component of $\operatorname{Int}(H_{1})\setminus H_{2}$. Consequently,
$f_{2}(C)\subset\operatorname{Int}(H_{1})$ for each connected component $C$ of
$\operatorname{Int}(H_{1})\setminus H_{2}$, so
$f_{2}(\operatorname{Int}(H_{1})\setminus
H_{2})\subset\operatorname{Int}(H_{1})$.

Finally, we prove (4). Pick a point $x\in\partial{\mathcal{Q}}$ and assume
that $x$ belongs exactly to $Y_{1},\ldots,Y_{s}$. Recall that the analytic
closure of $\partial{\mathcal{Q}}_{x}$ is $Y_{1,x}\cup\cdots\cup Y_{s,x}$ and
there exists an open semialgebraic neighborhood $U\subset M$ of $x$ equipped
with a Nash diffeomorphism $u:U\to{\mathbb{R}}^{d}$ such that $u(x)=0$,
$u(Y_{j}\cap U)=\\{{\tt x}_{j}=0\\}$ and $u({\mathcal{Q}}\cap U)=\\{{\tt
x}_{1}\geq 0,\ldots,{\tt x}_{s}\geq 0\\}$.

Observe that $\eta:=u\circ f\circ u^{-1}=\eta_{\ell}\circ\cdots\circ\eta_{1}$
where $\eta_{k}:=u\circ f_{k}\circ u^{-1}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$
is either a Nash diffeomorphism or a Nash map that has local representations
$(y_{1},\ldots,y_{d})\mapsto(y_{1}^{2},y_{2},\ldots,y_{d})$ at each point
$x\in Y_{k}$. In addition, these local representations preserve the local
representations of $Y_{1},\ldots,Y_{s}$, so $Y_{1},\ldots,Y_{s}$ correspond to
coordinates’ hyperplanes in these (local) coordinates. By Lemma 5.12 we may
assume that $\eta=\sigma_{s}\circ\cdots\circ\sigma_{1}$ is a composition of
Nash maps of the type $\sigma_{m}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$,

$(x_{1},\ldots,x_{d})\mapsto(x_{1}h_{1m},\ldots,x_{m-1}h_{m-1,m},x_{m}^{2}h_{mm}^{2},x_{m+1}h_{m+1,m},\ldots,x_{s}h_{sm},g_{m})$

where $h_{jm}:{\mathbb{R}}^{d}\to{\mathbb{R}}$ is a Nash function that does
not vanish around $\\{{\tt x}_{j}=0\\}$ and
$g_{m}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d-s}$ satisfies $g_{m}|_{\\{{\tt
x}_{1}=0,\ldots,{\tt x}_{s}=0\\}}:\\{{\tt x}_{1}=0,\ldots,{\tt
x}_{s}=0\\}\to{\mathbb{R}}^{d-s}$ is a Nash diffeomorphism. Thus, there exist
Nash functions $f_{j}:{\mathbb{R}}^{d}\to{\mathbb{R}}$ that do not vanish
around $\\{{\tt x}_{j}=0\\}$ and a Nash map
$g:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d-s}$ such that $g|_{\\{{\tt
x}_{1}=0,\ldots,{\tt x}_{s}=0\\}}:\\{{\tt x}_{1}=0,\ldots,{\tt
x}_{s}=0\\}\to{\mathbb{R}}^{d-s}$ is a Nash diffeomorphism satisfying

$\eta:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d},\
(x_{1},\ldots,x_{d})\mapsto(x_{1}^{2}f_{1}^{2},\ldots,x_{s}^{2}f_{s}^{2},g).$

Observe that the map

$\psi:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d},\
(x_{1},\ldots,x_{d})\mapsto(x_{1}f_{1},\ldots,x_{s}f_{s},g)$

is a Nash diffeomorphism around the origin because the determinant of its
Jacobian matrix at the origin

$J_{\psi}(x)=\left[\begin{array}[]{ccc|c}f_{1}(0)&\cdots&0&0\\\
\vdots&\ddots&\vdots&\vdots\\\ 0&\cdots&f_{s}(0)&0\\\
\hline\cr*&\ldots&*&J_{g}(0)\end{array}\right]$

is $f_{1}(0)\cdots f_{s}(0)J_{g}(0)\neq 0$. After shrinking the open
semialgebraic neighborhood $U\subset M$ of $x$, there exists a Nash
diffeomorphism $\phi:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ such that
$\psi\circ\phi=\operatorname{id}_{{\mathbb{R}}^{d}}$. As
$\eta=\theta\circ\psi$, where

$\theta:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d},\
(x_{1},\ldots,x_{d})\mapsto(x_{1}^{2},\cdots,x_{s}^{2},x_{s+1},\ldots,x_{d}),$

we deduce that $\eta\circ\phi=\theta$, so $f$ has local representation
$(y_{1},\ldots,y_{d})\mapsto(y_{1}^{2},\ldots,y_{s}^{2},y_{s+1},\ldots,y_{d})$
at $x\in\partial{\mathcal{Q}}$. Recall that $s\geq 1$ corresponds to the
number of irreducible components of $Y$ that passes through $x$, as required.
∎

###  5.7. The Nash double of a Nash manifold with corners.

Let ${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ be a $d$-dimensional Nash manifold
with corners and $M$ a Nash envelope of ${\mathcal{Q}}$ such that the Nash
closure $Y$ of $\partial{\mathcal{Q}}$ is a Nash normal-crossings divisor of
$M$ and ${\mathcal{Q}}\cap Y=\partial{\mathcal{Q}}$. Let
$Y_{1},\ldots,Y_{\ell}$ be the irreducible components of $Y$ and
$h_{i}:M\to{\mathbb{R}}$ Nash equations of $Y_{i}$ for $i=1,\ldots,\ell$ such
that $d_{x}h_{i}:T_{x}M\to{\mathbb{R}}$ is surjective for $i=1,\ldots,\ell$.
By Lemma 5.4 the sets $H_{i}:=h_{i}^{-1}([0,+\infty))$ are Nash manifolds with
boundary $Y_{i}$ that contain ${\mathcal{Q}}$ as a closed subset. Write (Lemma
5.6) ${\mathcal{Q}}=H_{1}\cap\ldots\cap H_{\ell}=\\{h_{1}\geq
0,\ldots,h_{\ell}\geq 0\\}\subset M$, $t:=(t_{1},\ldots,t_{\ell})$ and define
the Nash double of ${\mathcal{Q}}$ as:

$D({\mathcal{Q}}):=\\{(x,t)\in M\times{\mathbb{R}}^{\ell}:\
t_{1}^{2}-h_{1}(x)=0,\ldots,t_{\ell}^{2}-h_{\ell}(x)=0\\}.$

Observe that $D({\mathcal{Q}})$ only depends on the Nash equations
$h_{1},\ldots,h_{\ell}$, which are unique up to multiplication by strictly
positive Nash functions. Thus, $D({\mathcal{Q}})$ is unique up to Nash
diffeomorphism. In addition, the reader can check that $D({\mathcal{Q}})$ is
the inverse image of the value $0$ under the Nash map
$f:M\times{\mathbb{R}}^{\ell}\to{\mathbb{R}}^{\ell},(x,t)\mapsto(t_{1}^{2}-h_{1}(x),\ldots,t_{\ell}^{2}-h_{\ell}(x))$
and that $0$ is a regular value for $f$, so $D({\mathcal{Q}})$ is a
$d$-dimensional Nash manifold.

######  Theorem 5.13 (Nash double of a Nash manifold with corners).

The Nash subset $D({\mathcal{Q}})$ of $M\times{\mathbb{R}}^{\ell}$ is a
$d$-dimensional Nash manifold that satisfies the following properties:

  * (i)

The projection $\pi:D({\mathcal{Q}})\to{\mathcal{Q}},\ (x,t)\mapsto x$ is
proper and surjective.

  * (ii)

The semialgebraic set ${\mathcal{Q}}^{\ell}:=D({\mathcal{Q}})\cap\\{t_{1}\geq
0,\ldots,t_{\ell}\geq 0\\}$ is a Nash manifold with corners, which is Nash
diffeomorphic to ${\mathcal{Q}}$ and the Nash closure of
$\partial{\mathcal{Q}}^{\ell}$ in $D({\mathcal{Q}})$ is the Nash normal-
crossings divisor $Y^{\ell}:=\bigcup_{i=1}^{\ell}\\{t_{i}=0\\}$ of
$D({\mathcal{Q}})$.

  * (iii)

The restriction
$\pi|_{{\mathcal{Q}}^{\ell}}:{\mathcal{Q}}^{\ell}\to{\mathcal{Q}}$ is a
semialgebraic homeomorphism.

  * (iv)

The restriction
$\pi|_{\operatorname{Int}({\mathcal{Q}}^{\ell})}:\operatorname{Int}({\mathcal{Q}}^{\ell})\to\operatorname{Int}({\mathcal{Q}})$
is a Nash diffeomorphism.

  * (v)

There exists an open semialgebraic neighborhood $W\subset D({\mathcal{Q}})$ of
${\mathcal{Q}}^{\ell}$ Nash diffeomorphic to $M$.

