $\sigma:M\times{\mathbb{S}}^{k-1}\to
M\times{\mathbb{S}}^{k-1},\,(a,b)\to(a,-b)$ induces a Nash involution on
$\widehat{M}$ without fixed points that maps $\widetilde{M}_{+}$ onto
$\widetilde{M}_{-}$.

######  Fact 3.7.

Consider the projection $\pi:M\times{\mathbb{S}}^{k-1}\to M$ onto the first
factor. Denote $\pi_{\epsilon}:=\pi|_{\widetilde{M_{\epsilon}}}$ for
$\epsilon=\pm$ and $\widehat{\pi}:=\pi|_{\widehat{M}}$. We have:

  * (i)

$\pi_{\epsilon}$ is proper, $\pi_{\epsilon}(\widetilde{M_{\epsilon}})=M$ and
${\mathcal{R}}=\pi^{-1}_{\epsilon}(N)$.

  * (ii)

The restriction $\pi_{\epsilon}|_{\Gamma_{\epsilon}}:\Gamma_{\epsilon}\to
M\setminus N$ is a Nash diffeomorphism.

  * (iii)

Consider the Nash map $f:=(f_{1},\ldots,f_{k}):M\to{\mathbb{R}}^{k}$ (whose
components generate $\mathcal{I}(N)$). Fix $q\in N$ and let $E_{q}$ be any
complementary linear subspace of $T_{q}N$ in $T_{q}M$. Then
$\pi_{\epsilon}^{-1}(q)=\\{q\\}\times{\mathbb{S}}_{q}^{d-e-1}$, where
${\mathbb{S}}_{q}^{d-e-1}$ denotes the sphere of dimension $d-e-1$ obtained
when intersecting ${\mathbb{S}}^{k-1}$ with the $(d-e)$-dimensional linear
subspace $d_{q}f(E_{q})$. In case $e=d-1$, each $q\in N$ has exactly two
preimages under $\pi_{\epsilon}$.

  * (iv)

The Nash maps $\pi_{\epsilon}$ and $\widehat{\pi}$ have local representations
of the type

$(x_{1},\ldots,x_{d})\mapsto(x_{1},\ldots,x_{e},x_{e+1},x_{e+1}x_{e+2},\ldots,x_{e+1}x_{d})$

in an open neighborhood of each point $p\in{\mathcal{R}}$. In case $e=d-1$ the
previous local representations are
$(x_{1},\ldots,x_{d})\mapsto(x_{1},\ldots,x_{d})$. In addition,
$d\pi_{p}(T_{p}\widehat{M})\not\subset T_{\pi(p)}N$ and if $e=d-1$, we have
$d\pi_{p}(T_{p}\widehat{M})=T_{\pi(p)}M$.

######  Fact 3.8.

Up to Nash diffeomorphisms compatible with the respective projections, the
pairs $(\widetilde{M_{\epsilon}},\pi_{\epsilon})$ and
$(\widehat{M},\widehat{\pi})$ do not depend on the generators
$f_{1},\ldots,f_{k}$ of $\mathcal{I}(N)$. Moreover, such Nash diffeomorphisms
are unique.

######  Definition 3.9.

The pair $(\widetilde{M}_{+},\pi_{+})$ is the drilling blow-up of the Nash
manifold $M$ with center the closed Nash submanifold $N\subset M$ and
$(\widehat{M},\widehat{\pi})$ is the twisted Nash double of
$(\widetilde{M}_{+},\pi_{+})$.

######  Remark 3.10.

Let $N\subset M^{\prime}\subset M$ be Nash manifolds such that $N,M^{\prime}$
are closed in $M$. Let $(\widetilde{M}_{+},\pi_{+})$ be the drilling blow-up
of $M$ with center $N$ and let $(\widehat{M},\widehat{\pi})$ be the twisted
Nash double of $(\widetilde{M}_{+},\pi_{+})$. Denote
$M^{\prime*}:=\operatorname{Cl}(\pi_{+}^{-1}(M^{\prime}\setminus N))$ and
$M^{\prime\bullet}:=\operatorname{Cl}(\widehat{\pi}(M^{\prime}\setminus N))$.
Then $(M^{\prime*},\pi_{+}|_{M^{\prime*}})$ is the drilling blow-up of
$M^{\prime}$ with center $N$ and
$(M^{\prime\bullet},\widehat{\pi}|_{M^{\prime\bullet}})$ is the twisted Nash
double of $(M^{\prime*},\pi_{+}|_{M^{\prime*}})$.

One only has to observe that a finite system of generators
$f_{1},\ldots,f_{k}\in{\mathcal{N}}(M^{\prime})$ of the ideal $\mathcal{I}(N)$
can be obtained considering a finite system of generators
$g_{1},\ldots,g_{k}\in{\mathcal{N}}(M)$ of the ideal $\mathcal{I}(N)$ and
defining $f_{j}:=g_{j}|_{M^{\prime}}$ for $j=1,\ldots,k$.

###  3.3. Relationship between drilling blow-up and classical blow-up.

Let $M\subset{\mathbb{R}}^{m}$ be a $d$-dimensional Nash manifold and
$N\subset M$ a closed $e$-dimensional Nash submanifold. Let
$f_{1},\ldots,f_{k}\in{\mathcal{N}}(M)$ be a system of generators of the ideal
$\mathcal{I}(N)$. Define

$\Gamma^{\prime}:=\\{(x,(f_{1}(x):\ldots:f_{k}(x)))\in
M\times{\mathbb{R}}{\mathbb{P}}^{k-1}:\ x\in M\setminus N\\}.$

The closure $B(M,N)$ of $\Gamma^{\prime}$ in
$M\times{\mathbb{R}}{\mathbb{P}}^{k-1}$ together with the restriction
$\pi^{\prime}$ to $B(M,N)$ of the projection
$M\times{\mathbb{R}}{\mathbb{P}}^{k-1}\to M$ is the classical blow-up of $M$
with center $N$ (in the Nash setting).

######  Corollary 3.11.

Let $(\widehat{M},\widehat{\pi})$ be the twisted Nash double of the drilling
blow-up $(\widetilde{M}_{+},\pi_{+})$. Let $\sigma:\widehat{M}\to\widehat{M},\
(a,b)\mapsto(a,-b)$ be the involution of $\widehat{M}$ without fixed points.
Consider the Nash map $\Theta:M\times{\mathbb{S}}^{k-1}\to
M\times{\mathbb{R}}{\mathbb{P}}^{k-1},(p,q)\to(p,[q])$ and its restriction
$\theta:=\Theta|_{\widehat{M}}:\widehat{M}\to B(M,N)$. We have:

  * (i)

$\theta(\widehat{M})=B(M,N)$, $\theta\circ\sigma=\theta$,
$\pi^{\prime}\circ\theta=\widehat{\pi}$ and
$\theta^{-1}(a,[b])=\\{(a,b),(a,-b)\\}$ for each $(a,[b])\in B(M,N)$.

  * (ii)

$\theta$ is an unramified two to one Nash covering of $B(M,N)$.

###  3.4. Algebraic description of drilling blow-up.

We analyze a general enough situation for which we can guarantee that the
twisted (Nash) double of the drilling blow-up is a non-singular real algebraic
set. Let $X\subset{\mathbb{R}}^{n}$ be a $d$-dimensional non-singular real
algebraic set and $Y\subset X$ a $e$-dimensional non-singular real algebraic
subset. Let $f_{1},\ldots,f_{r}\in{\mathbb{R}}[{\tt x}]$ be a system of
generators of the ideal $\mathcal{I}(Y)$ of polynomials vanishing identically
on $Y$ and denote $u:=(u_{1},\ldots,u_{r})\in{\mathbb{S}}^{r-1}$. Fix
$\epsilon=\pm$ and let $\widetilde{X}_{\epsilon}$ be the (topological) closure
of

$\Gamma_{\epsilon}:=\Big{\\{}(x,u)\in(X\setminus Y)\times{\mathbb{S}}^{r-1}:\
{\rm rk}\begin{pmatrix}u_{1}&\cdots&u_{r}\\\
f_{1}(x)&\cdots&f_{r}(x)\end{pmatrix}=1,\,\epsilon(u_{1}f_{1}(x)+\cdots+u_{r}f_{r}(x))>0\Big{\\}}$

in $X\times{\mathbb{S}}^{r-1}$. Let $\pi:X\times{\mathbb{S}}^{r-1}\to X$ be
the projection (onto the first factor) and $\widehat{X}$ the (topological)
closure of

$\Gamma:=\Big{\\{}(x,u)\in(X\setminus Y)\times{\mathbb{S}}^{r-1}:\ {\rm
rk}\begin{pmatrix}u_{1}&\cdots&u_{r}\\\
f_{1}(x)&\cdots&f_{r}(x)\end{pmatrix}=1\Big{\\}}.$

Observe that $\widehat{X}=\widetilde{X}_{+}\cup\widetilde{X}_{-}$ is the union
of the irreducible components of the real algebraic set

$\Big{\\{}(x,u)\in X\times{\mathbb{S}}^{r-1}:\ {\rm
rk}\begin{pmatrix}u_{1}&\cdots&u_{r}\\\
f_{1}(x)&\cdots&f_{r}(x)\end{pmatrix}=1\Big{\\}}$

different from $Y\times{\mathbb{S}}^{r-1}$. If we denote
$\widehat{\pi}:=\pi|_{\widehat{X}}$, it holds that
$(\widehat{X},\widehat{\pi})$ is the twisted Nash double of the drilling blow-
up $(\widetilde{X}_{+},\pi_{+})$ of $X$ with center $Y$ (where
$\pi_{+}:=\pi|_{\widetilde{X}_{+}}$). Proceeding similarly to the proof of
[BCR, Prop.3.5.11(i)], one shows that $\widehat{X}$ is a non-singular real
algebraic set. In addition, proceeding similarly to the proof of [BCR,
Prop.3.5.11(ii)], one shows that there exists a finite open cover of $Y$ by
Zariski open subsets $U_{i}$ such that $\pi^{-1}(U_{i})$ is Nash diffeomorphic
to $U_{i}\times{\mathbb{S}}^{d-e-1}$ via $\pi$ (use also Fact 3.7(iii)). It
holds
$\widetilde{X}_{\epsilon}=\widehat{X}\cap\\{\epsilon(u_{1}f_{1}+\cdots+u_{r}f_{r})\geq
0\\}$ and the Zariski closure of $\widetilde{X}_{\epsilon}$ is contained in
$\widehat{X}$.

