# Integral curves and Jacobian Conjecture

Jean-Yves Charbonnel  Jean-Yves Charbonnel, Université Paris Cité - CNRS
Institut de Mathématiques de Jussieu - Paris Rive Gauche
UMR 7586
Groupes, représentations et géométrie
Bâtiment Sophie Germain
Case 7012
75205 Paris Cedex 13, France<EMAIL_ADDRESS>
###### Abstract.

In this note, we are interested in the Jacobian Conjecture. For that purpose,
we consider some vector fields depending on an étale polynomial map. From
results of semialgebraic geometry with the consideration of some integral
curves of these vector fields, we deduce that an étale polynomial endomorphism
is a birational morphism. So, by a well known result, an étale polynomial
endomorphism is a polynomial automorphism.

###### Key words and phrases:

Étale morphism, integral curves, semialgebraic sets and maps, connected
components

###### 1991 Mathematics Subject Classification:

14A10, 14E05, 34A05

###### Contents

  1. 1 Introduction
  2. 2 Semialgebraic sets and maps
  3. 3 Vector fields and integral curves
  4. 4 Special integral curves
  5. 5 Polynomial automorphism
  6. 6 Conclusion

##  1\. Introduction

In this note, we are interested in the jacobian conjecture, that is: for $n$
positive integer, an étale polynomial endomorphism of ${\mathbb{C}}^{n}$ is a
polynomial automorphism. Some results are known for a long time. For example,
an étale polynomial endomorphism is an automorphism if it is proper or
birational [Bass 82, Theorem 2.1]. According to the results of L.M.
Dru$\dot{\rm z}$kowski [Dru83, Theorem 3], it is sufficient to prove that an
odd étale polynomial endomorphism is a polynomial automorphism.

###  1.1. Sketch of proof

Let $n$ be a positive integer and $\Phi$ an odd étale polynomial endomorphism
of ${\mathbb{C}}^{n}$. For a well defined Zariski open subset $U$ of
${\mathbb{C}}^{n}$, the restriction of $\Phi$ to $\Phi^{-1}(U)$ is an étale
covering over $U$. For $x$ in ${\mathbb{C}}^{n}$, let $X_{x}$ be the vector
field on ${\mathbb{C}}^{n}$,

$y\longmapsto\Phi^{\prime}(y)^{-1}(x)$

with $\Phi^{\prime}(y)$ the differential of $\Phi$ at $y$. If $\gamma$ is an
integral curve of $X_{x}$ with initial datum $\gamma(0)=y$, then for all $t$
in the interval of definition of $\gamma$,

$\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(t)=\Phi(y)+tx.$

Denote by $\gamma_{x}$ the maximal integral curve of $X_{x}$ such that
$\gamma_{x}(0)=0$. The first key step of the proof is the following
proposition:

######  Proposition 1.1.

Let $\Gamma_{{\mathbb{R}}}$ be the subset of elements $(x,t,y)$ of
${\mathbb{C}}^{n}\times{\mathbb{R}}\times{\mathbb{C}}^{n}$ such that
$\Phi(y)=tx$ and $G$ the graph of the map $(x,t)\mapsto\gamma_{x}(t)$.

(i) The closure $\overline{G}$ of $G$ in
${\mathbb{C}}^{n}\times{\mathbb{R}}\times{\mathbb{C}}^{n}$ is the connected
component $\Gamma_{0}$ of $\Gamma_{{\mathbb{R}}}$, containing
${\mathbb{C}}^{n}\times\\{0\\}\times{\mathbb{C}}^{n}$.

(ii) Let $\widetilde{\Omega}$ be the image of $\Gamma_{0}$ by the projection
$(x,t,y)\mapsto(x,t)$. Then $\widetilde{\Omega}$ is an open subset of
${\mathbb{C}}^{n}\times{\mathbb{R}}$ and $\Gamma_{0}$ is the graph of a
continuous semialgebraic map $\widetilde{\gamma}$ from $\widetilde{\Omega}$ to
${\mathbb{C}}^{n}$.

(iii) For $(x,t)$ in ${\mathbb{C}}^{n}\times{\mathbb{R}}$, if $tx$ is in $U$
then $(x,t)$ is in $\widetilde{\Omega}$.