###### Proof.

The proof is conducted in several steps:

Step 1. Let $(D(H_{1}),\pi_{1})$ be the Nash double of $H_{1}:=\\{h_{1}\geq
0\\}$. Recall that

$M^{1}:=D(H_{1})=\\{(x,t_{1})\in M\times{\mathbb{R}}:\
t_{1}^{2}-h_{1}(x)=0\\}\quad\text{and}\quad\pi_{1}:D(H_{1})\to H_{1},\
(x,t_{1})\to x.$

The Nash map $\pi_{1}$ is proper by Lemma 5.8(iv). Consider the
$(d-1)$-dimensional Nash submanifolds $Y_{1}^{1}:=Y_{1}\times\\{0\\}$ and
$Y_{i}^{1}:=D(Y_{i}\cap H_{1})=(Y_{i}\times{\mathbb{R}})\cap D(H_{1})$ (see
(5.1)) of $M^{1}$ for $i=2,\ldots,\ell$. Consider the Nash functions

$h_{i}^{1}:M^{1}\to{\mathbb{R}},\ (x,t_{1})\mapsto
h_{i}^{1}(x,t_{1}):=\begin{cases}t_{1}&\text{if $i=1$,}\\\ h_{i}(x)&\text{if
$i=2,\ldots,\ell$.}\end{cases}$

The tangent space to $M^{1}$ at a point $(x,t_{1})\in M^{1}$ is:

$T_{(x,t_{1})}M^{1}=\\{(v,s_{1})\in T_{x}M\times{\mathbb{R}}:\
d_{x}h_{1}(v)-2t_{1}s_{1}=0\\}.$

We claim: $h_{i}^{1}:M^{1}\to{\mathbb{R}}$ is a Nash equation of $Y_{i}^{1}$
in $M^{1}$ such that
$d_{(x,t_{1})}h_{i}^{1}:T_{(x,t_{1})}M^{1}\to{\mathbb{R}}$ is surjective for
each $(x,t_{1})\in Y_{i}^{1}$ and $i=1,\ldots,\ell$.

Suppose first $i=1$. Then
$\\{h_{1}^{1}=0\\}=\\{h_{1}(x)=0,t_{1}=0\\}=Y_{1}\times\\{0\\}=Y_{1}^{1}$, so
$h_{i}^{1}$ is a Nash equation of $Y_{i}^{1}$ in $M^{1}$. In addition, for
$(x,0)\in Y_{1}^{1}=Y_{1}\times\\{0\\}$ we have

$T_{(x,0)}M^{1}=\\{(v,s_{1})\in T_{x}M\times{\mathbb{R}}:\
d_{x}h_{1}(v)=0\\}=T_{x}Y_{1}\times{\mathbb{R}}$

and $d_{(x,0)}h_{1}^{1}:T_{x}Y_{1}\times{\mathbb{R}}\to{\mathbb{R}},\
(v,s_{1})\mapsto s_{1}$ is surjective.

Suppose next $i=2,\ldots,\ell$ and observe that

$Y_{i}^{1}=\\{(x,t_{1})\in M\times{\mathbb{R}}:\
t_{1}^{2}-h_{1}(x)=0,h_{i}(x)=0\\}=\\{(x,t_{1})\in Y_{i}\times{\mathbb{R}}:\
t_{1}^{2}-h_{1}(x)=0\\},$

so $h_{i}^{1}=0$ is a Nash equation of $Y_{i}^{1}$ in $M^{1}$. Recall that

$T_{(x,t_{1})}M^{1}=\\{(v,s_{1})\in T_{x}M\times{\mathbb{R}}:\
d_{x}h_{1}(v)-2t_{1}s_{1}=0\\}$

and $d_{(x,t_{1})}h_{i}^{1}:T_{(x,t_{1})}M^{1}\to{\mathbb{R}},\
(v,s_{1})\mapsto d_{x}h_{i}(v)$. Let us check that it is surjective:

Case 1: $t_{1}\neq 0$. We pick a vector $v\in T_{x}M$ such that
$d_{x}h_{i}(v)\neq 0$ and $(v,\frac{d_{x}h_{1}(v)}{2t_{1}})\in
T_{(x,t_{1})}M^{1}$. We have

$d_{(x,t_{1})}h_{i}^{1}\Big{(}v,\frac{d_{x}h_{1}(v)}{2t_{1}}\Big{)}=d_{x}h_{i}(v)\neq
0.$

Case 2: $t_{1}=0$. We have $T_{(x,0)}M^{1}=\\{(v,s)\in
T_{x}M\times{\mathbb{R}}:\ d_{x}h_{1}(v)=0\\}$. As $Y_{1}\cup Y_{i}$ is a
normal-crossings divisor, $d_{x}h_{1}$ and $d_{x}h_{i}$ are linearly
independent, so there exists $v\in T_{x}M$ such that $d_{x}h_{i}(v)\neq 0$ and
$d_{x}h_{1}(v)=0$. Observe that $(v,0)\in T_{(x,0)}M^{1}$ and
$d_{(x,t_{1})}h_{i}^{1}(v,0)=d_{x}h_{i}^{1}(v)\neq 0$.

Consequently, $d_{(x,t_{1})}h_{i}^{1}:T_{(x,t_{1})}M^{1}\to{\mathbb{R}}$ is
surjective (in both cases), as claimed.

By Theorem 5.10 there exist an open semialgebraic neighborhood $N^{0}\subset
M$ of $H_{1}$ and a Nash embedding $\phi_{1}:N^{0}\to M^{1}$ such that
$\phi_{1}(H_{1})=H_{1+}$ and $W^{1}:=\phi_{1}(N^{0})$ is an open semialgebraic
subset of $M^{1}$ that contains the Nash manifold with corners
$\phi_{1}({\mathcal{Q}})$ as a closed subset. In addition,
$\phi_{1}|_{Y_{1}}=\operatorname{id}_{Y_{1}\times\\{0\\}}$,
$\phi_{1}(Y_{i})\subset Y_{i}^{1}$ for $i=2,\ldots,\ell$ and
$\phi_{1}|_{H_{1}}:H_{1}\to H_{1+}$ is close to $(\pi_{1}|_{H_{1+}})^{-1}$.
Consider the Nash subset $Y^{1}:=\bigcup_{i=1}^{\ell}Y_{i}^{1}$ of $M^{1}$,
whose irreducible components are Nash submanifolds of $M^{1}$. We have
$\phi_{1}(Y)=Y^{1}\cap W^{1}$. In addition,
$\pi_{1-}|_{\operatorname{Int}(H_{1-})}:\operatorname{Int}(H_{1-})\to\operatorname{Int}(H_{1})$
is a Nash diffeomorphism and
$\pi_{1-}(Y^{1}\cap\operatorname{Int}(H_{1-}))=Y\cap\operatorname{Int}(H_{1})$.
As $Y$ is a Nash normal-crossings divisor of $M$ and
$M^{1}=D(H_{1})=W^{1}\cup\operatorname{Int}(H_{1-})$, we conclude that $Y^{1}$
is a Nash normal-crossings divisor of $M^{1}$.

Denote $H_{1}^{1}:=H_{1+}$ and
$H_{i}^{1}:=\pi_{1}^{-1}(H_{i})=\\{h_{i}^{1}\geq 0\\}$ for $i=2,\ldots,\ell$.
By Lemma 5.6 the intersection

${\mathcal{Q}}^{1}:=\bigcap_{i=1}^{\ell}H_{i}^{1}=\\{(x,t_{1})\in
M\times{\mathbb{R}}:t_{1}^{2}-h_{1}(x)=0,t_{1}\geq 0,h_{2}(x)\geq
0,\ldots,h_{\ell}(x)\geq 0\\}\subset M^{1}$

is a Nash manifold with corners and the Nash closure in $M^{1}$ of
$\partial{\mathcal{Q}}^{1}$ is $Y^{1}$. Observe that
$\pi_{1}({\mathcal{Q}}^{1})={\mathcal{Q}}$, the restriction
$\pi_{1}|_{{\mathcal{Q}}^{1}}:{\mathcal{Q}}^{1}\to{\mathcal{Q}}$ is a
semialgebraic homeomorphism and the restriction
$\pi_{1}|_{\operatorname{Int}({\mathcal{Q}}^{1})}:\operatorname{Int}({\mathcal{Q}}^{1})\to\operatorname{Int}({\mathcal{Q}})$
is a Nash diffeomorphism. We claim:
$\phi_{1}({\mathcal{Q}})={\mathcal{Q}}^{1}$. As $\phi_{1}$ is a Nash
diffeomorphism, it is enough to prove: $\phi_{1}(H_{1}\cap
H_{i})=H_{1}^{1}\cap H_{i}^{1}=H_{1+}\cap\pi_{1}^{-1}(H_{i})$ for
$i=2,\ldots,\ell$.

As $\phi_{1}:N^{0}\to W^{1}$ is a Nash diffeomorphism, it is enough to check

$\phi_{1}(\operatorname{Int}(H_{1})\cap\operatorname{Int}(H_{i}))=\operatorname{Int}(H_{1+})\cap\pi_{1}^{-1}(\operatorname{Int}(H_{i})).$

Observe that $\operatorname{Int}(H_{1})\cap\operatorname{Int}(H_{i})$ is a
union of connected components of $N^{0}\setminus(Y_{1}\cup Y_{i})$. In
addition, $\phi_{1}(Y_{1})=Y_{1}\times\\{0\\}$ and $\phi_{1}(Y_{i})\subset
D(Y_{i}\cap H_{1})$, so

$\phi_{1}(Y_{i}\cap\operatorname{Int}(H_{1}))\subset D(Y_{i}\cap
H_{1})\cap\operatorname{Int}(H_{1+}).$

As
$\phi_{1}|_{\operatorname{Int}(H_{1})}:\operatorname{Int}(H_{1})\to\operatorname{Int}(H_{1+})$
is a Nash diffeomorphism, $\phi_{1}(Y_{i}\cap\operatorname{Int}(H_{1}))$ is a
closed Nash submanifold of the Nash manifold $D(Y_{i}\cap
H_{1})\cap\operatorname{Int}(H_{1+})$ of the same dimension. As $\phi_{1}$ is
close to $(\pi_{1}|_{H_{1},+})^{-1}$, both Nash manifolds have the same number
of connected components and

$\phi_{1}(Y_{1}\cap\operatorname{Int}(H_{1}))=D(Y_{i}\cap
H_{1})\cap\operatorname{Int}(H_{1+}).$

Note that $\operatorname{Int}(H_{1+})\cap\pi_{1}^{-1}(H_{i})$ is a union of
connected components of $\operatorname{Int}(H_{1+})\setminus D(Y_{i}\cap
H_{1})$. As $\phi_{1}$ is close to $(\pi_{1}|_{H_{1+}})^{-1}$, we conclude
that the Nash diffeomorphism $\phi_{1}$ provides a bijection between the
connected components of
$\operatorname{Int}(H_{1})\cap\operatorname{Int}(H_{i})$ and the connected
components of $\operatorname{Int}(H_{1+})\cap\pi_{1}^{-1}(H_{i})$.
Consequently,
$\phi_{1}(\operatorname{Int}(H_{1})\cap\operatorname{Int}(H_{i}))=\operatorname{Int}(H_{1+})\cap\pi_{1}^{-1}(H_{i})$,
as claimed.