If ${\mathcal{S}}\subset X$ is a pure dimensional semialgebraic set of
dimension $d$, then
$\operatorname{Reg}({\mathcal{S}})=\operatorname{Sth}({\mathcal{S}})$ and
${\mathcal{T}}:=\operatorname{Cl}(\pi_{+}^{-1}({\mathcal{S}}\setminus
Y))\cap\pi_{+}^{-1}({\mathcal{S}})$ is a pure dimensional semialgebraic set of
dimension $d$ such that
$\operatorname{Reg}({\mathcal{T}})=\operatorname{Sth}({\mathcal{T}})$. This is
because both $X$ and $\widehat{X}$ are non-singular real algebraic sets of
dimension $d$ and
$\pi_{+}^{-1}|_{\widetilde{X}_{+}}:\widetilde{X}_{+}\setminus\pi_{+}^{-1}(Y)\to
X\setminus Y$ is a Nash diffeomorphism.

######  Remark 3.12.

If $X$ is irreducible, then $\widehat{X}$ does not need to be irreducible
(even if $X$ is connected).

Consider for instance the plane $X:={\mathbb{R}}^{2}$ and the line $Y:=\\{{\tt
x}=0\\}$. Then $\mathcal{I}(Y)=({\tt x})$ and $\widehat{X}=\\{{\tt
z}=-1\\}\cup\\{{\tt z}=1\\}\subset{\mathbb{R}}^{3}$, which is disconnected.
However, $\widehat{X}$ is the Zariski closure of $\widetilde{X}_{+}=\\{{\tt
x}\geq 0,{\tt z}=1\\}\cup\\{{\tt x}\leq 0,{\tt z}=-1\\}$ (resp.
$\widetilde{X}_{-}=\\{{\tt x}\leq 0,{\tt z}=1\\}\cup\\{{\tt x}\geq 0,{\tt
z}=-1\\}$).

However, if $X$ is irreducible, at least we have the following result that we
prove below:

######  Lemma 3.13.

If $X$ is irreducible, then $\widehat{X}$ is the Zariski closure of
$\widetilde{X}_{\epsilon}$ for $\epsilon=\pm 1$.

####  3.4.1. Relationship with algebraic blow-up

Denote $[z]:=[z_{1}:\cdots:z_{r}]\in{\mathbb{R}}{\mathbb{P}}^{r-1}$ and define

$\Gamma^{\prime}:=\\{(x,[z])\in(X\setminus
Y)\times{\mathbb{R}}{\mathbb{P}}^{r-1}:\ {\rm
rk}\begin{pmatrix}z_{1}&\cdots&z_{r}\\\
f_{1}(x)&\cdots&f_{r}(x)\end{pmatrix}=1\\}.$

The closure $B(X,Y)$ of $\Gamma^{\prime}$ in
$X\times{\mathbb{R}}{\mathbb{P}}^{r-1}$ together with the restriction
$\pi^{\prime}$ to $B(X,Y)$ of the projection
$X\times{\mathbb{R}}{\mathbb{P}}^{r-1}\to Y$ (onto the first factor) is the
(algebraic) blow-up of $X$ with center $Y$. Consider the polynomial map
$\Theta:X\times{\mathbb{S}}^{r-1}\to
X\times{\mathbb{R}}{\mathbb{P}}^{r-1},(x,z)\to(x,[z])$ and its restriction
$\theta:=\Theta|_{\widehat{X}}:\widehat{X}\to B(X,Y)$. We have:

  * (i)

$\theta(\widehat{X})=B(X,Y)$, $\pi^{\prime}\circ\theta=\widehat{\pi}$ and
$\theta^{-1}(x,[z])=\\{(x,z),(z,-z)\\}$ for each $(x,[z])\in B(X,Y)$.

  * (ii)

$\theta$ is an unramified two to one polynomial covering of $B(X,Y)$.

###### Proof of Lemma 3.13.

As $X$ is irreducible, $T:=B(X,Y)$ is irreducible (because $X\setminus Y$ and
$T\setminus\pi^{\prime-1}(Y)$ are biregularly diffeomorphic via
$\pi^{\prime}|_{T\setminus\pi^{\prime-1}(Y)}$ [BCR, Prop.3.5.8]).

The polynomial map $\theta:\widehat{X}\to T$ is an unramified two to one
polynomial covering. If $\widehat{X}$ is irreducible, $\widehat{X}$ is the
Zariski closure of $\widetilde{X}_{\epsilon}$ for $\epsilon=\pm 1$, because
$\dim(\widehat{X})=\dim(\widetilde{X}_{\epsilon})$. Thus, let us assume
$\widehat{X}$ is reducible. Let $Z_{1},\ldots,Z_{\ell}$ be irreducible
components of $\widehat{X}=\widetilde{X}_{+}\cup\widetilde{X}_{-}$.

As $\widehat{X}$ is pure dimensional and non-singular, we deduce $Z_{i}$ is
pure dimensional and non-singular, so it is a union of connected components of
$\widehat{X}$. Thus, $Z_{i}$ is an open and closed subset of $\widehat{X}$. As
$\theta$ is an open and closed map and each $Z_{i}$ is an open and closed
subset of $\widehat{X}$, we deduce $\theta(Z_{i})$ is a union of connected
components of $T$.

Let $\widetilde{\widehat{X}}$ be the complexification of $\widehat{X}$,
$\widetilde{Z}_{i}$ the complexification of $Z_{i}$ and $\widetilde{T}$ the
complexification of $T$. Observe that
$\widetilde{\widehat{X}}=\bigcup_{i=1}^{\ell}\widetilde{Z}_{i}$.

Consider the polynomial extension
$\widetilde{\theta}:\widetilde{\widehat{X}}\to\widetilde{T}$ of $\theta$ to
$\widetilde{\widehat{X}}$. Its image is contained in $\widetilde{T}$, because
$\theta(\widehat{X})=T$. Observe that
$\widetilde{\theta}|_{\widetilde{Z}_{i}}$ is a dominant polynomial map,
because $\theta$ is a local diffeomorphism and $Z_{i}$ is a union of connected
components of $\widehat{X}$. In particular
$\widetilde{\theta}(\widetilde{Z}_{i})$ is a Zariski open subset of the
irreducible algebraic set $\widetilde{T}$.

Denote with ${\mathcal{R}}(\widetilde{Z}_{i})$ the field of rational functions
on $\widetilde{Z}_{i}$ and with ${\mathcal{R}}(\widetilde{T})$ the field of
rational functions on $\widetilde{T}$. The map
$\widetilde{\theta}|_{\widetilde{Z}_{i}}^{*}:{\mathcal{R}}(\widetilde{T})\to{\mathcal{R}}(\widetilde{Z}_{i}),\
f\mapsto f\circ\widetilde{\theta}|_{\widetilde{Z}_{i}}$ is a homomorphism of
fields of the same transcendence degree $d=\dim(X)$ over ${\mathbb{C}}$.
Consequently, ${\mathcal{R}}(\widetilde{Z}_{i})$ is an algebraic extension of
${\mathcal{R}}(\widetilde{T})$ of finite degree $m_{i}$. By [Ha, Prop.7.16]
the number of points in a general fiber of
$\widetilde{\theta}|_{\widetilde{Z}_{i}}$ is equal to $m_{i}$. As
$(\widetilde{\theta}|_{\widetilde{Z}_{i}})^{-1}(q)\setminus(\theta|_{Z_{i}})^{-1}(q)$
is even for each $q\in\theta(Z_{i})$, we deduce that the fibers of
$\theta|_{Z_{i}}:Z_{i}\to\theta(Z_{i})\subset T$ all have the same
cardinality, which is either one or two. Thus,
$\theta|_{Z_{i}}:Z_{i}\to\theta(Z_{i})\subset T$ is either a polynomial
diffeomorphism or an unramified two to one polynomial covering for
$i=1,\ldots,\ell$. We distinguish two cases:

Case 1. If $\theta|_{Z_{i}}:Z_{i}\to\theta(Z_{i})\subset T$ is an unramified
two to one polynomial covering, then
$Z_{i}\cap\widetilde{X}_{\epsilon}\neq\varnothing$ and $Z_{i}$ is contained in
the Zariski closure of $\widetilde{X}_{\epsilon}$.

Case 2. If $\theta|_{Z_{i}}:Z_{i}\to\theta(Z_{i})\subset T$ is a polynomial
diffeomorphism, let us prove: $\theta(Z_{i})=T$, $\ell=2$ and
$\theta|_{Z_{i}}:Z_{i}\to T$ is a polynomial diffeomorphism for $i=1,2$.

Suppose that $\theta(Z_{i})\neq T$. Then there exists a connected component
$M$ of $T$ such that $\theta(Z_{i})\cap M=\varnothing$ (recall that
$\theta(Z_{i})$ is a union of connected components of $T$). As $T$ is pure
dimensional of dimension $d$, we deduce that $M$ has dimension $d$. As
$\theta|_{Z_{i}}:Z_{i}\to\theta(Z_{i})$ is a polynomial diffeomorphism, we
deduce that $m_{i}$ is odd. Pick a point $q\in M$ such that
$(\theta|_{Z_{i}})^{-1}(q)$ has exactly $m_{i}$ points. As the cardinality of
$(\widetilde{\theta}|_{\widetilde{Z}_{i}})^{-1}(q)\setminus(\theta|_{Z_{i}})^{-1}(q)$
is even and $m_{i}$ is odd, the cardinality of
$(\theta|_{Z_{i}})^{-1}(q)=\varnothing$ is also odd, which is a contradiction.
Consequently, $\theta(Z_{i})=T$.

As $\theta:\widehat{X}\to T$ is an unramified two to one polynomial covering,
we deduce that it is trivial, that is, $\ell=2$ and $\theta|_{Z_{i}}:Z_{i}\to
T$ is a polynomial diffeomorphism for $i=1,2$.