Denote by $\Omega_{*}$ the subset of elements $(x,t)$ of
${\mathbb{C}}^{n}\times{\mathbb{C}}$ such that $tx$ is in $U$ and $\gamma_{*}$
the map from $\Omega_{*}$ to ${\mathbb{C}}^{n}$ such that
$\gamma_{*}(x,t)=\widetilde{\gamma}(tx,1)$. The second key step of the proof
is the following proposition:

######  Proposition 1.2.

Let $\Gamma$ be the subset of elements $(x,t,y)$ of
${\mathbb{C}}^{n}\times{\mathbb{C}}\times{\mathbb{C}}^{n}$ such that
$\Phi(y)=tx$ and $G_{*}$ the graph of $\gamma_{*}$.

(i) The closure $\overline{G_{*}}$ of $G_{*}$ in
${\mathbb{C}}^{n}\times{\mathbb{C}}\times{\mathbb{C}}^{n}$ is the irreducible
component $\Gamma_{*}$ of $\Gamma$, containing
${\mathbb{C}}^{n}\times\\{0\\}\times{\mathbb{C}}^{n}$.

(ii) Let $\widetilde{\Omega_{*}}$ be the image of $\Gamma_{*}$ by the
projection $(x,t,y)\mapsto(x,t)$. Then $\widetilde{\Omega}$ is a Zariski open
subset of ${\mathbb{C}}^{n}\times{\mathbb{C}}$ and $\Gamma_{*}$ is the graph
of a regular map $\widetilde{\gamma_{*}}$ from $\widetilde{\Omega_{*}}$ to
${\mathbb{C}}^{n}$.

Setting $\varphi(x)=\widetilde{\gamma_{*}}(x,1)$ for $x$ in $U$, the map
$\varphi$ so defined is a regular map from $U$ to ${\mathbb{C}}^{n}$ such that
$\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\varphi(x)=x$ for all $x$ in $U$.
As a result, $\varphi\raisebox{0.85358pt}{\scriptsize$\circ$}\Phi(y)=y$ for
all $y$ in $\Phi^{-1}(U)$. Then $\Phi$ is a birational map, whence a
polynomial automorphism.

###  1.2. Notations and Definition

$\bullet$ For a set $E$, $|E|$ is its cardinality when it is finite and
$+\infty$ when it is not finite.

$\bullet$ The topology on ${\mathbb{C}}^{n}$ is the usual topology and $\|.\|$
is an euclidean norm on ${\mathbb{C}}^{n}$.

$\bullet$ For $n$ positive integer, ${\mathrm{H}}_{n}$ is the group of
homotheties of ${\mathbb{C}}^{n}$ with ratios in ${\mathbb{R}}^{*}_{+}$.

$\bullet$ For $\Phi$ a holomorphic endomorphism of ${\mathbb{C}}^{n}$ and $x$
in ${\mathbb{C}}^{n}$, its differential at $x$ is denoted by
$\Phi^{\prime}(x)$.

$\bullet$ In this note, $\Phi$ is an odd étale polynomial endomorphism. In
particular, $\Phi(0)=0$. For all $x$ in ${\mathbb{C}}^{n}$, there exists an
open neighborhood $\omega$ of $x$ in ${\mathbb{C}}^{n}$ such that the
restiction of $\Phi$ to $\omega$ is a holomorhic isomorphism onto the open
neighborhood $\Phi(\omega)$ of $\Phi(x)$ in ${\mathbb{C}}^{n}$. Denote by
$\tau$ the inverse of the restriction of $\Phi$ to $\omega$ and call the pair
$(\omega,\tau)$ an adapted pair to $x$.

##  2\. Semialgebraic sets and maps

We recall some results of semialgebraic geometry (see [Cos02]). Let $E$ and
$F$ be finite dimensional real vector spaces. A subset of $E$ is semialgebraic
if it is a finite union of subsets defined by a system of a finite number of
equalities and inequalities of polynomials. Denoting by ${\mathfrak{S}}(E)$
the set of semialgebraic subsets of $E$, ${\mathfrak{S}}(E)$ is stable under
finite union and finite intersection. For $S$ in ${\mathfrak{S}}(E)$, its
closure in $E$, its interior and $E\setminus S$ are in ${\mathfrak{S}}(E)$.