Summarizing, the objects we have constructed in this step are the following:

  * •

The $d$-dimensional Nash manifold $M^{1}:=D(H_{1})$ and the projection
$\pi_{1}:D(H_{1})\to H_{1},\ (x,t_{1})\mapsto x$.

  * •

The Nash manifolds with boundary $H_{1}^{1}:=H_{1+}$ and
$H_{i}^{1}:=\pi_{1}^{-1}(H_{i})=\\{h_{i}^{1}\geq 0\\}$ for $i=2,\ldots,\ell$.
The boundary of $H_{1}^{1}$ is $Y_{1}^{1}:=Y_{1}\times\\{0\\}$ and the
boundary of $H_{i}^{1}$ is $Y_{i}^{1}:=D(Y_{i}\cap H_{1})$ for
$i=2,\ldots,\ell$.

  * •

The Nash normal-crossings divisor $Y^{1}:=\bigcup_{i=1}^{\ell}Y_{i}^{1}$ of
$M^{1}$.

  * •

The Nash equations

$h_{i}^{1}:M^{1}\to{\mathbb{R}},\ (x,t_{1})\mapsto\begin{cases}t_{1}&\text{if
$i=1$}\\\ h_{i}(x)&\text{if $i=2,\ldots,\ell$,}\end{cases}$

which satisfy: $d_{(x,t_{1})}h_{i}^{1}:T_{(x,t_{1})}M^{1}\to{\mathbb{R}}$ is
surjective for each $(x,t_{1})\in Y_{i}^{1}$ and $i=1,\ldots,\ell$.

  * •

The Nash manifold with corners
${\mathcal{Q}}^{1}:=\bigcap_{i=1}^{\ell}H_{i}^{1}$, which is mapped by the
projection $\pi_{1}$ onto ${\mathcal{Q}}$. In addition, the restriction
$\pi_{1}|_{{\mathcal{Q}}^{1}}:{\mathcal{Q}}^{1}\to{\mathcal{Q}}$ is a
semialgebraic homeomorphism and the restriction
$\pi_{1}|_{\operatorname{Int}({\mathcal{Q}}^{1})}:\operatorname{Int}({\mathcal{Q}}^{1})\to\operatorname{Int}({\mathcal{Q}})$
is a Nash diffeomorphism.

  * •

The open semialgebraic neighborhoods $N^{0}\subset M$ of $H_{1}$ and
$W^{1}\subset M^{1}$ of ${\mathcal{Q}}^{1}$ and the Nash diffeomorphism
$\phi_{1}:N^{0}\to W^{1}$ that maps ${\mathcal{Q}}$ onto ${\mathcal{Q}}^{1}$,
$Y_{1}$ onto $Y_{1}^{1}$ and $Y_{i}$ inside $Y_{i}^{1}$ for $i=2,\ldots,\ell$.

Step 2. Let $(D(H_{2}^{1}),\pi_{2})$ be the Nash double of
$H_{2}^{1}=\\{h_{2}^{1}\geq 0\\}$. We have

$M^{2}:=D(H_{2}^{1})=\\{(x,t_{1},t_{2})\in(M\times{\mathbb{R}})\times{\mathbb{R}}:\
t_{1}^{2}-h_{1}(x)=0,t_{2}^{2}-h_{2}^{1}(x,t_{1})=0\\}=\\\
\\{(x,t_{1},t_{2})\in M\times{\mathbb{R}}^{2}:\
t_{1}^{2}-h_{1}(x)=0,t_{2}^{2}-h_{2}(x)=0\\}$

and $\pi_{2}:D(H_{2}^{1})\to H_{2}^{1},\ (x,t_{1},t_{2})\mapsto(x,t_{1})$. The
Nash map $\pi_{2}$ is proper by Lemma 5.8(iv). Consider the
$(d-1)$-dimensional Nash submanifolds $Y_{1}^{2}:=D(Y_{1}^{1}\cap H_{2}^{1})$,
$Y_{2}^{2}=Y_{2}^{1}\times\\{0\\}$ and $Y_{i}^{2}:=D(Y_{i}^{1}\cap
H_{2}^{1})=(Y_{i}^{1}\times{\mathbb{R}})\cap D(H_{2}^{1})$ (see (5.1)) of
$M^{2}$ for $i=3,\ldots,\ell$. Define

$h_{i}^{2}:M^{2}\to{\mathbb{R}},\ (x,t_{1},t_{2})\mapsto
h_{i}^{2}(x,t_{1},t_{2})=\begin{cases}h_{1}^{1}(x,t_{1})=t_{1}&\text{if
$i=1$,}\\\ t_{2}&\text{if $i=2$,}\\\ h_{i}^{1}(x,t_{1})=h_{i}(x)&\text{if
$i=3,\ldots,\ell$.}\\\ \end{cases}$

The tangent space to $M^{2}$ at a point $(x,t_{1},t_{2})\in M^{2}$ is:

$T_{(x,t_{1},t_{2})}M^{1}=\\{(v,s_{1},s_{2})\in T_{x}M\times{\mathbb{R}}^{2}:\
d_{x}h_{1}(v)-2t_{1}s_{1}=0,d_{x}h_{2}(v)-2t_{2}s_{2}=0\\}.$

The reader can check (analogously as we have done in Step 1):
$h_{i}^{2}:M^{2}\to{\mathbb{R}}$ is a Nash equation of $Y_{i}^{2}$ in $M^{2}$
such that
$d_{(x,t_{1},t_{2})}h_{i}^{2}:T_{(x,t_{1},t_{2})}M^{2}\to{\mathbb{R}}$ is
surjective for each $(x,t_{1},t_{2})\in Y_{i}^{2}$ and $i=1,\ldots,\ell$.

By Theorem 5.10 there exist an open semialgebraic neighborhood $N^{1}\subset
M^{1}$ of $H_{2}^{1}$ and a Nash embedding $\phi_{2}:N^{1}\to M^{2}$ such that
$\phi_{2}(H_{2}^{1})=H_{2+}^{1}$ and $W^{2}:=\phi_{2}(N^{1})$ is an open
semialgebraic subset of $M^{2}$ that contains the Nash manifold with corners
$\phi_{2}({\mathcal{Q}}^{1})$ as a closed subset. In addition,
$\phi_{2}|_{Y_{2}^{1}}=\operatorname{id}_{Y_{2}^{1}\times\\{0\\}}$,
$\phi_{2}(Y_{i}^{1})\subset Y_{i}^{2}$ for $i=1,3,\ldots,\ell$ and
$\phi_{2}|_{H_{2}^{1}}:H_{2}^{1}\to H_{2+}^{1}$ is close to
$(\pi_{2}|_{H_{2+}^{1}})^{-1}$. Consider the Nash subset
$Y^{2}:=\bigcup_{i=1}^{\ell}Y_{i}^{2}$ of $M^{2}$, whose irreducible
components are Nash submanifolds of $M^{2}$. We have $\phi_{2}(Y^{1}\cap
N^{1})=Y^{2}\cap W^{2}$. In addition,
$\pi_{2-}|_{\operatorname{Int}(H_{2-}^{1})}:\operatorname{Int}(H_{2-}^{1})\to\operatorname{Int}(H_{2}^{1})$
is a Nash diffeomorphism and
$\pi_{2-}(Y^{2}\cap\operatorname{Int}(H_{2-}^{1}))=Y^{1}\cap\operatorname{Int}(H_{2}^{1})$.
As $Y^{1}$ is a Nash normal-crossings divisor of $M^{1}$ and
$M^{2}=D(H_{2}^{1})=W^{2}\cup\operatorname{Int}(H_{2-}^{1})$, we conclude that
$Y^{2}$ is a Nash normal-crossings divisor of $M^{2}$.

Denote $H_{2}^{2}:=H_{2+}^{1}$ and
$H_{i}^{2}:=\pi_{2}^{-1}(H_{i}^{1})=\\{h_{i}^{2}\geq 0\\}$ for
$i=1,3,\ldots,\ell$. By Lemma 5.6 the intersection

${\mathcal{Q}}^{2}:=\bigcap_{i=1}^{\ell}H_{i}^{2}=\\{(x,t_{1},t_{2})\in
M\times{\mathbb{R}}^{2}:t_{1}^{2}-h_{1}(x)=0,t_{2}^{2}-h_{2}(x)=0,t_{1}\geq
0,t_{2}\geq 0,\\\ h_{3}(x)\geq 0,\ldots,h_{\ell}(x)\geq 0\\}\subset M^{2}$

is a Nash manifold with corners and the Nash closure in $M^{2}$ of
$\partial{\mathcal{Q}}^{2}$ is $Y^{2}$. Observe that
$\pi_{2}({\mathcal{Q}}^{2})={\mathcal{Q}}^{1}$, the restriction
$\pi_{2}|_{{\mathcal{Q}}^{2}}:{\mathcal{Q}}^{2}\to{\mathcal{Q}}^{1}$ is a
semialgebraic homeomorphism and the restriction
$\pi_{2}|_{\operatorname{Int}({\mathcal{Q}}^{2})}:\operatorname{Int}({\mathcal{Q}}^{2})\to\operatorname{Int}({\mathcal{Q}}^{1})$
is a Nash diffeomorphism. In particular,
$(\pi_{1}\circ\pi_{2})({\mathcal{Q}}^{2})={\mathcal{Q}}$. The reader can check
(analogously as we have done in Step 1):
$\phi_{2}({\mathcal{Q}}^{1})={\mathcal{Q}}^{2}$. Consequently,
$(\phi_{2}\circ\phi_{1})({\mathcal{Q}})={\mathcal{Q}}^{2}$.