We deduce either:

  * •

Each restriction $\theta|_{Z_{i}}:Z_{i}\to\theta(Z_{i})\subset T$ is an
unramified two to one polynomial covering for $i=1,\ldots,\ell$ and the
Zariski closure of $\widetilde{X}_{\epsilon}$ is
$\bigcup_{i=1}^{\ell}Z_{i}=\widehat{X}$, or

  * •

$\ell=2$ and $\theta|_{Z_{i}}:Z_{i}\to T$ is a polynomial diffeomorphism for
$i=1,2$.

Suppose that we are under the hypothesis of the second case and pick a point
$y\in(\pi^{\prime})^{-1}(Y)\subset T$. Observe that
$\theta^{-1}(y)\subset\widetilde{X}_{+}\cap\widetilde{X}_{-}$, so
$\widetilde{X}_{\epsilon}\cap Z_{i}\neq\varnothing$ for $\epsilon=\pm$ and
$i=1,2$. As $\widetilde{X}_{\epsilon}$ is pure dimensional of dimension $d$
and each $Z_{i}$ is pure dimensional of dimension $d$ and open and closed in
$\widehat{X}=Z_{1}\sqcup Z_{2}$, we conclude that each $Z_{i}$ is contained in
the Zariski closure of $\widetilde{X}_{\epsilon}$, so $\widehat{X}$ is the
Zariski closure of $\widetilde{X}_{\epsilon}$ for $\epsilon=\pm$, as required.
∎

##  4\. Desingularization of semialgebraic sets

In this section we prove Theorems 1.4 and 1.6. We begin recalling the concept
of checkerboard set [Fe2, Thm.8.4].

###  4.1. Checkerboard sets

If ${\mathcal{S}}\subset{\mathbb{R}}^{m}$ is a semialgebraic set, we denote
$\partial{\mathcal{S}}:=\operatorname{Cl}({\mathcal{S}})\setminus\operatorname{Reg}({\mathcal{S}})$.
As ${\mathcal{S}}$ is closed, we have
$\partial{\mathcal{S}}=\operatorname{Sing}({\mathcal{S}})$. If ${\mathcal{Q}}$
is a Nash manifold with corners such that
$\overline{{\mathcal{Q}}}^{\operatorname{zar}}$ is a non-singular real
algebraic set, then
$\operatorname{Int}({\mathcal{Q}})=\operatorname{Reg}({\mathcal{Q}})$ and the
differences ${\mathcal{Q}}\setminus\operatorname{Int}({\mathcal{Q}})$ and
${\mathcal{Q}}\setminus\operatorname{Reg}({\mathcal{Q}})$ define the same
semialgebraic set $\partial{\mathcal{Q}}$. In other situations, one should be
careful with the possible ambiguity in the definition of the set
$\partial{\mathcal{Q}}$. In this section such ambiguity does not appear
because all the Zariski closures of all involved Nash manifolds with corners
are non-singular.

######  Definition 4.1.

A pure dimensional semialgebraic set ${\mathcal{T}}\subset{\mathbb{R}}^{n}$ is
a checkerboard set if it satisfies the following properties:

  * •

$\overline{{\mathcal{T}}}^{\operatorname{zar}}$ is a non-singular real
algebraic set.

  * •

$\overline{\partial{\mathcal{T}}}^{\operatorname{zar}}$ is a normal-crossings
divisor of $\overline{{\mathcal{T}}}^{\operatorname{zar}}$.

  * •

$\operatorname{Reg}({\mathcal{T}})=\operatorname{Sth}({\mathcal{T}})$ is
connected.

Recall that any checkerboard set is connected by analytic paths [Fe2, Main
Thm.1.4, Lem.8.2]. In [Fe2, Thm.8.4] the following crucial result is proved.

######  Theorem 4.2 ([Fe2, Thm.8.4]).

Let ${\mathcal{S}}\subset{\mathbb{R}}^{m}$ be a semialgebraic set of dimension
$d\geq 2$ connected by analytic paths and denote
$X:=\overline{{\mathcal{S}}}^{\operatorname{zar}}$. Then there exists a
checkerboard set ${\mathcal{T}}\subset{\mathbb{R}}^{n}$ of dimension $d$ and a
proper regular map $f:Y:=\overline{{\mathcal{T}}}^{\operatorname{zar}}\to X$
such that $f({\mathcal{T}})={\mathcal{S}}$.

As the map $f$ is proper, if the semialgebraic set ${\mathcal{S}}$ is compact,
we may assume that also the checkerboard set ${\mathcal{T}}$ is compact (see
the proof of [Fe2, Thm.8.4]). Even if it is not explicitly quoted in the
statement of [Fe2, Thm.8.4], there is actually proved much more. Looking at
this proof, we can reformulate the statement of Theorem 4.2 in the following
way:

######  Theorem 4.3.

Let ${\mathcal{S}}\subset{\mathbb{R}}^{m}$ be a semialgebraic set connected by
analytic paths of dimension $d\geq 2$ and denote
$X:=\overline{{\mathcal{S}}}^{\operatorname{zar}}$. Then there exists a
checkerboard set ${\mathcal{T}}\subset{\mathbb{R}}^{n}$ of dimension $d$ and a
proper polynomial map $f:Y:=\overline{{\mathcal{T}}}^{\operatorname{zar}}\to
X$ such that $f({\mathcal{T}})={\mathcal{S}}$ and the restriction
$f|_{{\mathcal{T}}}:{\mathcal{T}}\to{\mathcal{S}}$ is also proper. Moreover,
there exists a semialgebraic set ${\mathcal{R}}\subset{\mathcal{S}}$ of
dimension strictly smaller than $d$ such that
$f^{-1}({\mathcal{R}})\subset\overline{\partial{\mathcal{T}}}^{\operatorname{zar}}$,
${\mathcal{S}}\setminus{\mathcal{R}}$ and ${\mathcal{T}}\setminus
f^{-1}({\mathcal{R}})$ are Nash manifolds and $f|_{{\mathcal{T}}\setminus
f^{-1}({\mathcal{R}})}:{\mathcal{T}}\setminus
f^{-1}({\mathcal{R}})\to{\mathcal{S}}\setminus{\mathcal{R}}$ is a Nash
diffeomorphism. In particular, if $X$ is compact, also $Y$ is compact.

###  4.2. Desingularization of closed checkerboard sets

In order to prove Theorem 1.4, in view of Theorem 4.3 we are reduced to assume
that ${\mathcal{T}}\subset{\mathbb{R}}^{n}$ is a closed checkerboard set. We
need some preliminary notations and results on checkerboard sets.

####  4.2.1. Definition and properties of the function $e$ of a checkerboard
set

Given a non-singular real algebraic set $X\subset{\mathbb{R}}^{n}$ of
dimension $d$ and a normal-crossings divisor $Z\subset X$, we denote

$\displaystyle\operatorname{Sing}_{0}(Z):=Z,$
$\displaystyle\operatorname{Sing}_{\ell}(Z):=\operatorname{Sing}(\operatorname{Sing}_{\ell-1}(Z))\quad\text{for
$1\leq\ell\leq d$}.$

Observe that if $\operatorname{Sing}_{\ell}(Z)\neq\varnothing$, then
$\dim(\operatorname{Sing}_{\ell}(Z))=d-\ell-1$. In addition, if
$\operatorname{Sing}_{\ell}(Z)=\varnothing$, then
$\operatorname{Sing}_{k}(Z)=\varnothing$ for $k\geq\ell$. In particular,
$\operatorname{Sing}_{d}(Z)=\varnothing$. Recall that if
$A_{x}\subset{\mathbb{R}}^{n}_{x}$ is a set germ at $x$, its analytic closure
$\overline{A}^{\rm an}_{x}$ is the smallest analytic set germ of
${\mathbb{R}}^{n}_{x}$ that contains $A_{x}$.

Let ${\mathcal{T}}\subset{\mathbb{R}}^{n}$ be a closed checkerboard set and
denote $X:=\overline{{\mathcal{T}}}^{\operatorname{zar}}$ and
$\partial{\mathcal{T}}:={\mathcal{T}}\setminus\operatorname{Reg}({\mathcal{T}})$.
For each point $x\in{\mathcal{T}}$ there exists a coordinate system $({\tt
u}_{1},\ldots,{\tt u}_{d})$ of the Nash manifold $X$ at $x$ and an integer
$0\leq r_{x}\leq d$ such that either $\overline{\partial{\mathcal{T}}}^{\rm
an}_{x}=\\{{\tt u}_{1}\cdots{\tt u}_{r_{x}}=0\\}_{x}$ if $r_{x}\geq 1$ or
$x\in\operatorname{Reg}({\mathcal{T}})={\mathcal{T}}\setminus\partial{\mathcal{T}}$
if $r_{x}=0$. We denote with $e_{x}:=e_{x}({\mathcal{T}})\leq r_{x}$ the
number of indices $1\leq i\leq r_{x}$ such that the germ
${\mathcal{T}}_{x}\setminus\\{{\tt u}_{i}=0\\}_{x}$ is disconnected. If
$r_{x}\leq 1$, then $e_{x}=0$. We have the following:

######  Lemma 4.4.

The value $e_{x}=0$ if and only if ${\mathcal{T}}_{x}$ is the germ at $x$ of a
Nash manifold with corners.

###### Proof.

The if implication is clear because after changing the sign of some of the
variables if necessary, we may assume either
$x\in\operatorname{Reg}({\mathcal{T}})$ or ${\mathcal{T}}_{x}=\\{{\tt
u}_{1}\geq 0,\ldots,{\tt u}_{r_{x}}\geq 0\\}_{x}$ for some $1\leq r_{x}\leq
d$, so $e_{x}=0$.