For $X$ subset of $E$ and $\varphi$ a map from $X$ to $F$, $\varphi$ is a
semialgebraic map if its graph is a semialgebraic subset of $E\times F$. By
Tarski-Seidenberg’s theorem, $X$ is a semialgebraic subset of $E$ and the
image of $\varphi$ is a semialgebraic subset of $F$.

######  Lemma 2.1.

Let $X$ be a subset of $E$, $Y$ a subset of $F$ and $G$ a real vector space of
finite dimension. If $\varphi$ is a semialgebraic map from $X$ to $Y$ and
$\psi$ a semialgebraic map from $Y$ to $G$ then
$\psi\raisebox{0.85358pt}{\scriptsize$\circ$}\varphi$ is a semialgebraic map
fom $X$ to $G$.

###### Proof.

Let $\Gamma_{\varphi}$ be the graph of $\varphi$ and $\Gamma_{\psi}$ the graph
of $\psi$. Denote by $\Gamma$ the intersection of $\Gamma_{\varphi}\times G$
and $E\times\Gamma_{\psi}$. Then the graph of
$\psi\raisebox{0.85358pt}{\scriptsize$\circ$}\varphi$ is the image of $\Gamma$
by the projection

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
31.85077pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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55.85077pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E\times
G}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(x,y,z)\longmapsto(x,z).$

By hypothesis, $\Gamma$ is a semialgebraic subset of $E\times F\times G$ and
by Tarski-Seidenberg’s theorem,
$\psi\raisebox{0.85358pt}{\scriptsize$\circ$}\varphi$ is a semialgebraic map.
∎

###  2.1. Some results on semialgebraic sets

According to [Cos02, Secion 2], a semialgebraic subset $S$ of
${\mathbb{R}}^{n}$ has a decomposition into a finite disjoint union of
semialgebraic subsets semialgebraically homeomorphic to open hypercubes
$]0,1[^{d}$ (by convention $]0,1[^{0}$ is a point). By [Cos02, Proposition
3.15], the dimension of $S$ is the maximum of these $d$ and is independent of
the decomposition. Moreover, by [Cos02, Theorem 2.23], $S$ has a finite number
of connected components and its connected components are semialgebraic.

######  Lemma 2.2.

Let $S$ be in ${\mathfrak{S}}({\mathbb{R}}^{n})$ and $\overline{S}$ its
closure in ${\mathbb{R}}^{n}$.

(i) The sets $S$ and $\overline{S}$ have the same dimension.

(ii) The dimension of $\overline{S}\setminus S$ is smaller than ${\rm
dim}\hskip 1.5ptS$. Moreover, $S$ contains a dense open subset of
$\overline{S}$.

(iii) The set $S$ has an empty interior if and only if ${\rm dim}\hskip
1.5ptS\leq n-1$.

(iv) If $S$ is a Zariski closed subset of ${\mathbb{R}}^{n}$, then ${\rm
dim}\hskip 1.5ptS$ is the dimension of $S$ as an algebraic subset.

(v) For $m$ positive integer and $\varphi$ semialgebraic map from $S$ to
${\mathbb{R}}^{m}$, ${\rm dim}\hskip 1.5pt\varphi(S)\leq{\rm dim}\hskip
1.5ptS$.

###### Proof.

Assertions (i) and (ii) are given by [Cos02, Proposition 3.16], Assertion (iv)
is given by [Cos02, Theorem 3.20] and Assertion (v) is given by [Cos02,
Theorem 3.18].

(iii) By definition, a nonempty semialgebraic open subset of
${\mathbb{R}}^{n}$ has dimension $n$. For $T$ semialgebraic subset $T$ of $S$,
${\rm dim}\hskip 1.5ptT\leq{\rm dim}\hskip 1.5ptS$ by [Cos02, Corollary 3.8].
Then $S$ has an empty interior if ${\rm dim}\hskip 1.5ptS\leq n-1$. Conversely
if ${\rm dim}\hskip 1.5ptS=n$, it contains a semialgebraic subset homeomorphic
to $]0,1[^{n}$. Hence $S$ has a nonempty interior. ∎

######  Proposition 2.3.

Let $X$ and $Y$ be semialgebraic subsets of ${\mathbb{R}}^{n}$ such that $0$
is in $Y$ and $Y$ has an empty interior in $X$. Then there exists a continuous
semialgebraic arc $\theta$,

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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{R}}^{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad
t\longmapsto\theta(t)$

such that $\theta(0)=0$ and $\theta(t)\in X\setminus Y$ for all $t$ in
$]0,1]$.