Recursive steps. We proceed recursively for $k=1,\ldots,\ell$ and we
construct:

  * •

The $d$-dimensional Nash manifold $M^{k}$ and the projection

$\pi_{k}:M^{k}\to H_{k}^{k-1},\
(x,t_{1},\ldots,t_{k})\mapsto(x,t_{1},\ldots,t_{k-1}),$

which is proper by Lemma 5.8(iv).

  * •

The Nash manifolds with boundary $H_{i}^{k}$ and the $(d-1)$-dimensional Nash
submanifolds $Y_{i}^{k}=\partial H_{i}^{k}$ of $M^{k}$.

  * •

The Nash normal-crossings divisor $Y^{k}:=\bigcup_{i=1}^{\ell}Y_{i}^{k}$ of
$M^{k}$.

  * •

The Nash equations

$h_{i}^{k}:M^{k}\to{\mathbb{R}},\
(x,t_{1},\ldots,t_{k})\mapsto\begin{cases}t_{i}&\text{if $i=1,\ldots,k$}\\\
h_{i}(x)&\text{if $i=k+1,\ldots,\ell$,}\end{cases}$

which satisfy:
$d_{(x,t_{1},\ldots,t_{k})}h_{i}^{k}:T_{(x,t_{1},\ldots,t_{k})}M^{k}\to{\mathbb{R}}$
is surjective for each $(x,t_{1},\ldots,t_{k})\in Y_{i}^{k}$ and
$i=1,\ldots,\ell$.

  * •

The Nash manifold with corners contained in $M^{k}$

$\begin{split}{\mathcal{Q}}^{k}:=\bigcap_{i=1}^{\ell}H_{i}^{k}=\\{(x,t_{1},\ldots,t_{k})&\in
M\times{\mathbb{R}}^{k}:\ t_{1}^{2}-h_{1}(x)=0,\ldots,t_{k}^{2}-h_{k}(x)=0,\\\
&t_{1}\geq 0,\ldots,t_{k}\geq 0,h_{k+1}(x)\geq 0,\ldots,h_{\ell}(x)\geq
0\\},\end{split}$

which is mapped by the projection $\pi_{k}$ onto ${\mathcal{Q}}^{k-1}$.
Consequently,
$(\pi_{1}\circ\cdots\circ\pi_{k})({\mathcal{Q}}^{k})={\mathcal{Q}}$. In
addition, the restriction
$\pi_{k}|_{{\mathcal{Q}}^{k}}:{\mathcal{Q}}^{k}\to{\mathcal{Q}}^{k-1}$ is a
semialgebraic homeomorphism and the restriction
$\pi_{k}|_{\operatorname{Int}({\mathcal{Q}}^{k})}:\operatorname{Int}({\mathcal{Q}}^{k})\to\operatorname{Int}({\mathcal{Q}}^{k-1})$
is a Nash diffeomorphism.

  * •

The open semialgebraic neighborhoods $N^{k-1}\subset M^{k-1}$ of $H_{k}^{k-1}$
and $W^{k}\subset M^{k}$ of ${\mathcal{Q}}^{k}$ and the Nash diffeomorphism
$\phi_{k}:N^{k-1}\to W^{k}$ that maps ${\mathcal{Q}}^{k-1}$ onto
${\mathcal{Q}}^{k}$, $Y_{k}^{k-1}$ onto $Y_{k}^{k}$ and $Y_{i}^{k-1}$ inside
$Y_{i}^{k}$ for $i\neq k$. Consequently,
$(\phi_{k}\circ\cdots\circ\phi_{1})({\mathcal{Q}})={\mathcal{Q}}^{k}$.

At this point observe that

$D({\mathcal{Q}})=M^{\ell}=\\{(x,t)\in
M\times{\mathbb{R}}^{\ell}:t_{1}^{2}-h_{1}(x)=0,\ldots,t_{\ell}^{2}-h_{\ell}(x)=0\\}$

is a $d$-dimensional Nash manifold and ${\mathcal{Q}}^{\ell}=\\{t_{1}\geq
0,\ldots,t_{\ell}\\}\subset D({\mathcal{Q}})$ is a Nash manifold with corners.
The Nash closure of the boundary

$\partial{\mathcal{Q}}^{\ell}=\bigcup_{i=1}^{\ell}\\{t_{1}\geq
0,\cdots,t_{i-1}\geq 0,t_{i}=0,t_{i+1}\geq 0,\cdots,t_{\ell}\geq 0\\}$

is the Nash normal-crossings divisor
$Y^{\ell}=\bigcup_{i=1}^{\ell}\\{t_{i}=0\\}$ of $D({\mathcal{Q}})$. After
shrinking $N^{0}$ if necessary, the composition
$\phi:=\phi_{\ell}\circ\cdots\circ\phi_{1}:N^{0}\to W:=\phi(N^{0})$ is a Nash
diffeomorphism such that $W$ is an open semialgebraic subset of
$D({\mathcal{Q}})$ and $\phi({\mathcal{Q}})={\mathcal{Q}}^{\ell}$. The
composition
$\pi:=\pi_{1}\circ\cdots\circ\pi_{\ell}:D({\mathcal{Q}})\to{\mathcal{Q}}$ is a
proper surjective Nash map such that the restriction
$\pi|_{{\mathcal{Q}}^{\ell}}:{\mathcal{Q}}^{\ell}\to{\mathcal{Q}}$ is a
semialgebraic homeomorphism and the restriction
$\pi|_{\operatorname{Int}({\mathcal{Q}}^{\ell})}:\operatorname{Int}({\mathcal{Q}}^{\ell})\to\operatorname{Int}({\mathcal{Q}})$
is a Nash diffeomorphism, as required. ∎

##  6\. Applications

In this section we prove several results proposed in the introduction as
consequences of Theorems 1.4 and 1.7.

###  6.1. Weak desingularization of closed semialgebraic sets.

We prove next Theorem 1.9.

###### Proof of Theorem 1.9.

By Theorem 1.4 applied to ${\mathcal{S}}$ and [Sh, Rmk.II.1.5] to
${\mathcal{S}}$ and $\varepsilon$ we may assume that:

  * •

${\mathcal{Q}}:={\mathcal{S}}$ is a connected Nash manifolds with corners,
which is closed in ${\mathbb{R}}^{n}$.

  * •

The Zariski closure $X\subset{\mathbb{R}}^{n}$ of ${\mathcal{Q}}$ is a
$d$-dimensional non-singular real algebraic set.

  * •

The Zariski closure $Y$ of $\partial{\mathcal{Q}}$ is a normal-crossings
divisor of $X$.

If ${\mathcal{S}}$ is compact, we may assume in addition by Theorem 1.6 that
$X$ is compact and connected.

By [FGR, Thm.1.11, 1.12] there exists an open semialgebraic neighborhood
$U\subset X$ of ${\mathcal{Q}}$ such that:

  * •

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

  * •

For every $x\in\partial{\mathcal{Q}}$ the analytic closure of the germ
$\partial{\mathcal{Q}}_{x}$ is $Y_{x}$.

Choose a strictly positive continuous semialgebraic function
$\varepsilon_{0}:X\to{\mathbb{R}}$ and an open semialgebraic neighborhood
$U_{1}\subset X$ of ${\mathcal{Q}}$ such that $\operatorname{Cl}(U_{1})\subset
U$. In case ${\mathcal{Q}}$ is compact, we may choose $\varepsilon_{0}$
constant. Define $M:=\\{x\in U_{1}:\
\operatorname{dist}(x,{\mathcal{Q}})<\varepsilon_{0}(x)\\}$, which is a Nash
manifold such that ${\mathcal{Q}}\subset M\subset\operatorname{Cl}(M)\subset
U$.

By [Sh, Thm.VI.2.1] and its proof there exists a ${\mathcal{C}}^{1}$
semialgebraic function $g:U\to{\mathbb{R}}$ such that $M$ is diffeomorphic to
$g^{-1}((-1,+\infty))$ and $H:=g^{-1}([0,+\infty))\subset M$ is a
${\mathcal{C}}^{1}$ semialgebraic manifold with boundary that contains
${\mathcal{Q}}$ in its interior. In case ${\mathcal{Q}}$ is compact, also $H$
is compact. We may assume in addition that the differential
$d_{x}g:T_{x}M\to{\mathbb{R}}$ is surjective for each $x\in g^{-1}(0)$ (see
Lemma 5.4). As $\operatorname{Cl}(M)\subset U$ is a closed subset of $X$,
there exists a ${\mathcal{C}}^{1}$ semialgebraic extension of
$g|_{\operatorname{Cl}(M)}$ to $X$ such that $G^{-1}(0)=g^{-1}(0)$. Let
$F:X\to{\mathbb{R}}$ be a Nash approximation of $G$ in the ${\mathcal{C}}^{1}$
semialgebraic topology (in case $X$ is compact, we may assume $F$ is a
polynomial). As $G$ is strictly positive on the closed semialgebraic set
${\mathcal{Q}}$, we may assume $F$ is strictly positive on ${\mathcal{Q}}$, so
$F^{-1}(0)\subset M\setminus{\mathcal{Q}}$. In addition, as we consider a
$\mathcal{C}^{1}$ approximation, we may assume $d_{x}F:T_{x}X\to{\mathbb{R}}$
is surjective for each $x\in F^{-1}(0)$. Thus,
${\mathcal{H}}_{\varepsilon_{0}}^{\prime}:=F^{-1}([0,+\infty))\subset M$ is a
Nash manifold with (non-singular) boundary $Z_{\varepsilon_{0}}:=F^{-1}(0)$,
which is a Nash submanifold of $X$ of dimension $d-1$. In case $X$ is compact,
also ${\mathcal{H}}_{\varepsilon_{0}}^{\prime}$ is compact and
$Z_{\varepsilon_{0}}:=F^{-1}(0)$ is a (compact) non-singular real algebraic
subset of $X$. As ${\mathcal{Q}}$ is connected, it is contained in one of the
connected components ${\mathcal{H}}_{\varepsilon_{0}}$ of
${\mathcal{H}}_{\varepsilon_{0}}^{\prime}$. Observe that
${\mathcal{H}}_{\varepsilon_{0}}$ is a Nash manifold with boundary that
contains ${\mathcal{Q}}$ in its interior
$\operatorname{Int}({\mathcal{H}}_{\varepsilon_{0}})$, which is connected.
Thus, $\operatorname{Int}({\mathcal{H}}_{\varepsilon_{0}})$ is a connected
component of $X\setminus Z_{\varepsilon_{0}}$.