Conversely, suppose $e_{x}=0$. Then, either
$x\in\operatorname{Reg}({\mathcal{T}})$ (so
${\mathcal{T}}_{x}=\operatorname{Reg}({\mathcal{T}})_{x}$ is the germ at $x$
of a Nash manifold) or $\overline{\partial{\mathcal{T}}}^{\rm an}_{x}=\\{{\tt
u}_{1}\cdots{\tt u}_{r_{x}}=0\\}_{x}$ for some $1\leq r_{x}\leq d$ and, after
changing the sign of some of the variables, we may assume
${\mathcal{T}}_{x}\subset\\{{\tt u}_{1}\geq 0,\ldots,{\tt u}_{r_{x}}\geq
0\\}_{x}$, because $e_{x}=0$. As
${\mathcal{T}}_{x}\setminus\overline{\partial{\mathcal{T}}}^{\rm
an}_{x}=\operatorname{Reg}({\mathcal{T}})_{x}\setminus\overline{\partial{\mathcal{T}}}^{\rm
an}_{x}$ is an open and closed germ,
${\mathcal{T}}_{x}\setminus\overline{\partial{\mathcal{T}}}^{\rm an}_{x}$ is a
union of connected components of $\\{{\tt u}_{1}\cdots{\tt u}_{r_{x}}\neq
0\\}_{x}$ contained in $\\{{\tt u}_{1}>0,\ldots,{\tt u}_{r_{x}}>0\\}_{x}$, so
${\mathcal{T}}_{x}\setminus\overline{\partial{\mathcal{T}}}^{\rm
an}_{x}=\\{{\tt u}_{1}>0,\ldots,{\tt u}_{r_{x}}>0\\}_{x}$. As
${\mathcal{T}}_{x}$ is closed and pure dimensional and
$\dim(\overline{\partial{\mathcal{T}}}^{\rm an}_{x})<\dim({\mathcal{T}}_{x})$,
we conclude ${\mathcal{T}}_{x}=\\{{\tt u}_{1}\geq 0,\ldots,{\tt u}_{r_{x}}\geq
0\\}_{x}$ is the germ at $x$ of a Nash manifold with corners, as required. ∎

It follows from the previous statement: A closed checkerboard set is a Nash
manifold with corners if and only if $e_{x}({\mathcal{T}})=0$ for each
$x\in{\mathcal{T}}$.

######  Lemma 4.5.

Let ${\mathcal{T}}\subset{\mathbb{R}}^{n}$ be a closed checkerboard set. Then
$e_{x}\neq 1$, for each $x\in\partial{\mathcal{T}}$.

###### Proof.

Let $X:=\overline{{\mathcal{T}}}^{\operatorname{zar}}$. For each
$x\in\partial{\mathcal{T}}$ there exist an open semialgebraic set $U\subset X$
equipped with a Nash diffeomorphism
$u:=(u_{1},\ldots,u_{d}):U\to{\mathbb{R}}^{d}$ and an integer $1\leq r_{x}\leq
d$ such that $u(x)=0$ and $\overline{\partial{\mathcal{T}}}^{{\rm
an}}_{x}=\\{{\tt u}_{1}\cdots{\tt u}_{r_{x}}=0\\}_{x}$.

Suppose that $e_{x}\geq 1$ for some $x\in\partial{\mathcal{T}}$. As $e_{x}\neq
0$, then $r_{x}\geq 2$, because otherwise ${\mathcal{T}}_{x}$ is the germ of a
Nash manifold with boundary and $e_{x}=0$. Up to rename the variables if
necessary, we may assume ${\mathcal{T}}_{x}\setminus\\{{\tt u}_{1}=0\\}_{x}$
is disconnected. Suppose that for each $2\leq i\leq r_{x}$ the germ
${\mathcal{T}}_{x}\setminus\\{{\tt u}_{i}=0\\}_{x}$ is connected. After
changing the signs of some of the variables if necessary, we may assume
${\mathcal{T}}_{x}\subset\\{{\tt u}_{2}\geq 0,\ldots,{\tt u}_{r_{x}}\geq
0\\}_{x}$. Proceeding as in the proof of Lemma 4.4, as
${\mathcal{T}}_{x}\setminus\overline{\partial{\mathcal{T}}}^{\rm
an}_{x}=\operatorname{Reg}({\mathcal{T}})_{x}\setminus\overline{\partial{\mathcal{T}}}^{\rm
an}_{x}$ is an open and closed germ,
${\mathcal{T}}_{x}\setminus\overline{\partial{\mathcal{T}}}^{\rm an}_{x}$ is a
union of connected components of $\\{{\tt u}_{1}\cdots{\tt u}_{r_{x}}\neq
0\\}_{x}$ contained in $\\{{\tt u}_{2}>0,\ldots,{\tt u}_{r_{x}}>0\\}_{x}$. As
${\mathcal{T}}_{x}$ is closed and pure dimensional and
$\dim(\overline{\partial{\mathcal{T}}}^{\rm an}_{x})<\dim({\mathcal{T}}_{x})$,
we have only two possibilities:

  * •

${\mathcal{T}}_{x}=\\{{\tt u}_{1}\geq 0,\ldots,{\tt u}_{r_{x}}\geq 0\\}_{x}$
(up to changing the sign of the germ ${\tt u}_{1}$ if necessary),

  * •

${\mathcal{T}}_{x}=\\{{\tt u}_{2}\geq 0,\ldots,{\tt u}_{r_{x}}\geq 0\\}_{x}$.

In the first case $e_{x}=0$, which contradicts the fact that
${\mathcal{T}}_{x}\setminus\\{{\tt u}_{1}=0\\}_{x}$ is disconnected, whereas
in the second case $\\{{\tt
u}_{1}=0\\}_{x}\not\subset\overline{\partial{\mathcal{T}}}^{{\rm an}}$, which
contradicts the fact that $\overline{\partial{\mathcal{T}}}^{{\rm an}}=\\{{\tt
u}_{1}\cdots{\tt u}_{r_{x}}=0\\}_{x}$. Thus, there exists $2\leq i\leq r_{x}$
such that ${\mathcal{T}}_{x}\setminus\\{{\tt u}_{i}=0\\}_{x}$ is disconnected,
so $e_{x}\geq 2$ as required. ∎

We show next that the function $e({\mathcal{T}})$ is a semialgebraic function.

######  Lemma 4.6 (Semialgebricity of $e({\mathcal{T}})$).

Let ${\mathcal{T}}\subset{\mathbb{R}}^{n}$ be a $d$-dimensional closed
checkerboard set and let $0\leq e\leq d$. The set
${\mathcal{T}}_{e}:=\\{x\in{\mathcal{T}}:\,e_{x}=e\\}$ is a semialgebraic set
and ${\mathcal{T}}_{0}$ is an open subset of ${\mathcal{T}}$. In addition, if
$Z$ is the Zariski closure of $\partial{\mathcal{T}}$ and $C$ is a connected
component of
$\operatorname{Sing}_{\ell}(Z)\setminus\operatorname{Sing}_{\ell+1}(Z)$ for
some $0\leq\ell\leq d-1$, then either $C\cap{\mathcal{T}}=\varnothing$ or
$C\subset{\mathcal{T}}$ and $e({\mathcal{T}})$ is constant on $C$.

###### Proof.

The boundary $\partial{\mathcal{T}}$ is a closed semialgebraic subset of the
Nash manifold $X:=\overline{{\mathcal{T}}}^{\operatorname{zar}}$. For each
$x\in\partial{\mathcal{T}}$ there exists a coordinate system $({\tt
u}_{1},\ldots,{\tt u}_{d})$ of $X$ at $x$ and an integer $1\leq r_{x}\leq d$
such that $\overline{\partial{\mathcal{T}}}^{\rm an}_{x}=\\{{\tt
u}_{1}\cdots{\tt u}_{r_{x}}=0\\}_{x}$. By [FGR, Prop.4.4, Prop.4.6] there
exist finitely many open semialgebraic sets $\\{U_{i}\\}_{i=1}^{s}$ equipped
with Nash diffeomorphisms ${\tt u}_{i}:=({\tt u}_{i1},\ldots,{\tt
u}_{id}):U_{i}\to{\mathbb{R}}^{d}$ and integers $r_{i}\geq 1$ such that
$\overline{\partial{\mathcal{T}}}^{{\rm an}}_{x}=\\{{\tt u}_{i1}\cdots{\tt
u}_{ir_{i}}=0\\}_{x}$ for all $x\in{\mathcal{T}}\cap U_{i}$.

Fix $i\in\\{1,\ldots,s\\}$ and $J\subset\\{1,\ldots,r_{i}\\}$. Reordering the
variables if necessary, we may assume $J=\\{1,\ldots,m\\}$ for some $1\leq
m\leq r_{i}$. Let $V$ be a connected component of $U_{i}\setminus\\{{\tt
u}_{i,m+1}\cdots{\tt u}_{i,r_{i}}=0\\}$. After changing the signs of some of
the variables if necessary, we may assume $V:=\\{{\tt u}_{i,m+1}>0,\ldots,{\tt
u}_{i,r_{i}}>0\\}$. Consider the semialgebraic set
${\mathcal{T}}^{\prime}:={\mathcal{T}}\cap U_{i}\cap V$ and the projection

$\pi_{i}:{\mathbb{R}}^{d}\equiv{\mathbb{R}}^{m}\times{\mathbb{R}}^{d-m}\to{\mathbb{R}}^{m}$

onto the first $m$ coordinates. We take coordinates $(x_{1},\ldots,x_{m})$ on
${\mathbb{R}}^{m}$ and $(x_{m+1},\ldots,x_{d})$ on ${\mathbb{R}}^{d-m}$.
Denote $\Lambda_{i}:=\\{{\tt x}_{i,m+1}>0,\ldots,{\tt
x}_{i,r_{i}}>0\\}\subset{\mathbb{R}}^{d-m}$. As $u_{i}({\mathcal{T}}\cap
U_{i})$ is the union of connected components of
${\mathbb{R}}^{d}\setminus\\{{\tt x}_{1}\ldots{\tt x}_{r_{i}}=0\\}$, there
exist $\varepsilon_{i,1},\ldots,\varepsilon_{i,k}\in\\{-1,1\\}^{r_{i}}$ such
that

$u_{i}({\mathcal{T}}\cap U_{i})=\bigcup_{p=1}^{k}\\{\varepsilon_{1p_{1}}{\tt
x}_{1}\geq 0,\ldots,\varepsilon_{ip_{r_{i}}}{\tt x}_{r_{i}}\geq 0\\}$

where
$\varepsilon_{ip}:=(\varepsilon_{ip_{1}},\ldots,\varepsilon_{ip_{r_{i}}})$.
Consequently,