###### Proof.

As $Y$ has an empty interior in $X$, $X\setminus Y$ contains a dense
semialgebraic open subset of $X$, not containing $0$, whence the proposition
by the Curve selection lemma [Cos02, Theorem 3.13]. ∎

###  2.2. On semialgebraic maps

Let $X$ be in ${\mathfrak{S}}({\mathbb{R}}^{n})$ and $Y$ in
${\mathfrak{S}}({\mathbb{R}}^{m})$. Denote by $\psi$ a semialgebraic map from
$X$ to $Y$.

######  Proposition 2.4.

For $i$ nonnegative integer, denote by $Y_{i}$ the subset of elements $y$ of
$Y$ such that $|\psi^{-1}(\\{y\\})|=i$.

(i) For $i$ nonnegative integer, $Y_{i}$ is semialgebraic.

(ii) If the fibers of $\psi$ are finite, for some positive integer $j$,
$Y_{j}$ is nonempty and $Y_{i}$ is empty for all $i$ bigger than $j$.

(iii) If $\psi$ is a local homeomorphism then the fibers of $\psi$ are finite
and $Y_{j}$ is a nonempty open subset of $Y$.

###### Proof.

(i) Let $Y^{\prime}_{i}$ be the subset of elements $y$ of $Y$ such that
$|\psi^{-1}(y)|\geq i$. Denote by $Z_{i}$ the subset of elements
$(y,{x}_{1}{,\ldots,\hskip 1.79993pt}{x}_{i})$ of $Y\times X^{i}$ such that
$y={\psi}(x_{1}){=\cdots=\hskip 1.79993pt}{\psi}(x_{i})$. Then $Z_{i}$ is a
semialgebraic subset of $Y\times X^{i}$ and so is the subset $Z^{\prime}_{i}$
of elements $(y,{x}_{1}{,\ldots,\hskip 1.79993pt}{x}_{i})$ of $Z_{i}$ such
that $x_{j}\neq x_{k}$ for $j\neq k$. Then $Y^{\prime}_{i}$ is a semialgebraic
subset of $Y$ as the image of $Z^{\prime}_{i}$ by the projection

$(y,{x}_{1}{,\ldots,\hskip 1.79993pt}{x}_{i})\longmapsto x.$

By definition, $Y_{i}=Y^{\prime}_{i}\setminus Y^{\prime}_{i+1}$. Hence $Y_{i}$
is semialgebraic.

(ii) Let $G$ be the graph of $\psi$ and $\varphi$ the map

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3.0pt}}}}}}\ignorespaces{\hbox{\kern 61.71748pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Y}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(x,y)\longmapsto
y.$

Then $\varphi$ is continuous as the restriction of a continuous map from
${\mathbb{R}}^{n}\times{\mathbb{R}}^{m}$ to ${\mathbb{R}}^{m}$. Moreover
$\varphi$ is semialgebraic and it has the same fibers as $\psi$. Then, by
Hardt’s semialgebraic triviality theorem [Cos02, Theorem 4.1], for some
positive integer $j$, $Y_{i}$ is empty for $i>j$, whence the assertion.

(iii) Let $y$ be in $Y$. Then $\psi^{-1}(\\{y\\})$ is semialgebraic of
dimension $0$ since $\psi$ is a local homeomorphism. As a result, the fibers
of $\psi$ are finite. Suppose now that $y$ is in $Y_{j}$. Let
${x}_{1}{,\ldots,\hskip 1.79993pt}{x}_{j}$ be the elements of
$\psi^{-1}(\\{y\\})$. By hypothesis, for $i=1,\ldots,m$, there exists a
neighborhood $\omega_{i}$ of $x_{i}$ such that the restriction of $\psi$ to
$\omega_{i}$ is a homeomorphism onto the neighborhood $\psi(\omega_{i})$ of
$y$ in $Y$. As ${x}_{1}{,\ldots,\hskip 1.79993pt}{x}_{j}$ are pairwise
different, we can choose ${\omega}_{1}{,\ldots,\hskip 1.79993pt}{\omega}_{j}$
pairwise disjoint. Denoting by $\omega$ the intersection of
${\psi}(\omega_{1}){,\ldots,\hskip 1.79993pt}{\psi}(\omega_{j})$, $\omega$ is
a neighborhood of $y$ in $Y$, contained in $Y_{j}$ by maximality of $j$,
whence the assertion. ∎

###  2.3. On semialgebraic curves

In this subsection we give some results on continuous semialgebraic curves.