In case ${\mathcal{Q}}$ is non-compact and $Z_{\varepsilon_{0}}$ is not a non-
singular algebraic subset of $X$, we apply [Fe2, Lem.C.1] to assume up to a
suitable Nash embedding of $X$ in some affine space ${\mathbb{R}}^{p}$ that
$X\subset{\mathbb{R}}^{p}$ is a non-singular algebraic set and there exists an
(algebraic) normal-crossings divisor $Y_{\varepsilon_{0}}$ such that the Nash
submanifold $Z_{\varepsilon_{0}}$ is a union of connected components of
$Y_{\varepsilon_{0}}$. By Hironaka’s resolution of singularities we may assume
that $Y_{\varepsilon_{0}}$ is a non-singular algebraic subset of $X$.

By Theorem 1.7 there exists (after shrinking $M$ at the beginning if needed) a
proper Nash map $h:\operatorname{Cl}(M)\to{\mathcal{Q}}$ such that
$h(M)=h({\mathcal{Q}})={\mathcal{Q}}$, the restriction
$h|_{{\mathcal{H}}_{\varepsilon_{0}}\setminus
h^{-1}({\mathcal{T}}_{\varepsilon_{0}})}:{\mathcal{H}}_{\varepsilon_{0}}\setminus
h^{-1}({\mathcal{T}}_{\varepsilon_{0}})\to{\mathcal{Q}}\setminus{\mathcal{T}}_{\varepsilon_{0}}$
is a Nash diffeomorphism, where
${\mathcal{T}}_{\varepsilon_{0}}:=\\{x\in{\mathcal{Q}}:\
\operatorname{dist}(x,\partial{\mathcal{Q}})<\varepsilon_{0}\\}$. As
${\mathcal{Q}}\subset{\mathcal{H}}_{\varepsilon_{0}}\subset M$, we have
$h({\mathcal{H}}_{\varepsilon_{0}})={\mathcal{Q}}$. As
${\mathcal{H}}_{\varepsilon_{0}}\subset\operatorname{Cl}(M)$ is closed and
$h:\operatorname{Cl}(M)\to{\mathcal{Q}}$ is proper,
$f:=h|_{{\mathcal{H}}_{\varepsilon_{0}}}:{\mathcal{H}}_{\varepsilon_{0}}\to{\mathcal{Q}}$
is also a proper map, as required. ∎

###  6.2. An alternative proof to Theorem 1.10

In order to give an alternative proof of Theorem 1.10 based on Theorem 1.9, we
need some preliminary results. We start with the following Lemma that extends
[Fe2, Lem.2.8] to Nash manifolds with boundary.

######  Lemma 6.1.

Let $H_{1}\subset{\mathbb{R}}^{m}$ and $H_{2}\subset{\mathbb{R}}^{n}$ be Nash
manifolds with (non-singular) boundary. Let $f:H_{1}\to H_{2}$ be a
semialgebraic homeomorphism. Then every continuous semialgebraic map
$g:H_{1}\to H_{2}$ close to $f$ in the ${\mathcal{C}}^{0}$ semialgebraic
topology, such that $g(\partial H_{1})\subset\partial H_{2}$, is surjective.

###### Proof.

As $f:H_{1}\to H_{2}$ is a semialgebraic homeomorphism, $f(\partial
H_{1})=\partial H_{2}$ (use invariance of domain). Consider the Nash doubles
$(D(H_{i}),\pi_{i})$ and $H_{i,\epsilon}=D(H_{i})\cap\\{\epsilon t\geq 0\\}$
where $\epsilon=\pm$. Recall that $\pi_{i\epsilon}:H_{i,\epsilon}\to H_{i}$ is
a semialgebraic homeomorphism. As $f(\partial H_{1})=\partial H_{2}$, the map

$F:D(H_{1})\to D(H_{2}),\ (x,t)\mapsto\begin{cases}(\pi_{2+})^{-1}\circ
f\circ\pi_{1+}(x,t)&\text{ if }t\geq 0,\\\ (\pi_{2-})^{-1}\circ
f\circ\pi_{1-}(x,t)&\text{ if }t\leq 0,\end{cases}$

is well-defined and semialgebraic. Observe that $F$ is bijective and

$F^{-1}:D(H_{2})\to D(H_{1}),\ (y,s)\mapsto\begin{cases}(\pi_{1+})^{-1}\circ
f^{-1}\circ\pi_{2+}(x,t)&\text{ if }t\geq 0,\\\ (\pi_{1-})^{-1}\circ
f^{-1}\circ\pi_{2-}(x,t)&\text{ if }t\leq 0.\end{cases}$

Let us check: $F$ is continuous. Once this is done, the same proof shows that
$F^{-1}$ is continuous, so $F$ is a semialgebraic homeomorphism.

As $F$ is continuous on both $H_{1,+}$, $H_{1,-}$ and $\pi_{1}|_{H_{1+}\cap
H_{1-}}=\pi_{1}|_{\partial H_{1}}=\operatorname{id}_{\partial H_{1}},$ we
conclude by the pasting lemma that $F$ is continuous on $D(H_{1})$.

Let $g:H_{1}\to H_{2}$ be a continuous semialgebraic map. If $g(\partial
H_{1})\subset\partial H_{2}$, the map

$G:D(H_{1})\to D(H_{2}),\ (x,t)\mapsto(\pi_{2\epsilon})^{-1}\circ
g\circ\pi_{1\epsilon}(x,t)$

is well-defined, continuous and semialgebraic for $\epsilon=\pm$. The proof is
analogous to the one for $F$. Next, by [Fe2, Lem.2.8] there exists a strictly
positive continuous semialgebraic function
$\varepsilon:D(H_{1})\to{\mathbb{R}}$ such that if $\|G-F\|<\varepsilon$, then
$G$ is surjective.

For $\epsilon=\pm$, consider the strictly positive continuous semialgebraic
functions

$\varepsilon_{\epsilon}:=\varepsilon\circ\pi_{1\epsilon}^{-1}:H_{1}\to{\mathbb{R}}$

and $\varepsilon^{*}=\min\\{\varepsilon_{+},\varepsilon_{-}\\}$. Let
$\delta:H_{1}\to{\mathbb{R}}$ be a strictly positive continuous semialgebraic
function such that if $f,g:H_{1}\to H_{2}$ are continuous semialgebraic maps
such that $\|g-f\|<\delta$, then $\|(\pi_{2\epsilon})^{-1}\circ
f-(\pi_{2\epsilon})^{-1}\circ g\|<\varepsilon^{*}$ for $\epsilon=\pm$ (see
[Sh, Rmk.II.1.5]). Thus, $\|(\pi_{2\epsilon})^{-1}\circ
f\circ\pi_{1\epsilon}-(\pi_{2\epsilon})^{-1}\circ
g\circ\pi_{1\epsilon}\|<\varepsilon^{*}\circ\pi_{1\epsilon}\leq\varepsilon$
for $\epsilon=\pm$. Consequently, $\|F-G\|<\varepsilon$, so $G$ is surjective.
Taking into account the definition of $G$, we conclude that $g$ is also
surjective, as required. ∎

####  6.2.1. Preliminary reductions to prove Theorem 1.10

The proof of Theorem 1.10 requires some additional preparation. As the image
of a compact semialgebraic set connected by analytic paths under a Nash map is
a compact semialgebraic set connected by analytic paths [Fe2, Main Thm.1.4],
we are reduced to prove: If ${\mathcal{S}}\subset{\mathbb{R}}^{n}$ is a
compact semialgebraic set connected by analytic paths of dimension $d\geq 2$,
there exists a Nash map $f:{\mathbb{R}}^{d}\to{\mathbb{R}}^{n}$ such that
$f(\overline{{\mathcal{B}}}_{d})={\mathcal{S}}$.

By Theorem 1.9 we may assume that $H:={\mathcal{S}}$ is a connected compact
Nash manifold with smooth boundary. Let $M:=D(H)$ be the Nash double of $H$
(which is a connected compact Nash manifold) and consider the proper
surjective Nash map $\pi:D(H)\to H$ introduced in Lemma 5.8. Thus, we may
assume that $M:={\mathcal{S}}$ is a connected compact Nash manifold.
Alternatively, we may use Theorem 1.4 and Theorem 5.13 to obtain a similar
reduction.

Consider the simplex $\Delta_{d}:=\\{{\tt x}_{1}\geq 0,\ldots,{\tt x}_{d}\geq
0,{\tt x}_{1}+\cdots+{\tt x}_{d}\leq 1\\}\subset{\mathbb{R}}^{d}$. In [FU,
Cor.2.8] we proved that the closed ball $\overline{{\mathcal{B}}}_{d}$ is a
polynomial image of the simplicial prism $\Delta_{d-1}\times[0,1]$. Thus, the
proof of Theorem 1.10 is reduced to show the following: If
$M\subset{\mathbb{R}}^{n}$ is a connected compact Nash manifold of dimension
$d\geq 2$, there exists a Nash map $f:{\mathbb{R}}^{d}\to{\mathbb{R}}^{n}$
such that $f(\sigma\times[a,b])=M$, where $\sigma\subset{\mathbb{R}}^{d-1}$ is
a $(d-1)$-dimensional simplex and $[a,b]\subset{\mathbb{R}}$ a compact
interval.

Let $\\{U_{i}\\}_{i=1}^{r}$ be a finite covering of $M$ equipped with Nash
diffeomorphisms $u_{i}:U_{i}\to{\mathbb{R}}^{d}$ such that
$M=\bigcup_{i=1}^{m}u_{i}^{-1}(\Delta_{d})$. Define

$\Delta_{d}^{\prime}:=\\{{\tt x}_{1}\geq-2,\ldots,{\tt x}_{d}\geq-2,{\tt
x}_{1}+\cdots+{\tt x}_{d}\leq d+1\\}$

and observe that
$\Delta_{d}\subset\overline{{\mathcal{B}}}_{d}\subset\operatorname{Int}(\Delta_{d}^{\prime})$.
Let $\psi:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ be an affine isomorphism such
that $\psi(\Delta_{d})=\Delta_{d}^{\prime}$.