$u_{i}({\mathcal{T}}^{\prime})=u_{i}({\mathcal{T}}\cap U_{i}\cap
V)=\bigcup_{p=1}^{k}\\{\varepsilon_{1p_{1}}{\tt x}_{1}\geq
0,\ldots,\varepsilon_{ip_{r_{i}}}{\tt x}_{r_{i}}\geq 0,{\tt
x}_{m+1}>0,\ldots,{\tt x}_{r_{i}}>0\\}$

Observe that

$\\{\varepsilon_{1p_{1}}{\tt x}_{1}\geq 0,\ldots,\varepsilon_{ip_{r_{i}}}{\tt
x}_{r_{i}}\geq 0,{\tt x}_{m+1}>0,\ldots,{\tt x}_{r_{i}}>0\\}\\\
=\begin{cases}\\{\varepsilon_{1p_{1}}{\tt x}_{1}\geq
0,\ldots,\varepsilon_{ip_{m}}{\tt x}_{m}\geq 0,{\tt x}_{m+1}>0,\ldots,{\tt
x}_{r_{i}}>0\\}&\text{if
}\varepsilon_{ip_{m+1}}=\cdots=\varepsilon_{ip_{r_{i}}}=1,\\\
\varnothing&\text{otherwise}.\end{cases}$

Thus,

$u_{i}({\mathcal{T}}^{\prime})=\bigcup_{\begin{subarray}{c}p\in\\{1,\ldots,k\\}\\\
(\varepsilon_{ip_{m+1}},\ldots,\varepsilon_{ip_{r_{i}}})=(1,\ldots,1)\end{subarray}}\\{\varepsilon_{1p_{1}}{\tt
x}_{1}\geq 0,\ldots,\varepsilon_{ip_{m}}{\tt x}_{m}\geq 0,{\tt
x}_{m+1}>0,\ldots,{\tt
x}_{r_{i}}>0\\}=\pi_{i}(u_{i}({\mathcal{T}}^{\prime}))\times\Lambda_{i}.$

Observe that $W:=\\{{\tt u}_{i1}=0,\ldots,{\tt u}_{im}=0,{\tt
u}_{i,m+1}>0,\ldots,{\tt u}_{ir_{i}}>0\\}\subset{\mathcal{T}}$ and for each
$x\in W$, we have
$e_{x}({\mathcal{T}})=e_{0}(\pi_{i}(u_{i}({\mathcal{T}}^{\prime})))$, so
$e_{x}({\mathcal{T}})$ is constant on $W$. As each $x\in{\mathcal{T}}\cap
U_{i}\cap\\{{\tt u}_{i1}\ldots{\tt u}_{ir_{i}}=0\\}$ belongs to a set of the
type $W_{J,\varepsilon}:=\\{{\tt u}_{ij}=0,j\in J\\}\cap\\{\varepsilon_{j}{\tt
u}_{ij}>0,j\not\in J\\}$ where $J=\\{1,\ldots,r_{i}\\}$ and
$\varepsilon_{j}\in\\{-1,1\\}$, the function $e({\mathcal{T}})$ provides a
semialgebraic partition of $\partial{\mathcal{T}}\cap U_{i}$ for each
$i=1,\ldots,s$. In particular, each set ${\mathcal{T}}_{e}$ is semialgebraic.
As the condition ‘to be a Nash manifold with corners’ is a local open
condition, we deduce ${\mathcal{T}}_{0}$ is an open semialgebraic subset of
${\mathcal{T}}$.

We have proved above that if $Z^{\prime}_{i}:=\\{{\tt u}_{i1}\cdots{\tt
u}_{ir_{i}}=0\\}$ and $C_{i}^{\prime}$ is a connected component of
$\operatorname{Sing}_{\ell}(Z^{\prime}_{i})\setminus\operatorname{Sing}_{\ell+1}(Z^{\prime}_{i})$
for some $0\leq\ell\leq d-1$, then either
$C_{i}^{\prime}\cap{\mathcal{T}}=\varnothing$ or
$C_{i}^{\prime}\subset{\mathcal{T}}$ and $e({\mathcal{T}})$ is constant on
$C_{i}^{\prime}$. Let $C$ be a connected component of
$\operatorname{Sing}_{\ell}(Z)\setminus\operatorname{Sing}_{\ell+1}(Z)$. As
${\mathcal{T}}$ is a checkerboard set, either $C\cap{\mathcal{T}}=\varnothing$
or $C\subset{\mathcal{T}}$. Assume we are in the second case. As we have
proved above, there exists a finite semialgebraic open covering
${\mathcal{W}}_{C}:=\\{W_{i}\\}_{i=1}^{p}$ of $C$ such that $e({\mathcal{T}})$
is constant on $W_{i}$. If $x,y\in C$, there exists
$W_{i_{1}},\ldots,W_{i_{q}}\in{\mathcal{W}}_{C}$ such that $x\in W_{i_{1}}$,
$y\in W_{i_{q}}$ and $W_{i_{j}}\cap W_{i_{j+1}}\neq\varnothing$ for
$j=1,\ldots,q-1$. As $e({\mathcal{T}})|_{W_{i_{j}}}$ is constant, we deduce
recursively that $e_{x}({\mathcal{T}})=e_{y}({\mathcal{T}})$, as required. ∎

We show next that the function $e({\mathcal{T}})$ is upper semi-continuous.

######  Lemma 4.7 (Upper semi-continuity of $e({\mathcal{T}})$).

Let ${\mathcal{T}}\subset{\mathbb{R}}^{d}$ be a $d$-dimensional closed
checkerboard set and let $x\in\partial{\mathcal{T}}$. Then $e_{x}\geq e_{y}$
for each $y\in{\mathcal{T}}$ close enough to $x$.

###### Proof.

Let $k$ be the maximum of the values $e\geq 0$ such that
$x\in\operatorname{Cl}({\mathcal{T}}_{e})$. It is enough to check that
$e_{x}\geq k$. Consider the Nash manifold
$X:=\overline{{\mathcal{T}}}^{\operatorname{zar}}$ and let $U\subset X$ be an
open semialgebraic neighborhood of $x$ equipped with a Nash diffeomorphism
$u:=(u_{1},\ldots,u_{d}):U\to{\mathbb{R}}^{d}$ such that $u(x)=0$ and
$\overline{\partial{\mathcal{T}}}^{\rm an}_{x}=\\{{\tt u}_{1}\cdots{\tt
u}_{r}=0\\}_{x}$. If $e_{x}<k$, we may assume ${\mathcal{T}}_{x}\subset\\{{\tt
u}_{k}\geq 0,\ldots,{\tt u}_{r}\geq 0\\}_{x}$ and
$\partial{\mathcal{T}}_{x}\subset\\{{\tt u}_{1}\cdots{\tt u}_{r}=0\\}_{x}$.
Shrinking $U$ if necessary, we have ${\mathcal{T}}\cap U\subset\\{{\tt
u}_{k}\geq 0,\ldots,{\tt u}_{r}\geq 0\\}$ and $\partial{\mathcal{T}}\cap
U\subset\\{{\tt u}_{1}\cdots{\tt u}_{r}=0\\}$. Thus, $e_{y}<k$ for each $y\in
U$, which is a contradiction because
$x\in\operatorname{Cl}({\mathcal{T}}_{k})$. ∎

######  Remark 4.8.

As $e({\mathcal{T}})$ is upper semi-continuous, the set $\bigcup_{k\geq
e}{\mathcal{T}}_{k}$ is a closed subset of $\partial{\mathcal{T}}$ for each
$1\leq e\leq d$.

####  4.2.2. Closed checkerboard sets and drilling blow-up.

We want to study now how the value $e_{x}({\mathcal{T}})$ changes after
performing a drilling blow-up. We show the following:

######  Lemma 4.9.

Let ${\mathcal{S}}\subset{\mathbb{R}}^{n}$ be a $d$-dimensional closed
checkerboard set. Denote $X:=\overline{{\mathcal{S}}}^{\operatorname{zar}}$
and $Z:=\overline{\partial{\mathcal{S}}}^{\operatorname{zar}}$. Let
$Z_{1},\ldots,Z_{r}$ be the irreducible components of $Z$ and $Y$ an
irreducible component of $Z_{1}\cap\cdots\cap Z_{\ell}$ for some
$2\leq\ell\leq r$. Let $(\widehat{X},\widehat{\pi})$ be the twisted Nash
double of the drilling blow-up $(\widetilde{X},\pi_{+})$ of $X$ with center
$Y$. Let ${\mathcal{T}}:=\operatorname{Cl}(\pi_{+}^{-1}({\mathcal{S}}\setminus
Y))=\pi_{+}^{-1}({\mathcal{S}})\cap\operatorname{Cl}(\pi_{+}^{-1}({\mathcal{S}}\setminus
Y))$ be the strict transform of ${\mathcal{S}}$ under $\pi_{+}$. Then
${\mathcal{T}}$ is a $d$-dimensional closed checkerboard set,
$e_{y}({\mathcal{T}})\leq e_{\pi_{+}(y)}({\mathcal{S}})$ for each
$y\in\partial{\mathcal{T}}$ and
$e_{y}({\mathcal{T}})<e_{\pi_{+}(y)}({\mathcal{S}})$ for each
$y\in\partial{\mathcal{T}}\cap\pi_{+}^{-1}(Y)$ such that
${\mathcal{S}}_{\pi_{+}(y)}\setminus Z_{i,\pi_{+}(y)}$ is not connected for
$i=1,\ldots,\ell$.

###### Proof.

As $\ell\geq 2$, we have $\dim(Y)\leq d-2$, so
$\operatorname{Reg}({\mathcal{S}})\setminus Y$ is connected, because
$\operatorname{Reg}({\mathcal{S}})$ is a connected $d$-dimensional Nash
manifold. As ${\mathcal{S}}$ and ${\mathcal{T}}$ are both pure dimensional, we
have

$\pi_{+}^{-1}(\operatorname{Reg}({\mathcal{S}})\setminus
Y)\subset\operatorname{Sth}({\mathcal{T}})=\operatorname{Reg}({\mathcal{T}})\subset\operatorname{Cl}(\pi_{+}^{-1}({\mathcal{S}}\setminus
Y))=\operatorname{Cl}(\pi_{+}^{-1}(\operatorname{Reg}({\mathcal{S}})\setminus
Y)).$

Thus, $\operatorname{Reg}({\mathcal{T}})$ is connected and ${\mathcal{T}}$ is
a checkerboard set, because (see §3.4): it is closed,
$\overline{{\mathcal{T}}}^{\operatorname{zar}}=\widehat{X}$ is a non-singular
real algebraic set and the Zariski closure of $\partial{\mathcal{T}}$ is a
union of irreducible components of
$\widehat{\pi}^{-1}(\overline{\partial{\mathcal{S}}}^{\operatorname{zar}})$,
which is a normal-crossings divisor by Remarks 3.4 and 3.10.