######  Lemma 2.5.

Let $\gamma$ be a continuous semialgebraic function from the interval $]a,b]$
to ${\mathbb{R}}$.

(i) Suppose that $\gamma$ has a boundary value when $t$ goes to $a$. Then
$\gamma$ has a continuous extension to $[a,b]$.

(ii) If $\gamma$ is not bounded, then for some positive rational number $r$
and a continuous semialgebraic function $\gamma_{0}$ on $[a,b]$ such that
$\gamma_{0}(a)\neq 0$, $\gamma(t)=(t-a)^{-r}\gamma_{0}(t)$ for all $t$ in
$]a,b]$.

###### Proof.

(i) Let $\Gamma$ be the graph of the function $\gamma$ and $\overline{\Gamma}$
its closure in $[a,b]\times{\mathbb{R}}$. By hypothesis,
$\overline{\Gamma}\setminus\Gamma$ is not empty and contained in
$\\{a\\}\times{\mathbb{R}}$ since $\gamma$ is continuous. Let $x$ be in
${\mathbb{R}}$ such that $(a,x)$ is in $\overline{\Gamma}$ and $\varepsilon$ a
positive real number. The intersection of $\overline{\Gamma}$ and
$[a,b]\times]x-\varepsilon,x+\varepsilon[$ is a semialgebraic subset of
${\mathbb{R}}^{2}$. Then it has a finite number of connected components by
[Cos02, Theorem 2.23]. Denote them by ${C}_{1}{,\ldots,\hskip
1.79993pt}{C}_{m}$. Since $(a,x)$ is in $\overline{\Gamma}$, for some positive
number $\eta$ and for some $i$ in $\\{1,\ldots,m\\}$, $]a,a+\eta]$ is
contained in the image of $C_{i}$ by the first projection since a
semialgebraic subset of ${\mathbb{R}}$ is a finite union of intervals and
isolated points. As a result, for all $t$ in $]a,a+\eta[$,
$|\gamma(t)-x|<\varepsilon$. Hence the extension $\overline{\gamma}$ of
$\gamma$ to $[a,b]$ such that $\overline{\gamma}(a)=x$ is continuous.

(ii) Suppose that $\gamma$ is not bounded. By (i),

$\lim_{t\rightarrow a}|\gamma(t)|=+\infty.$

Moreover, the function $t\mapsto|\gamma(t)|$ is a semialgebraic function. Let
${P}_{1}{,\ldots,\hskip 1.79993pt}{P}_{m}$ be the polynomials corresponding to
the equalities of a system defining the graph of the function $|\gamma|$.
Denote by $Q$ the polynomial

$Q(t,s):=\prod_{i=1}^{m}P_{i}(t+a,s).$

Then, for some prime divisor $Q_{*}$ of $Q$, $Q_{*}(t,|\gamma(t+a)|)=0$ for
all $t$, different from $0$, in a neighborhood of $0$ in $]0,b-a]$. So, by a
Puiseux’s expansion, for some positive rational number $r$ and positive real
number $c$,

$\lim_{t\rightarrow a}(t-a)^{r}|\gamma(t)|=c.$

Setting:

$\gamma_{0}(t)=(t-a)^{r}\gamma(t)$

for $t$ in $]a,b]$, the function $\gamma_{0}$ on $]a,b]$ is bounded,
continuous and semialgebraic. Then, by (i), $\gamma_{0}$ has a continuous
expansion to $[a,b]$ such that $\gamma_{0}(a)\neq 0$. ∎

######  Corollary 2.6.

Let $E$ be a finite dimensional real vector space and $\gamma$ a continuous
semialgebraic map from the interval $[a,b]$ to $E$. For $k$ nonnegative
integer, there exist an increasing sequence ${r}_{1}{<\cdots<\hskip
1.79993pt}{r}_{k+1}$ of positive rational numbers and ${x}_{1}{,\ldots,\hskip
1.79993pt}{x}_{k}$ in $E$ such that the map

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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{E}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad
t\longmapsto(t-a)^{-r_{k+1}}(\gamma(t)-(\gamma(a)+(t-a)^{r_{1}}x_{1}+\cdots+(t-a)^{r_{k}}x_{k}))$

is bounded, continuous and semialgebraic.