######  Lemma 6.2.

Consider the $(d-1)$-simplex $\sigma:=\\{{\tt x}_{1}\geq 0,\ldots,{\tt
x}_{d-1}\geq 0,{\tt x}_{1}+\cdots+{\tt x}_{d-1}\leq
1\\}\subset{\mathbb{R}}^{d-1}$. Then the surjective Nash map
$f:\sigma\times[0,1]\to\Delta_{d}^{\prime},\ (x,t)\mapsto\psi(((1-t)x,t))$
restricts to a Nash diffeomorphism
$f|_{\sigma\times[0,1)}:\sigma\times[0,1)\to\Delta_{d}^{\prime}\setminus\\{\psi((0,\ldots,0,1))\\}$.

###### Proof.

As $\psi$ is an affine isomorphism such that
$\psi(\Delta_{d})=\Delta_{d}^{\prime}$, it is enough to prove that the
surjective Nash map $\sigma\times[0,1]\to\Delta_{d},\,(x,t)\mapsto((1-t)x,t)$
restricts to a Nash diffeomorphism

$\varphi:\sigma\times[0,1)\to\Delta_{d}\setminus\\{(0,\ldots,0,1)\\}.$

This is straightforward, because the Nash map

$\phi:\Delta_{d}\setminus\\{(0,\ldots,0,1)\\}\to\sigma\times[0,1),\
(x_{1},\ldots,x_{d})\mapsto\Big{(}\frac{x_{1}}{1-x_{d}},\ldots,\frac{x_{d-1}}{1-x_{d}},x_{d}\Big{)}$

is the inverse of $\varphi$. ∎

We are ready to prove Theorem 1.10.

####  6.2.2. Proof of Theorem 1.10

We keep the notations quoted above. In view of the contruction made in [CF,
§2.3], to obtain a global Nash map $f:{\mathbb{R}}^{d}\to{\mathbb{R}}^{n}$
such that $f(\sigma\times[0,2r-1])=M$, it is enough to prove: The connected
compact Nash manifold $M\subset{\mathbb{R}}^{n}$ is the image of
$\sigma\times[0,2r-1]$ under a Nash map
$f:\sigma\times[0,2r-1]\to{\mathbb{R}}^{n}$.

Let us construct first a surjective continuous semialgebraic function

$h:\sigma\times[0,2r-1]\to
M=\bigcup_{i=1}^{r}u_{i}^{-1}(\Delta_{d})=\bigcup_{i=1}^{r}u_{i}^{-1}(\overline{{\mathcal{B}}}_{d})=\bigcup_{i=1}^{r}u_{i}^{-1}(\Delta_{d}^{\prime})$

that is Nash on $\sigma\times(\bigcup_{i=1}^{r}(2(i-1),2(i-1)+1)$ and
satisfies

$u_{i}^{-1}(\overline{{\mathcal{B}}}_{d})\subset
h(\Delta_{d}\times(2(i-1),2(i-1)+1)),$

for each $i=1,\ldots,r$.

Let $f:\sigma\times[0,1]\to\Delta_{d}^{\prime}$ be the surjective Nash map
introduced in Lemma 6.2 and define

$h_{i}:\sigma\times[2(i-1),2(i-1)+1]\to u_{i}^{-1}(\Delta_{d}^{\prime}),\
(x,t)\mapsto u_{i}^{-1}(f(x,t-2(i-1))),$

which is a surjective Nash map whose restriction to
$\sigma\times\\{2(i-1)+1\\}$ is constant and its restriction to
$\sigma\times[2(i-1),2(i-1)+1)$ is a Nash diffeomorphism onto its image. Write
$\\{p_{i}\\}:=h_{i}(\sigma\times\\{2(i-1)+1\\})$. Let
$\alpha_{i}:[2(i-1)+1,\tfrac{1}{2}+2(i-1)+1]\to M$ be a continuous
semialgebraic map such that $\alpha_{i}(2(i-1)+1)=p_{i}$ and
$\alpha_{i}(\tfrac{1}{2}+2(i-1)+1)=u_{i+1}^{-1}(0)$. Consider the continuous
semialgebraic map

$g_{i}:\sigma\times[2(i-1)+1,\tfrac{1}{2}+2(i-1)+1]\to M,\
(x,t)\mapsto\alpha_{i}(t).$

Let $b_{1},\ldots,b_{d}$ be the vertices of $\Delta_{d}^{\prime}$ different
from $\psi(0,\ldots,0,1)$. Define $x_{d}:=1-x_{1}-\cdots-x_{d-1}$ and consider
the continuous semialgebraic map

$\displaystyle
g_{i}^{\prime}:\sigma\times\Big{[}\tfrac{1}{2}+2(i-1)+1,2(i-1)+2\Big{]}$
$\displaystyle\to{\mathbb{R}}^{d},$ $\displaystyle(x,t)$ $\displaystyle\mapsto
u_{i+1}^{-1}\Big{(}\sum_{i=1}^{d}x_{i}b_{i}2\Big{(}t-\tfrac{1}{2}-2(i-1)-1\Big{)}\Big{)},$

which satisfies $g_{i}^{\prime}(x,\tfrac{1}{2}+2(i-1)+1)=u_{i+1}^{-1}(0)$ and
$g_{i}^{\prime}(x,2(i-1)+2)=u_{i+1}^{-1}(\sum_{i=1}^{d}x_{i}b_{i})$.

Define $h:\sigma\times[0,2r-1]\to M$, as

$(x,t)\mapsto\begin{cases}h_{i}(x,t)&\text{if
$(x,t)\in\sigma\times[2(i-1),2(i-1)+1]$,}\\\ g_{i}(x,t)&\text{if
$(x,t)\in\sigma\times[2(i-1)+1,\tfrac{1}{2}+2(i-1)+1]$,}\\\
g_{i}^{\prime}(x,t)&\text{if
$(x,t)\in\sigma\times[\tfrac{1}{2}+2(i-1)+1,2(i-1)+2]$,}\\\
h_{r}(x,t)&\text{if $(x,t)\in\sigma\times[2(r-1),2r-1]$}\end{cases}$

for $i=1,\ldots,r-1$. Observe that $h$ is a continuous semialgebraic map such
that for each $i=1,\ldots,r$:

  * •

the restriction $h|_{\sigma\times(2(i-1),2(i-1)+1)}$ is a Nash diffeomorphism
onto its image.

  * •

$h(\sigma\times[2(i-1),2(i-1)+1])=u_{i}^{-1}(\Delta_{d}^{\prime})$.

  * •

$u_{i}^{-1}(\overline{{\mathcal{B}}}_{d})\subset
u_{i}^{-1}(\operatorname{Int}(\Delta_{d}^{\prime}))\subset
h(\sigma\times(2(i-1),2(i-1)+1))$.

We conclude that the restriction of $h$ to
$\sigma\times(\bigcup_{i=1}^{r}(2(i-1),2(i-1)+1)$ is Nash and surjective,
because $M=\bigcup_{i=1}^{m}u_{i}^{-1}(\overline{{\mathcal{B}}}_{d})$. In
addition,
$h_{i}^{-1}(u_{i}^{-1}(\overline{{\mathcal{B}}}_{d}))\subset\operatorname{Int}(\sigma)\times(2(i-1),2(i-1)+1)$,
so the sets $h_{i}^{-1}(u_{i}^{-1}(\overline{{\mathcal{B}}}_{d}))$ (for
$i=1,\ldots,r$) are pairwise disjoint. Let $H:{\mathbb{R}}^{d}\to M$ be a
continuous semialgebraic extension of $h$ to ${\mathbb{R}}^{d}$. Define
$X_{1}:=\bigcup_{i=1}^{r}(u_{i}\circ
h_{i})^{-1}(\partial\overline{{\mathcal{B}}}_{d})$, which is a Nash subset of
$\Omega:=\operatorname{Int}(\sigma)\times\bigcup_{i=1}^{r}(2(i-1),2(i-1)+1)$
and observe that $H$ is Nash on the open semialgebraic set $\Omega$. By [Sh,
Thm.II.5.2] there exists a Nash map $F:{\mathbb{R}}^{d}\to M$ close to $H$
such that $F|_{(u_{i}\circ
h_{i})^{-1}(\partial\overline{{\mathcal{B}}}_{d})}=H|_{(u_{i}\circ
h_{i})^{-1}(\partial\overline{{\mathcal{B}}}_{d})}$ for $i=1,\ldots,r$. As the
restriction $H|_{(u_{i}\circ
h_{i})^{-1}(\overline{{\mathcal{B}}}_{d})}=h_{i}|_{h_{i}^{-1}(u_{i}^{-1}(\overline{{\mathcal{B}}}_{d}))}:h_{i}^{-1}(u_{i}^{-1}(\overline{{\mathcal{B}}}_{d}))\to
u_{i}^{-1}(\overline{{\mathcal{B}}}_{d})$ is by Lemma 6.2 a Nash
diffeomorphism, the restriction $F|_{(u_{i}\circ
h_{i})^{-1}(\overline{{\mathcal{B}}}_{d})}$ is by Lemma 6.1 surjective for
$i=1,\ldots,r$. Thus,
$M=\bigcup_{i=1}^{m}u_{i}^{-1}(\overline{{\mathcal{B}}}_{d})\subset
F(\sigma\times[0,2r-1])\subset M$, so $F(\sigma\times[0,2r-1])=M$, as
required. ∎

###  6.3. Nash structure of orientable compact smooth manifolds

It is well-known that the compact orientable surface of genus $g$ admits a
Nash model. We show in the following example how to deduce straightforwardly
this fact from Theorem 1.7.