As
$\pi_{+}|_{\widetilde{X}\setminus\pi_{+}^{-1}(Y)}:\widetilde{X}\setminus\pi_{+}^{-1}(Y)\to
X\setminus Y$ is a Nash diffeomorphism, it holds
$e_{y}({\mathcal{T}})=e_{\pi_{+}(y)}({\mathcal{S}})$ for each
$y\in\partial{\mathcal{T}}\setminus\pi_{+}^{-1}(Y)$. Let us see what happens
at the points of $\partial{\mathcal{T}}\cap\pi_{+}^{-1}(Y)$. Fix a point
$y\in\partial{\mathcal{T}}\cap\pi_{+}^{-1}(Y)$ and denote $x:=\pi_{+}(y)\in
Y$.

Assume that the irreducible components of $Z$ that contain $x$ are
$Z_{1},\ldots,Z_{r^{\prime}}$ for some $2\leq\ell\leq r^{\prime}\leq r$. Let
$U\subset X$ be an open semialgebraic neighborhood of $x$ equipped with a Nash
diffeomorphism $u:=(u_{1},\ldots,u_{d}):U\to{\mathbb{R}}^{d}$ such that
$u(Z\cap U)=\\{{\tt u}_{1}\cdots{\tt u}_{r^{\prime}}=0\\}$ and $u(Y\cap
U)=\\{{\tt u}_{1}=0,\ldots,{\tt u}_{\ell}=0\\}$. Write $e:=\dim(Y)=d-\ell$ and
assume that $e_{x}({\mathcal{S}})=k\leq r$. Reordering the variables and
changing their signs if necessary, we may assume

${\mathcal{S}}\cap U\subset\\{{\tt u}_{1}\geq 0,\ldots,{\tt u}_{m}\geq 0,{\tt
u}_{\ell+1}\geq 0,\ldots,{\tt u}_{s}\geq 0\\}$ (4.1)

for some $0\leq m\leq\ell$ and $\ell\leq s\leq r^{\prime}$ and both $m$ and
$s$ are maximal satisfying (4.1). If $m=0$, then ${\mathcal{S}}\cap
U\subset\\{{\tt u}_{\ell+1}\geq 0,\ldots,{\tt u}_{s}\geq 0\\}$, whereas if
$s=\ell$, then ${\mathcal{S}}\cap U\subset\\{{\tt u}_{1}\geq 0,\ldots,{\tt
u}_{m}\geq 0\\}$. As $e_{x}({\mathcal{S}})=k$, we have
$k=(\ell-m)+(r^{\prime}-s)$. By Remarks 3.4 we can choose (local) coordinates
in $\widehat{X}$ such that $\pi_{+}$ behaves (with the already taken (local)
coordinates in $X$) as the Nash map

$g_{+}:[0,+\infty)\times{\mathbb{S}}^{d-e-1}\times{\mathbb{R}}^{e}\to{\mathbb{R}}^{d},\
(\rho,w,z)\mapsto(\rho w,z),$

where $w:=(w_{1},\ldots,w_{\ell})$ and
$z:=(z_{\ell+1},\ldots,z_{d})\in{\mathbb{R}}^{e}={\mathbb{R}}^{d-\ell}$. We
have

$\displaystyle g_{+}^{-1}(u(Z\cap U))$ $\displaystyle=\\{\rho^{\ell}{\tt
w}_{1}\cdots{\tt w}_{\ell}{\tt z}_{\ell+1}\cdots{\tt z}_{r^{\prime}}=0,{\tt
w}_{1}^{2}+\cdots+{\tt w}_{\ell}^{2}=1\\},$ $\displaystyle g_{+}^{-1}(u(Y\cap
U))$ $\displaystyle=\\{\rho{\tt w}_{1}=0,\ldots,\rho{\tt w}_{\ell}=0,{\tt
w}_{1}^{2}+\cdots+{\tt w}_{\ell}^{2}=1\\}=\\{\rho=0,{\tt
w}_{1}^{2}+\cdots+{\tt w}_{\ell}^{2}=1\\},$ $\displaystyle
g_{+}^{-1}(u({\mathcal{S}}\cap U\setminus Y))$
$\displaystyle\subset\\{\rho\geq 0,{\tt w}_{1}\geq 0,\ldots,{\tt w}_{m}\geq
0,{\tt z}_{\ell+1}\geq 0,\ldots,{\tt z}_{s}\geq 0,{\tt w}_{1}^{2}+\cdots+{\tt
w}_{\ell}^{2}=1\\}.$

If $m=0$, then ${\mathcal{S}}\cap U\subset\\{{\tt u}_{\ell+1}\geq
0,\ldots,{\tt u}_{s}\geq 0\\}$ and

$g_{+}^{-1}(u({\mathcal{S}}\cap U\setminus Y))\subset\\{\rho\geq 0,{\tt
z}_{\ell+1}\geq 0,\ldots,{\tt z}_{s}\geq 0,{\tt w}_{1}^{2}+\cdots+{\tt
w}_{\ell}^{2}=1\\}.$

Thus, $e_{y}({\mathcal{T}})\leq\ell-1+r^{\prime}-s=k-1<k=e_{x}({\mathcal{S}})$
for each $y\in g^{-1}(x)$. The condition $m=0$ means that
${\mathcal{S}}_{x}\setminus Z_{i,x}$ is not connected for $i=1,\ldots,\ell$.

We assume in the following $m\geq 1$. Observe that ${\mathcal{S}}_{x}\setminus
Z_{i,x}$ is not connected for $i=m+1,\ldots,\ell$. Let us show
$e_{y}({\mathcal{T}})\leq e_{x}({\mathcal{S}})$ for each $y\in g^{-1}(x)$. It
may happen that for some $y\in g^{-1}(x)$ the previous inequality is strict
even if ${\mathcal{S}}_{x}\setminus Z_{i,x}$ is connected for the indices
$i=1,\ldots,m$. If some ${\tt w}_{i}(y)\neq 0$, this variable has no relevance
in the description of ${\mathcal{T}}$ locally around $y$ and ${\tt w}_{i}$
behaves as $\pm\sqrt{1-\sum_{j\neq i}{\tt w}_{j}^{2}}$. Analogously, if some
${\tt z}_{j}(y)\neq 0$, this variable has no relevance in the description of
${\mathcal{T}}$ locally around $y$. As ${\tt w}_{1}^{2}+\cdots+{\tt
w}_{\ell}^{2}=1$, there exists an index $1\leq i\leq\ell$ such that ${\tt
w}_{i}(y)\neq 0$:

Case 1. If $1\leq i\leq m$, we may assume $i=m$. Thus, ${\tt w}_{m}(y)>0$ and
${\tt w}_{m}=+\sqrt{1-\sum_{j\neq m}{\tt w}_{j}^{2}}$. We consider (local)
coordinates $(\rho,{\tt w}_{1},\ldots,{\tt w}_{m-1},{\tt w}_{m+1},\ldots,{\tt
w}_{\ell},{\tt z}_{\ell+1},\ldots,{\tt z}_{s},{\tt z}_{s+1},\ldots,{\tt
z}_{r^{\prime}},{\tt z}_{r^{\prime}+1},\ldots{\tt z}_{d})$ and

$e_{y}({\mathcal{T}})\leq\ell-(m-1+1)+(r^{\prime}-s)=(\ell-m)+(r^{\prime}-s)=k=e_{x}({\mathcal{S}}).$

Case 2. If $m+1\leq i\leq\ell$, we may assume $i=\ell$. Thus, ${\tt
w}_{\ell}(y)\neq 0$ and ${\tt w}_{\ell}=\pm\sqrt{1-\sum_{j=1}^{\ell-1}{\tt
w}_{i}^{2}}$. We consider (local) coordinates $(\rho,{\tt w}_{1},\ldots,{\tt
w}_{m},{\tt w}_{m+1},\ldots,{\tt w}_{\ell-1},{\tt z}_{\ell+1},\ldots,{\tt
z}_{s},{\tt z}_{s+1},\ldots,{\tt z}_{r^{\prime}},{\tt
z}_{r^{\prime}+1},\ldots{\tt z}_{d})$ and

$e_{y}({\mathcal{T}})\leq
1+(\ell-1)-(m+1)+(r^{\prime}-s)=(\ell-m)+(r^{\prime}-s)-1=k-1<k=e_{x}({\mathcal{S}}),$

as required. ∎

####  4.2.3. Closed checkerboard sets and Nash manifolds with corners.

We are ready to prove Theorem 1.4. After the previous preparation (in
particular Theorem 4.3) we are reduced to the case when ${\mathcal{S}}$ is a
$d$-dimensional closed checkerboard set.

###### Proof of Theorem 1.4.

Let us construct the Nash manifold with corners ${\mathcal{Q}}$ first. Let
${\mathcal{S}}\subset{\mathbb{R}}^{m}$ be a $d$-dimensional closed
checkerboard set and denote
$X_{0}:=\overline{{\mathcal{S}}}^{\operatorname{zar}}$. Let
$Z:=\overline{\partial{\mathcal{S}}}^{\operatorname{zar}}$ and
$Z_{1},\ldots,Z_{r}$ be its irreducible components. Define
$e:=\max\\{e_{x}({\mathcal{S}}):\ x\in\partial{\mathcal{S}}\\}$.

If $e=0$, we conclude by Lemma 4.4 that ${\mathcal{S}}$ is already a Nash
manifold with corners. Otherwise, by Lemma 4.5 $e\geq 2$. By Remark 4.8
${\mathcal{S}}_{e}$ is a closed subset of $\partial{\mathcal{S}}$. By Lemma
4.6 ${\mathcal{S}}_{e}$ is a union of connected components of the
semialgebraic sets
$\operatorname{Sing}_{\ell}(Z)\setminus\operatorname{Sing}_{\ell+1}(Z)$ for
$1\leq\ell\leq d-1$ (recall that $e\geq 2$).