###### Proof.

Considering the coordinates of $\gamma$ in a basis of $E$, it is sufficient to
prove the corollary for $E={\mathbb{R}}$. The map
$t\mapsto\gamma(t)-\gamma(a)$ has a zero at $a$. Moreover, the set of zeros of
this map in $[a,b]$ is a closed semialgebraic subset of $[a,b]$. As a result,
it is a union of a finite number of closed intervals and isolated points.
Hence $\gamma(t)-\gamma(a)=0$ for all $t$ in a neighborhood of $a$ in $[a,b]$
or $\gamma(t)-\gamma(a)\neq 0$ for all $t$ different from $a$ in a
neighborhood of $a$ in $[a,b]$. In the first case, there is nothing to prove.
Suppose that the second case is satisfied. For some positive number $\eta$
such that $a+\eta\leq b$, the function

$t\longmapsto\frac{1}{\gamma(t)-\gamma(a)}$

on $]a,a+\eta]$ is a continuous semialgebraic function and it is not bounded.
Hence, by Lemma 2.5(ii), for some positive rational number $r_{1}$ and for
some continuous semialgebraic function $\gamma_{1}$ on $[a,b]$ such that
$\gamma_{1}(a)\neq 0$,

$\gamma(t)=\gamma(a)+(t-a)^{r_{1}}\gamma_{1}(t)$

for all $t$ in $[a,b]$, whence the corollary by induction on $k$. ∎

######  Remark 2.7.

By Corollary 2.6 and an induction argument, we can find an increasing sequence
${r}_{1}{<\hskip 1.79993pt}{r}_{2}<\cdots$ of positive rational numbers and a
sequence ${x}_{1}{,\hskip 1.79993pt}{x}_{2},\ldots$ in $E\setminus\\{0\\}$
satisfying one of the two following conditions:

  * (1)

if the sequences are finite of cardinality $k$, then the function

$t\longmapsto\gamma(t)-(\gamma(a)+(t-a)^{r_{1}}x_{1}+\cdots+(t-a)^{r_{k}}x_{k})$

is equal to $0$ in a neighborhood of $a$ in $[a,b]$,

  * (2)

if the first sequence is not finite so is the second one and for all positive
integer $k$, the map

$t\longmapsto(t-a)^{-r_{k+1}}(\gamma(t)-(\gamma(a)+(t-a)^{r_{1}}x_{1}+\cdots+(t-a)^{r_{k}}x_{k}))$

is bounded and semialgebraic.

In the second case, by [Duv89], as the rational numbers ${r}_{1}{,\hskip
1.79993pt}{r}_{2},\ldots$ are the exponents of a Puiseux’s serie, for some
positive integer $m$, ${mr}_{1}{,\hskip 1.79993pt}{mr}_{2},\ldots$ are
positive integers so that the sequence ${r}_{1}{,\hskip
1.79993pt}{r}_{2},\ldots$ is not bounded.

##  3\. Vector fields and integral curves

For $x$ in ${\mathbb{C}}^{n}$, denote by $X_{x}$ the polynomial vector field
on ${\mathbb{C}}^{n}$,

$y\longmapsto X_{x}(y):=\Phi^{\prime}(y)^{-1}(x).$

######  Definition 3.1.

An integral curve of the vector field $X_{x}$ is a derivable map $\gamma$ from
an open interval $I$ of ${\mathbb{R}}$ to ${\mathbb{C}}^{n}$ such that its
derative $\gamma^{\prime}$ satisfies the equality

$\gamma^{\prime}(t)=X_{x}(\gamma(t))$

for all $t$ in $I$.

###  3.1. Elementary properties

Let $x$ be in ${\mathbb{C}}^{n}$.

######  Lemma 3.2.

Denote by $\gamma$ a continuous map from an open interval $I$ of
${\mathbb{R}}$ to ${\mathbb{C}}^{n}$. Then $\gamma$ is an integral curve of
$X_{x}$ if and only if for some $t_{0}$ in $I$

$\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(t)=\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(t_{0})+(t-t_{0})x$

for all $t$ in $I$.