######  Example 6.3.

Let ${\mathcal{P}}_{n}\subset{\mathbb{R}}^{2}$ be a convex polygon with $n$
edges. As ${\mathcal{P}}_{n}$ is convex, there exist polynomials
$h_{1},\ldots,h_{n}\in{\mathbb{R}}[{\tt x},{\tt y}]$ of degree 1 such that
${\mathcal{P}}_{n}=\\{h_{1}\geq 0,\ldots,h_{n}\geq 0\\}$. Fix
$2+\frac{1}{2}(1-(-1)^{n})\leq s\leq n$ and let
${\mathcal{J}}:=\\{J_{k}\\}_{k=1}^{s}$ be a partition of the set
$\\{1,\ldots,n\\}$ such that

${\mathcal{P}}_{n}\cap\\{h_{i}=0\\}\cap\\{h_{j}=0\\}=\varnothing$ (6.1)

for each $i,j\in J_{k}$ with $i\neq j$. For each $k=1,\ldots,s$ define the
polynomial $h_{J_{k}}:=\prod_{i\in J_{k}}h_{i}$. As the partition
${\mathcal{J}}$ satisfies (6.1), it holds that ${\mathcal{P}}_{n}$ is a
connected component of the semialgebraic set $\\{x\in{\mathbb{R}}^{2}:\
h_{J_{1}}(x)\geq 0,\ldots,h_{J_{s}}(x)\geq 0\\}$. Thus,

$D_{s}({\mathcal{P}}_{n}):=\\{(x,t)\in{\mathcal{P}}_{n}\times{\mathbb{R}}^{s}:\
t_{1}^{2}-h_{J_{1}}(x)=0,\ldots,t_{s}^{2}-h_{J_{s}}(x)=0\\}\subset{\mathbb{R}}^{s+2}$

is a connected compact Nash surface, which is in addition a connected
component of the (maybe singular) real algebraic set

$X_{s,n}=\\{(x,t)\in{\mathbb{R}}^{2}\times{\mathbb{R}}^{s}:\
t_{1}^{2}-h_{J_{1}}(x)=0,\ldots,t_{s}^{2}-h_{J_{s}}(x)=0\\}\subset{\mathbb{R}}^{s+2}.$

We claim: $D_{s}({\mathcal{P}}_{n})\subset\operatorname{Reg}(X_{s,n})$.

Pick $(x,t)\in D_{s}({\mathcal{P}}_{n})$ and consider the Jacobian matrix

$J_{s,n}(x,t):=\begin{pmatrix}2t_{1}&0&\cdots&0&\frac{\partial
h_{J_{1}}}{\partial{\tt x}_{1}}(x)&\frac{\partial h_{J_{1}}}{\partial{\tt
x}_{2}}(x)\\\ 0&2t_{2}&\cdots&0&\frac{\partial h_{J_{2}}}{\partial{\tt
x}_{1}}(x)&\frac{\partial h_{J_{2}}}{\partial{\tt x}_{2}}(x)\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\ 0&0&\cdots&2t_{s}&\frac{\partial
h_{J_{s}}}{\partial{\tt x}_{1}}(x)&\frac{\partial h_{J_{s}}}{\partial{\tt
x}_{2}}(x)\end{pmatrix},$

which has rank $\leq s$. As $(x,t)\in D_{s}({\mathcal{P}}_{n})$, then
$x\in{\mathcal{P}}_{n}$ and there exists at most two indices $k,\ell$ such
that $x\in\\{h_{J_{k}}=0,h_{J_{\ell}}=0\\}$. If such is the case, the vectors
$(\frac{\partial h_{J_{k}}}{\partial{\tt x}_{1}}(x),\frac{\partial
h_{J_{k}}}{\partial{\tt x}_{2}}(x))$ and $(\frac{\partial
h_{J_{\ell}}}{\partial{\tt x}_{1}}(x),\frac{\partial
h_{J_{\ell}}}{\partial{\tt x}_{2}}(x))$ are linearly independent. If there
exists only one index $k$ such that $x\in\\{h_{J_{k}}=0\\}$, the vector
$(\frac{\partial h_{J_{k}}}{\partial{\tt x}_{1}}(x),\frac{\partial
h_{J_{k}}}{\partial{\tt x}_{2}}(x))$ is non-zero because the partition
${\mathcal{J}}$ satisfies (6.1). As $t_{k}^{2}-h_{J_{k}}(x)=0$ for
$k=1,\ldots,s$, the matrix $J_{s,n}(x,t)$ has rank $s$, so
$(x,t)\in\operatorname{Reg}(X_{s,n})$, as claimed.

As $D_{s}({\mathcal{P}}_{n})$ is obtained recursively by doubling orientable
Nash manifolds with boundary, $D_{s}({\mathcal{P}}_{n})$ is an orientable Nash
surface. By the (smooth) classification of surfaces (see for instance [H,
Ch.9]) $D_{s}({\mathcal{P}}_{n})$ is diffeomorphic to a connected sum of $g$
tori and the genus $g$ completely characterize its diffeomorphism class.

For each
$\varepsilon:=(\varepsilon_{1},\ldots,\varepsilon_{s})\in\\{-1,1\\}^{s}$ the
set $D_{s}({\mathcal{P}}_{n})\cap\\{\varepsilon_{1}t_{1}\geq
0,\ldots,\varepsilon_{s}t_{s}\geq 0\\}$ is Nash diffeomorphic to
${\mathcal{P}}_{n}$. Thus, $D_{s}({\mathcal{P}}_{n})$ is obtained
(topologically) by glueing $2^{s}$ copies of ${\mathcal{P}}_{n}$, one for each
choice of $\varepsilon\in\\{-1,1\\}^{s}$. The polygon ${\mathcal{P}}_{n}$ has
a (natural) structure of CW complex with $n$ vertices, $n$ edges and 1 face.
This CW complex structure induces a CW complex structure on
$D_{s}({\mathcal{P}}_{n})$ with $2^{s-2}n$ vertices (because each vertex
belongs exactly to 4 polygons of the CW complex), $2^{s-1}n$ edges (because
each edge belongs exactly to 2 polygons of the CW complex) and $2^{s}$ faces.
The Euler characteristic of $D_{s}({\mathcal{P}}_{n})$ is
$2^{s-2}n-2^{s-1}n+2^{s}=2^{s-2}(4-n)$ and its genus is
$g=\frac{2-2^{s-1}(4-n)}{2}=2^{s-3}(n-4)+1$, that is,
$D_{s}({\mathcal{P}}_{n})$ is diffeomorphic to the connected sum of
$g=2^{s-3}(n-4)+1$ tori, whenever the number $2^{s-3}(n-4)+1$ is a non-
negative integer. In particular, the compact orientable surface of genus
$g\geq 1$ is diffeomorphic to the Nash surfaces
$D_{2}({\mathcal{P}}_{2g+2})\subset{\mathbb{R}}^{4}$ and
$D_{3}({\mathcal{P}}_{g+3})\subset{\mathbb{R}}^{5}$. Note that the genus $g$
of the surface $D_{s}({\mathcal{P}}_{n})$ depends both on $n$ and $s$ (see
Table 1).

| $s=2$ | $s=3$ | $s=4$ | $s=5$ | $s=6$ | $s=7$
---|---|---|---|---|---|---
$n=3$ | – | 0 | – | – | – | –
$n=4$ | 1 | 1 | 1 | – | – | –
$n=5$ | – | 2 | 3 | 5 | – | –
$n=6$ | 2 | 3 | 5 | 9 | 17 | –
$n=7$ | – | 4 | 7 | 13 | 25 | 49
Table 1. The genus of the Nash surface $D_{s}({\mathcal{P}}_{n})$ for $n\leq
7$.

By Lemma 2.8 there exist irreducible non-singular real algebraic sets $X$ with
at most two connected components that are both Nash diffeomorphic to either
$D_{2}({\mathcal{P}}_{2g+2})\subset{\mathbb{R}}^{4}$ or
$D_{3}({\mathcal{P}}_{g+3})\subset{\mathbb{R}}^{5}$. Consequently, our
procedure provides algebraic sets with at most two connected components
(mutually diffeomorphic) such that their connected components are models for
the orientable compact smooth surfaces.

###  6.4. Nash approximation.

We make use of Theorem 5.13 to prove Theorem 1.12. Let
${\mathcal{Q}}\subset{\mathbb{R}}^{n}$ be a Nash manifold with corners, let
$M\subset{\mathbb{R}}^{n}$ be a Nash envelope of ${\mathcal{Q}}$ and $Y\subset
M$ a Nash normal-crossings divisor such that ${\mathcal{Q}}\cap
Y=\partial{\mathcal{Q}}$. Let $h_{1},\ldots,h_{\ell}\in\mathcal{N}(M)$ be Nash
equations for the irreducible components $Y_{1},\ldots,Y_{\ell}$ of $Y$, that
is, $Y_{i}:=\\{h_{i}=0\\}$ and $d_{x}h_{i}:T_{x}M\to{\mathbb{R}}$ is
surjective for each $x\in Y_{i}$. We have ${\mathcal{Q}}=\\{h_{1}\geq
0,\ldots,h_{\ell}\geq 0\\}$ and consider the Nash double of ${\mathcal{Q}}$,
that is, the Nash manifold $D({\mathcal{Q}}):=\\{(x,t)\in
M\times{\mathbb{R}}^{\ell}:\
t_{1}^{2}-h_{1}(x)=0,\ldots,t_{\ell}^{2}-h_{\ell}(x)=0\\}$ together with the
projection $\pi:D({\mathcal{Q}})\to M,(x,t)\mapsto x$. It holds
$\pi(D({\mathcal{Q}}))={\mathcal{Q}}$ and if we define
$\pi_{+}^{-1}:{\mathcal{Q}}\to D({\mathcal{Q}}),\
x\mapsto(x,\sqrt{h_{1}(x)},\ldots,\sqrt{h_{\ell}(x)})$, then $\pi_{+}^{-1}$ is
a semialgebraic homeomorphism onto its image
${\mathcal{Q}}^{+}:=D({\mathcal{Q}})\cap\\{t_{1}\geq 0,\ldots,t_{\ell}\geq
0\\}$ such that
$\pi_{+}^{-1}\circ\pi|_{{\mathcal{Q}}^{+}}=\operatorname{id}_{{\mathcal{Q}}^{+}}$
and $\pi\circ\pi_{+}^{-1}=\operatorname{id}_{{\mathcal{Q}}}$.