Pick a point $x\in{\mathcal{S}}_{e}$ and assume that $Z_{1},\ldots,Z_{e}$ are
the irreducible components of $Z$ such that the germ
${\mathcal{S}}_{x}\setminus Z_{i,x}$ is not connected for $i=1,\ldots,e$. Then
there exists an open semialgebraic neighborhood $U\subset X$ of $x$ equipped
with a Nash diffeomorphism $u:=(u_{1},\ldots,u_{d}):U\to{\mathbb{R}}^{d}$ such
that $u(x)=0$, $Z\cap U=\\{{\tt u}_{1}\cdots{\tt u}_{r}=0\\}$, $Z_{i}\cap
U=\\{{\tt u}_{i}=0\\}$ and ${\mathcal{S}}\cap U\subset\\{{\tt u}_{e+1}\geq
0,\ldots,{\tt u}_{r}\geq 0\\}$. Thus, $\\{{\tt u}_{1}=0,\ldots,{\tt
u}_{e}=0,{\tt u}_{e+1}\geq 0,\ldots,{\tt u}_{r}\geq
0\\}\subset{\mathcal{S}}_{e}.$ By Lemma 4.7 the connected component
${\mathcal{C}}$ of $(Z_{1}\cap\cdots\cap
Z_{e})\setminus\bigcup_{i=e+1}^{r}Z_{i}$ that contains $\\{{\tt
u}_{1}=0,\ldots,{\tt u}_{e}=0,{\tt u}_{e+1}\geq 0,\ldots,{\tt u}_{r}\geq 0\\}$
is contained in ${\mathcal{S}}_{e}$. Thus, the Zariski closure of
${\mathcal{C}}$ is the irreducible component of $Z_{1}\cap\cdots\cap Z_{e}$
that contains $x$. As we can repeat the previous argument for each
$x\in{\mathcal{S}}_{e}$, we conclude that the Zariski closure of
${\mathcal{S}}_{e}$ is a union of irreducible components of the real algebraic
set
$\bigcup_{\\{i_{1},\ldots,i_{e}\\}\subset\\{1,\ldots,r\\}}\bigcap_{j=1}^{e}Z_{i_{j}}$.
As $e\geq 2$, we have
$\dim(\overline{{\mathcal{S}}_{e}}^{\operatorname{zar}})\leq d-2$.

In addition, for each $x\in{\mathcal{S}}_{e}$ there exist irreducible
components $Z_{i_{1}},\ldots,Z_{i_{e}}$ of $Z$ such that the germ
${\mathcal{S}}_{x}\setminus Z_{i_{j},x}$ is not connected for $j=1,\ldots,e$.
We proceed by double induction on $e$ and the number $m$ of irreducible
components of the Zariski closure of
$\overline{{\mathcal{S}}_{e}}^{\operatorname{zar}}$ of ${\mathcal{S}}_{e}$.

Let $W$ be an irreducible component of
$\overline{{\mathcal{S}}_{e}}^{\operatorname{zar}}$, which has dimension $\leq
d-2$. Let $(\widehat{X}_{0},\widehat{\pi})$ be the twisted Nash double of the
drilling blow-up $(\widetilde{X}_{0},\pi_{+})$ of $X_{0}$ with center $W$,
which is by Subsection 3.4 a non-singular real algebraic set. Let

${\mathcal{T}}:=\operatorname{Cl}(\pi_{+}^{-1}({\mathcal{S}}\setminus
W))=\pi_{+}^{-1}({\mathcal{S}})\cap\operatorname{Cl}(\pi_{+}^{-1}({\mathcal{S}}\setminus
W))$

be the strict transform of ${\mathcal{S}}$ under $\pi_{+}$ (recall that
${\mathcal{S}}$ is closed). As ${\mathcal{S}}$ is pure dimensional and
${\mathcal{S}}_{e}\subset\overline{\partial{\mathcal{S}}}^{\operatorname{zar}}$
has dimension strictly smaller, ${\mathcal{S}}\setminus W$ is dense in
${\mathcal{S}}$, so

$\pi_{+}({\mathcal{T}})=\pi_{+}(\operatorname{Cl}(\pi^{-1}_{+}({\mathcal{S}}\setminus
W)))=\operatorname{Cl}(\pi_{+}(\pi_{+}^{-1}({\mathcal{S}}\setminus
W)))=\operatorname{Cl}({\mathcal{S}}\setminus W)={\mathcal{S}},$

because $\pi_{+}:\widetilde{X}_{0}\to X_{0}$ is proper and surjective. As
$e\geq 2$, we can apply Lemma 4.9 and we deduce: ${\mathcal{T}}$ is a
checkerboard set, $e_{y}({\mathcal{T}})\leq e_{\pi_{+}(y)}({\mathcal{S}})$ for
each $y\in\partial{\mathcal{T}}$ and
$e_{y}({\mathcal{T}})<e_{\pi_{+}(y)}({\mathcal{S}})$ for each
$y\in\partial{\mathcal{T}}\cap\pi_{+}^{-1}(W)$ such that
${\mathcal{S}}_{\pi_{+}(y)}\setminus Z_{i_{j},\pi_{+}(y)}$ is not connected
for $j=1,\ldots,e$.

If $\max\\{e({\mathcal{T}})_{y}:\ y\in\partial{\mathcal{T}}\\}<e$, by
induction hypothesis the statement holds for ${\mathcal{T}}$, so it also holds
for ${\mathcal{S}}$. If $\max\\{e({\mathcal{T}})_{y}:\
y\in\partial{\mathcal{T}}\\}=e$, the Zariski closure of ${\mathcal{T}}_{e}$ is
contained in
$\operatorname{Cl}(\widehat{\pi}^{-1}(\overline{{\mathcal{S}}_{e}}^{\operatorname{zar}}\setminus
W))$ and it has $m-1$ irreducible components. As by Lemma 4.4 $e=0$ if and
only if ${\mathcal{T}}$ is a Nash manifold with corners, our inductive
argument is consistent. Thus, by induction hypothesis the statement holds for
${\mathcal{T}}$, so it also holds for ${\mathcal{S}}$.

Let ${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ be the Nash manifold with corners
obtained by our inductive process. As ${\mathcal{Q}}$ is a checkerboard set,
$\operatorname{Reg}({\mathcal{Q}})$ is connected, so also
${\mathcal{Q}}=\operatorname{Cl}(\operatorname{Reg}({\mathcal{Q}}))$ is
connected. We have constructed ${\mathcal{Q}}$, starting from ${\mathcal{S}}$,
with a finite number of drilling blow-ups. Namely, we have constructed a
finite number of tuples
$\\{{\mathcal{T}}_{i},(\widetilde{X}_{{\mathcal{T}}_{i}},\pi_{+,i}),(\widehat{X}_{{\mathcal{T}}_{i}},\widehat{\pi}_{i})\\}_{i=0}^{s}$
where:

  * •

${\mathcal{T}}_{0}:={\mathcal{S}}$,
$\widetilde{X}_{{\mathcal{T}}_{0}}:=X_{0}=\overline{{\mathcal{S}}}^{\operatorname{zar}}$,
$\pi_{+,0}:=\operatorname{id}_{X_{0}}$,
$\widehat{X}_{{\mathcal{T}}_{0}}:=X_{0}$ and
$\widehat{\pi}_{0}:=\operatorname{id}_{X_{0}}$.

  * •

$(\widetilde{X}_{{\mathcal{T}}_{i+1}},\pi_{+,i+1})$ is the drilling blow-up of
the irreducible non-singular real algebraic set
$\overline{{\mathcal{T}}}_{i}^{\operatorname{zar}}$ with center a (suitable)
irreducible non-singular real algebraic subset $W_{i}$ of
$\overline{{\mathcal{T}}}_{i}^{\operatorname{zar}}$ of dimension $\leq d-2$.

  * •

$(\widehat{X}_{{\mathcal{T}}_{i+1}},\widehat{\pi}_{i+1})$ is the twisted Nash
double of $(\widetilde{X}_{{\mathcal{T}}_{i+1}},\pi_{+,i+1})$ and it is a non-
singular real algebraic set of dimension $d$. By Lemma 3.13
$\widehat{X}_{{\mathcal{T}}_{i+1}}$ is the Zariski closure of
$\widetilde{X}_{{\mathcal{T}}_{i+1}}$.

  * •

${\mathcal{T}}_{i+1}:=\operatorname{Cl}(\pi_{+,i+1}^{-1}({\mathcal{T}}_{i}\setminus
W_{i}))$ is the strict transform of ${\mathcal{T}}_{i}$ under $\pi_{+,i+1}$
and $\pi_{+,i+1}({\mathcal{T}}_{i+1})={\mathcal{T}}_{i}$. By Lemma 4.9
${\mathcal{T}}_{i+1}$ is a checkerboard set.

  * •

$\pi_{+,i+1}^{-1}(W_{i})\cap{\mathcal{T}}_{i+1}\subset\partial{\mathcal{T}}_{i+1}$,
$\pi_{+,i+1}^{-1}(\partial{\mathcal{T}}_{i}\setminus
W_{i})\subset\partial{\mathcal{T}}_{i+1}$ and
$\pi_{+,i+1}^{-1}(\operatorname{Reg}({\mathcal{T}}_{i})\setminus
W_{i})\subset\operatorname{Reg}({\mathcal{T}}_{i+1})$ (use Fact 3.7(ii) and
§3.4). Consequently,
$\operatorname{Reg}({\mathcal{T}}_{i+1})=\pi_{+,i+1}^{-1}(\operatorname{Reg}({\mathcal{T}}_{i})\setminus
W_{i})$ and
$\partial{\mathcal{T}}_{i+1}=\pi_{+,i+1}^{-1}(\partial{\mathcal{T}}_{i}\setminus
W_{i})\cup(\pi_{+,i+1}^{-1}(W_{i})\cap{\mathcal{T}}_{i+1})$.

  * $(*)$

$\pi_{+,i+1}(\partial{\mathcal{T}}_{i+1})=\partial{\mathcal{T}}_{i}\cup(W_{i}\cap{\mathcal{T}}_{i})$.