###### Proof.

Suppose that $\gamma$ is an integral curve of $X_{x}$. Then
$\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma$ is an integral curve of
the vector field $z\mapsto\Phi^{\prime}(z)(X_{x}(z))$. By definition,
$\Phi^{\prime}(z)(X_{x}(z))=x$ for all $z$ in ${\mathbb{C}}^{n}$. As a result,
by the unicity of the integral curves with a given initial datum,

$\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(t)=\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(t_{0})+(t-t_{0})x$

for all $(t,t_{0})$ in $I\times I$. Conversely, suppose that $\gamma$ is a
continuous map from $I$ to ${\mathbb{C}}^{n}$ such that for some $t_{0}$ in
$I$,

$\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(t)=\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(t_{0})+(t-t_{0})x\qquad\forall
t\in I.$

Let $s$ be in $I$ and $(\omega,\tau)$ an adapted pair to $\gamma(s)$. Then,
for all $t$ in $\gamma^{-1}(\omega)$,

$\gamma(t)=\tau(\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(t_{0})+(t-t_{0})x).$

As a result, $\gamma$ is a derivable map and
$\Phi^{\prime}(\gamma(t))(\gamma^{\prime}(t))=x$ for all $t$ in $I$, whence
the assertion by definition of $X_{x}$. ∎

Denote by ${\mathcal{C}}_{x}$ the set of integral curves of $X_{x}$ whose
interval of definition contains $0$. For $\gamma$ and $\delta$ in
${\mathcal{C}}_{x}$, we say that $\gamma$ is smaller than $\delta$ if the
interval of definition of $\gamma$ is contained in the interval of definition
$\delta$ and if $\delta$ is an expansion of $\gamma$. This order is inductive
so that any element of ${\mathcal{C}}_{x}$ is smaller than a maximal one.

###  3.2. On the integral curves

Denote by $\Psi$ the map

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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50.10028pt\raise 5.87pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.87pt\hbox{$\scriptstyle{\Psi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 83.36696pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
53.36696pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 83.36696pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}\times{\mathbb{C}}\times{\mathbb{C}}^{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(x,t,z)\longmapsto(x,t,\Phi(z)).$

As $\Phi$ is étale so is $\Psi$. Let $\Delta$ be the graph of the map

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72.35014pt\raise 0.0pt\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}}{\hbox{\kern
42.35014pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 72.35014pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(x,t)\longmapsto
tx.$

######  Lemma 3.3.

Denote by $\Gamma$ the inverse image of $\Delta$ by $\Psi$ and for $x$ in
${\mathbb{C}}^{n}$ denote by $\Gamma_{x}$ the intersection of $\Gamma$ with
$\\{x\\}\times{\mathbb{C}}\times{\mathbb{C}}^{n}$.

(i) The subset $\Gamma$ of
${\mathbb{C}}^{n}\times{\mathbb{C}}\times{\mathbb{C}}^{n}$ is a smooth
algebraic variety of dimension $n+1$, containing
${\mathbb{C}}^{n}\times\\{0,0\\}$.

(ii) For $x$ in ${\mathbb{C}}^{n}$, $\Gamma_{x}$ is a smooth algebraic variety
of dimension $1$, containing $(x,0,0)$.

###### Proof.

(i) As $\Psi(x,0,0)=(x,0,0)$, $(x,0,0)$ is in $\Gamma$ for all $x$ in
${\mathbb{C}}^{n}$. Let $(x,t,y)$ be in $\Gamma$. Since $\Psi$ is étale and
$\Delta$ is smooth of dimension $n+1$, the tangent space of $\Gamma$ at
$(x,t,y)$ has dimension at most $n+1$. Moreover, the image of some open
neighborhood of $(x,t,y)$ in
${\mathbb{C}}^{n}\times{\mathbb{C}}\times{\mathbb{C}}^{n}$ by $\Psi$ is an
open neighborhood of $(x,t,\Phi(y))$ in
${\mathbb{C}}^{n}\times{\mathbb{C}}\times{\mathbb{C}}^{n}$. As a result,
$\Gamma$ has dimension $n+1$. Hence $\Gamma$ is a smooth algebraic variety,
equidimensional of dimension $n+1$.