###### Proof of Theorem 1.12.

By [DK] there exist an open semialgebraic neighborhood
$U\subset{\mathbb{R}}^{m}$ of ${\mathcal{S}}$ (in which ${\mathcal{S}}$ is
closed) and a semialgebraic retraction $\nu:U\to{\mathcal{S}}$. Consider the
continuous semialgebraic function $H:=\pi_{+}^{-1}\circ h\circ\nu:U\to
D({\mathcal{Q}})$. Let $\varepsilon_{0}:{\mathcal{S}}\to{\mathbb{R}}$ be a
strictly positive continuous semialgebraic function and let
$\varepsilon:U\to{\mathbb{R}}$ be a strictly positive continuous semialgebraic
extension of $\varepsilon_{0}$ to $U$. We consider the following commutative
diagram:

$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H}$$\scriptstyle{\nu}$$\textstyle{{\mathcal{Q}}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D({\mathcal{Q}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{{\mathcal{S}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{{\mathcal{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{+}^{-1}}$

As the map
$\pi_{*}:\mathcal{S}^{0}(U,D({\mathcal{Q}}))\to\mathcal{S}^{0}(U,{\mathcal{Q}}),H\mapsto\pi\circ
H$ is continuous by [Sh, Rmk.II.1.5], there exists $\delta:U\to{\mathbb{R}}$
such that if $G\in\mathcal{S}^{0}(U,D({\mathcal{Q}}))$ and $\|H-G\|<\delta$,
then $\|\pi\circ H-\pi\circ G\|<\varepsilon$. Let $G:U\to D({\mathcal{Q}})$ be
a Nash map such that $\|H-G\|<\delta$, so $\|\pi\circ H-\pi\circ
G\|<\varepsilon$. As $H|_{{\mathcal{S}}}=\pi_{+}^{-1}\circ h$, we deduce

$\|h-\pi\circ G|_{{\mathcal{S}}}\|=\|\operatorname{id}_{{\mathcal{Q}}}\circ
h-\pi\circ G|_{{\mathcal{S}}}\|=\|\pi\circ\pi_{+}^{-1}\circ h-\pi\circ
G|_{{\mathcal{S}}}\|=\|\pi\circ H|_{{\mathcal{S}}}-\pi\circ
G|_{{\mathcal{S}}}\|<\varepsilon|_{{\mathcal{S}}}=\varepsilon_{0},$

as required. ∎

## References

  * [A1] S.S. Abhyankar: Resolution of singularities of embedded algebraic surfaces. Pure and Applied Mathematics, 24 Academic Press, New York-London (1966).
  * [A2] S.S. Abhyankar: Weighted expansions for canonical desingularization. Lect. Notes Math.910, Berlin Heidelberg New York: Springer (1982).
  * [A3] S.S. Abhyankar: Good points of a hypersurface. Adv. in Math. 68 (1988), no. 2, 87–256.
  * [AK] S. Akbulut, H. King: A relative Nash theorem. Trans. Amer. Math. Soc. 267 (1981), no. 2, 465–481.
  * [BFR] E. Baro, J.F. Fernando, J.M. Ruiz: Approximation on Nash sets with monomial singularities. Adv. Math. 262 (2014), 59–114.
  * [BM1] E. Bierstone, P.D. Milman: Uniformization of analytic spaces. J. Amer. Math. Soc. 2 (1989), no. 4, 801–836.
  * [BM3] E. Bierstone, P.D. Milman: Canonical desingularization in characteristic zero by blow up the maximum strata of a local invariant. Invent. Math. 128 (1997), no. 2, 207–302.
  * [BM2] E. Bierstone, P.D. Milman: Functoriality in resolution of singularities.Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 609–639.
  * [BP] E. Bierstone, A. Parusiński: Global smoothing of a subanalytic set.Duke Math. J. 167 (2018), no.16, 3115–3128.
  * [BCR] J. Bochnak, M. Coste, M.-F. Roy: Real algebraic geometry. Ergeb. Math.  36, Springer-Verlag, Berlin (1998).
  * [CF] A. Carbone, J.F. Fernando: Surjective Nash maps between semialgebraic sets. Preprint (2023). arXiv:2306.00401
  * [C] H. Clemens: Curves in generic hypersurfaces. Ann. Sci. Ecole Norm. Sup. 19 (1986), 629–636.
  * [DK] H. Delfs, M. Knebusch: Separation, retractions and homotopy extension in semialgebraic spaces. Pacific J. Math. 114 (1984), no. 1, 47–71.
  * [Ef] G.A. Efroymson: The Extension Theorem for Nash Functions. Real Algebraic Geometry and Quadratic Forms (Rennes, 1981). Lecture Notes in Mathematics, vol. 959, Springer, Berlin (1982), 343–357.
  * [E] D. Eisenbud: Open Problems in Computational Algebraic Geometry and commutative Algebra, in Computational Algebraic Geometry and Commutative Algebra, Cortona (1991), (ed. D. Eisenbud and L. Robbiano) Cambridge University Press, Cambridge (1993), 49–71.
  * [Fe1] J.F. Fernando: On the substitution theorem for rings of semialgebraic functions. J. Inst. Math. Jussieu 14 (2015), no. 4, 857–894.
  * [Fe2] J.F. Fernando: On Nash images of Euclidean spaces. Adv. Math. 331 (2018), 627–719.
  * [Fe3] J.F. Fernando: On a Nash curve selection lemma through finitely many points. Preprint RAAG (2023).
  * [FFQU] J.F. Fernando, G. Fichou, R. Quarez, C. Ueno: On regulous and regular images of Euclidean spaces, Q. J. Math. 69 (2018), no. 4, 1327–1351.
  * [FG] J.F. Fernando, J.M. Gamboa: On open and closed morphisms between semialgebraic sets. Proc. Amer. Math. Soc. 140 (2012), no. 4, 1207–1219.
  * [FGR] J.F. Fernando, J.M. Gamboa, J.M. Ruiz: Finiteness problems on Nash manifolds and Nash sets. J. Eur. Math. Soc. (JEMS) 16 (2014), no. 3, 537–570.
  * [FGU] J.F. Fernando, J.M. Gamboa, C. Ueno: Polynomial, regular and Nash images of Euclidean spaces. Ordered algebraic structures and related topics, Contemp. Math., 697, Amer. Math. Soc., Providence, RI (2017), 145–167.
  * [FGh1] J.F. Fernando, R. Ghiloni: Differentiable approximation of continuous semialgebraic maps. Selecta Math. (N.S.) 25 (2019), no. 3, Paper No. 46, 30 pp.
  * [FGh2] J.F. Fernando, R. Ghiloni: Smooth approximations in PL geometry. Amer. J. Math. 144 (2022), no. 4, 967–1007.
  * [FU] J.F. Fernando, C. Ueno: On polynomial images of a closed ball. J. Math. Soc. Japan 75 (2023), no. 2, 679–733.
  * [G] J.M. Gamboa: Algebraic images of the real plane. Reelle algebraische Geometrie, June, $10^{\rm th}-16^{\rm th}$ (1990), Oberwolfach.
  * [GS] R. Ghiloni, E. Savi: The topology of real algebraic sets with isolated singularities is determined by the field of rational numbers. Preprint (2023). arXiv:2302.04142
  * [Ha] J. Harris: Algebraic geometry. A first course. Corrected reprint of the 1992 original. Graduate Texts in Mathematics, 133. Springer-Verlag, New York, (1995).
  * [H] M.W. Hirsch: Differential topology. Graduate Texts in Mathematics, 33. Springer–Verlag, New York–Heidelberg–Berlin (1976).
  * [Hi1] H. Hironaka: Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964) 205–326.
  * [Hi2] H. Hironaka: Stratification and flatness. Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976). Sijthoff and Noordhoff, Alphen aan den Rijn (1977), 199–265.
  * [HPV] J. Hubbard, P. Papadopol, V. Veselov: A compactification of Hénon mappings in ${\mathbb{C}}^{2}$ as dynamical systems. Acta Math. 184 (2000), no. 2, 203–270.
  * [Ko] J. Kollár: Lectures on resolution of singularities. Annals of Mathematics Studies, 166. Princeton University Press, Princeton, NJ, (2007).
  * [L] J. Lipman: Introduction to resolution of singularities. Algebraic geometry (Proc. Sympos. Pure Math., 29, Humboldt State Univ., Arcata, Calif., 1974), pp. 187–230. Amer. Math. Soc., Providence, R.I., (1975).
  * [M] T. Mostowski: Some properties of the ring of Nash functions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 2, 245–266.
  * [ORR] E. Outerelo, J. A. Rojo, and J. M. Ruiz, Topología diferencial: un curso de iniciación, Sanz y Torres, S.L., Madrid, (2020).
  * [Sh] M. Shiota: Nash manifolds. Lecture Notes in Mathematics, 1269. Springer-Verlag, Berlin (1987).
  * [S] F.P. Di Stefano: Oriented blow-up. Ann. Univ. Ferrara Sez. VII (N.S.) 24 (1979), 25–35.
  * [St] J. Stasica: Smooth points of a semialgebraic set. Ann. Pol. Math. 82 (2003), no.2, 149–153.
  * [Vi] O. Villamayor: Constructiveness of Hironaka’s resolution. Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 1–32.
  * [V] C. Voisin: On a conjecture of Clemens on rational curves on hypersurfaces. J. Differential Geom. 44 (1996), 200–214.
  * [Wh] H. Whitney: Differentiable manifolds. Ann. of Math. (2) 37 (1936), no. 3, 645–680.
  * [Z1] O. Zariski: Algebraic varieties over ground fields of characteristic zero. Amer. J. Math. 62 (1940), 187–221.
  * [Z2] O. Zariski: A simplified proof for the resolution of singularities of an algebraic surface. Ann. of Math. (2) 43 (1942), 583–593.
  * [Z3] O. Zariski: Reduction of the singularities of algebraic three dimensional varieties. Ann. of Math. (2) 45 (1944), 472–542.