  * $(**)$

$\pi_{+,i+1}|_{\operatorname{Reg}({\mathcal{T}}_{i+1})}:\operatorname{Reg}({\mathcal{T}}_{i+1})\to\operatorname{Reg}({\mathcal{T}}_{i})\setminus(W_{i}\cap{\mathcal{T}}_{i})$
is a Nash diffeomorphism (Fact 3.7(ii)).

  * •

${\mathcal{T}}_{s}={\mathcal{Q}}$.

As ${\mathcal{T}}_{i}$ is connected by analytic paths,
$\overline{{\mathcal{T}}}_{i}^{\operatorname{zar}}$ is by [Fe2, Lem.7.3] an
irreducible component of the pure dimensional non-singular real algebraic set
$\widehat{X}_{{\mathcal{T}}_{i}}$ of dimension $d$. Thus,
$\overline{{\mathcal{T}}}_{i}^{\operatorname{zar}}$ is a pure dimensional
irreducible non-singular real algebraic set of dimension $d$. Consequently,
the Nash manifold with corners ${\mathcal{Q}}={\mathcal{T}}_{s}$ is (by Lemma
4.9) a checkerboard set, $X:=\overline{{\mathcal{Q}}}^{\operatorname{zar}}$ is
a $d$-dimensional irreducible non-singular real algebraic set and
$Y:=\overline{\partial{\mathcal{Q}}}^{\operatorname{zar}}$ is a normal-
crossings divisor of $X$. Thus, (i) and (ii) hold. As all the involved
polynomials maps $\widehat{\pi}_{i}$ are proper, if $X_{0}$ is compact, then
$X$ is also compact.

Consider the polynomial map
$f:=\widehat{\pi}_{1}\circ\ldots\circ\widehat{\pi}_{s}:\widehat{X}_{{\mathcal{T}}_{s}}\to
X_{0}$. By Fact 3.7(i) $f:\widehat{X}_{{\mathcal{T}}_{s}}\to X_{0}$ is
composition of proper maps, so it is proper. Moreover, as ${\mathcal{S}}$ is
closed and ${\mathcal{Q}}$ is obtained from ${\mathcal{S}}$ after a finite
number of drilling blow-ups taking strict transforms in each step,
${\mathcal{Q}}$ is a closed subset of $X$. Thus, also the restriction
$f|_{{\mathcal{Q}}}:{\mathcal{Q}}\to X_{0}$ is proper. In addition, as
$\widehat{\pi}_{i+1}({\mathcal{T}}_{i+1})={\mathcal{T}}_{i}$ for
$i=0,\ldots,s-1$, we conclude that $f({\mathcal{Q}})={\mathcal{S}}$.

Let us show (iv). By property $(**)$ applied inductively we deduce

$f(\operatorname{Reg}({\mathcal{Q}}))\subset\operatorname{Reg}({\mathcal{T}}_{0})\setminus((W_{0}\cap{\mathcal{T}}_{0})\cup\bigcup_{k=1}^{s-1}(\pi_{+,1}\circ\cdots\circ\pi_{+,k})(W_{k}\cap{\mathcal{T}}_{k})),$

whereas by property $(*)$ applied recursively we have

$f(\partial{\mathcal{Q}})=\partial{\mathcal{T}}_{0}\cup(W_{0}\cap{\mathcal{T}}_{0})\cup\bigcup_{k=1}^{s-1}(\pi_{+,1}\circ\cdots\circ\pi_{+,k})(W_{k}\cap{\mathcal{T}}_{k}),$
(4.2)

so $f(\operatorname{Reg}({\mathcal{Q}}))\cap
f(\partial{\mathcal{Q}})=\varnothing$. Thus, as
${\mathcal{S}}=f({\mathcal{Q}})=f(\operatorname{Reg}({\mathcal{Q}}))\cup
f(\partial{\mathcal{Q}})$, we deduce
$f(\operatorname{Reg}({\mathcal{Q}}))={\mathcal{S}}\setminus
f(\partial{\mathcal{Q}})$ and
$f^{-1}(f(\partial{\mathcal{Q}}))=\partial{\mathcal{Q}}$. Property $(**)$
implies
$f|_{\operatorname{Reg}({\mathcal{Q}})}=(\pi_{+,1}\circ\cdots\circ\pi_{+,s})|_{\operatorname{Reg}({\mathcal{Q}})}:\operatorname{Reg}({\mathcal{Q}})\to{\mathcal{S}}\setminus
f(\partial{\mathcal{Q}})$ is a Nash diffeomorphism (because it is a
composition of finitely many Nash diffeomorphisms). By (4.2) we have
$\partial{\mathcal{S}}\subset f(\partial{\mathcal{Q}})$. The semialgebraic set
${\mathcal{R}}:=f(\partial{\mathcal{Q}})\subset{\mathcal{S}}$ is closed,
because $f$ is proper and $\partial{\mathcal{Q}}$ is closed. As
$\partial{\mathcal{Q}}$ has dimension not greater than $d-1$, we have
$\dim({\mathcal{R}})\leq d-1<d$. The semialgebraic sets
${\mathcal{S}}\setminus{\mathcal{R}}$ and ${\mathcal{Q}}\setminus
f^{-1}({\mathcal{R}})$ are Nash manifolds, because
${\mathcal{S}}\setminus{\mathcal{R}}$ is an open semialgebraic subset of the
Nash manifold
$\operatorname{Reg}({\mathcal{S}})={\mathcal{S}}\setminus\partial{\mathcal{S}}$
and ${\mathcal{Q}}\setminus
f^{-1}({\mathcal{R}})={\mathcal{Q}}\setminus\partial{\mathcal{Q}}=\operatorname{Reg}({\mathcal{Q}})$.
Consequently,
$f|_{\operatorname{Reg}({\mathcal{Q}})}:\operatorname{Reg}({\mathcal{Q}})={\mathcal{Q}}\setminus
f^{-1}({\mathcal{R}})\to{\mathcal{S}}\setminus{\mathcal{R}}$ is a Nash
diffeomorphism, as required. ∎

${\mathcal{Q}}$${\mathcal{S}}$$f|_{{\mathcal{Q}}}$ Figure 4.1.
Desingularization of the closed checkerboard set ${\mathcal{S}}$ (right) by
the Nash manifold with corners ${\mathcal{Q}}$ (left).

###  4.3. Desingularization of general checkerboard sets

Let ${\mathcal{S}}\subset{\mathbb{R}}^{m}$ be a semialgebraic set connected by
analytic paths. To prove Theorem 1.6 we introduce Nash quasi-manifolds with
corners, that is, semialgebraic sets ${\mathcal{T}}\subset{\mathbb{R}}^{n}$
whose closure is a Nash manifold with corners
${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ and
${\mathcal{Q}}\setminus{\mathcal{T}}$ is a union of some of the strata of
certain (Nash) stratification of $\partial{\mathcal{Q}}$, that we introduce
below. Let us recall the definition of (Nash) stratification of a
semialgebraic set.

######  Definition 4.10.

Let ${\mathcal{S}}\subset{\mathbb{R}}^{m}$ be a semialgebraic set. A (Nash)
stratification of ${\mathcal{S}}$ is a finite semialgebraic partition
$\\{{\mathcal{S}}_{\alpha}\\}_{\alpha\in A}$ of ${\mathcal{S}}$, where each
${\mathcal{S}}_{\alpha}$ is a connected Nash submanifold of ${\mathbb{R}}^{m}$
and the following property is satisfied: if
${\mathcal{S}}_{\alpha}\cap\operatorname{Cl}({\mathcal{S}}_{\beta})\neq\varnothing$
and $\alpha\neq\beta$, then
${\mathcal{S}}_{\alpha}\subset\operatorname{Cl}({\mathcal{S}}_{\beta})$ and
$\dim({\mathcal{S}}_{\alpha})<\dim({\mathcal{S}}_{\beta})$. The
${\mathcal{S}}_{\alpha}$ are called the strata of the (Nash) stratification
and if $d:=\dim({\mathcal{S}}_{\alpha})$, then ${\mathcal{S}}_{\alpha}$ is a
$d$-stratum.

The condition $\dim({\mathcal{S}}_{\alpha})<\dim({\mathcal{S}}_{\beta})$
follows from [BCR, Prop.2.8.13], because
${\mathcal{S}}_{\alpha}\subset\operatorname{Cl}({\mathcal{S}}_{\beta})\setminus{\mathcal{S}}_{\beta}$.

######  Definition 4.11.

Given a $d$-dimensional semialgebraic set
${\mathcal{S}}\subset{\mathbb{R}}^{m}$, we consider the following
semialgebraic partition of ${\mathcal{S}}$. Recall that
$\operatorname{Sth}({\mathcal{S}})$ is the set of points $x\in{\mathcal{S}}$
at which the germ ${\mathcal{S}}_{x}$ is the germ of a Nash manifold (Section
2.1). Define $\Gamma_{1}:=\operatorname{Sth}({\mathcal{S}})$ and
$\Gamma_{k}:=\operatorname{Sth}({\mathcal{S}}\setminus\bigcup_{j=1}^{k-1}\Gamma_{j})$
for $k\geq 2$. Let $s\geq 1$ be the largest index $k$ such that
$\Gamma_{k}\neq\varnothing$. For each $k\geq 1$ let $\Gamma_{k\ell}$ (for
$\ell=1,\ldots,r_{k}$) be the connected components of $\Gamma_{k}$. The
collection ${\mathfrak{G}}({\mathcal{S}}):=\\{\Gamma_{k\ell}:\ 1\leq k\leq
s,1\leq\ell\leq r_{k}\\}$ is a partition of ${\mathcal{S}}$. We say that
${\mathfrak{G}}({\mathcal{S}})$ is compatible with a semialgebraic set
${\mathcal{T}}\subset{\mathcal{S}}$ if ${\mathcal{T}}$ is the union of some of
the $\Gamma_{k\ell}$.

######  Examples 4.12.

(i) The semialgebraic partition ${\mathfrak{G}}({\mathcal{S}})$ of a
semialgebraic set ${\mathcal{S}}\subset{\mathbb{R}}^{n}$ is not in general a