(ii) Let $x$ be in ${\mathbb{C}}^{n}$. By definition, $\Gamma_{x}$ is the
inverse image by $\Psi$ of the graph $\Delta_{x}$ of the map

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55.44601pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(x,t)\longmapsto
tx.$

As the restriction of $\Psi$ to
$\\{x\\}\times{\mathbb{C}}\times{\mathbb{C}}^{n}$ is an étale endomorphism of
$\\{x\\}\times{\mathbb{C}}\times{\mathbb{C}}^{n}$, $\Gamma_{x}$ is a smooth
algebraic variety of dimension $1$ since so is $\Delta_{x}$. Moreover,
$(x,0,0)$ is in $\Gamma_{x}$. ∎

######  Proposition 3.4.

Let $y$ be in ${\mathbb{C}}^{n}$ and $\gamma$ in ${\mathcal{C}}_{x}$ such that
$\gamma(0)=y$. Then $\gamma$ is a semialgebraic map from an interval of
${\mathbb{R}}$ to ${\mathbb{C}}^{n}$.

###### Proof.

Let $I$ be the interval of definition of $\gamma$. According to Lemma 3.2,

$\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(t)=\Phi(y)+tx$

for all $t$ in $I$. Denote by $\Delta_{\gamma}$ the graph of $\gamma$ and set:

$\Delta_{I}:=\\{(t,z)\in I\times{\mathbb{C}}^{n}\,|\,\Phi(z)=\Phi(y)+tx\\}.$

Since $\Phi$ is a polynomial map, $\Delta_{I}$ is a semialgebraic subset of
${\mathbb{R}}\times{\mathbb{C}}^{n}$. Moreover, $\Delta_{\gamma}$ is a
connected closed subset of $\Delta_{I}$. For all $t$ in $I$, denote by
$(\omega_{t},\tau_{t})$ an adapted pair to $\gamma(t)$. Let $\delta_{t}$ be
the inverse image of $\omega_{t}$ by $\gamma$. Then, for $(s,z)$ in the
intersection of $\delta_{t}\times\omega_{t}$ and $\Delta_{I}$,

$\Phi(z)=\Phi(y)+sx\quad\text{and}\quad\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(s)=\Phi(y)+sx,$

whence $z=\gamma(s)$. As a result, $\Delta_{\gamma}$ is an open subset of
$\Delta_{I}$. Then $\Delta_{\gamma}$ is a connected component of $\Delta_{I}$.
So, by [Cos02, Theorem 2.23], $\Delta_{\gamma}$ is a semialgebraic subset of
${\mathbb{R}}\times{\mathbb{C}}^{n}$ since so is $\Delta_{I}$. ∎

######  Corollary 3.5.

Let $y$ be in ${\mathbb{C}}^{n}$ and $\gamma$ in ${\mathcal{C}}_{x}$ such that
$\gamma(0)=y$. Suppose that $\gamma$ is defined over the relatively compact
interval $]a,b[$ of ${\mathbb{R}}$. If $\|\gamma\|$ is bounded then there
exists $\delta$ in ${\mathcal{C}}_{x}$, bigger than $\gamma$ and such that its
domain of definition contains $[a,b]$.

###### Proof.

By Proposition 3.4 and Lemma 2.5(i), $\gamma$ has a continuous extension to
$[a,b]$. Denote again by $\gamma$ this extension. For some positive real
number $\eta$, there exists an integral curve $\gamma_{a}$ of $X_{x}$ defined
over $]a-\eta,a+\eta[$ and such that $\gamma_{a}(a)=\gamma(a)$. Let
$(\omega,\tau)$ be an adapted pair to $\gamma(a)$. For some positive real
number $\eta^{\prime}$, smaller than $\eta$, for all $t$ in
$]a,a+\eta^{\prime}[$, $\gamma(t)$ is in $\omega$ and
$\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(t)=\Phi(y)+tx$, whence

$\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(a)=\Phi(y)+ax\quad\text{and}\quad\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(t)=\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(a)+(t-a)x.$

By Lemma 3.2, for all $t$ in $[a,a+\eta[$,

$\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma_{a}(t)=\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(a)+(t-a)x.$

As a result, for some $t$ in $]a,a+\eta^{\prime}[$, $\gamma(t)=\gamma_{a}(t)$.
So, by the unicity of the integral curve with initial datum, $\gamma_{a}$ and
$\gamma$ have the same restriction to $[a,a+\eta[$. As a result, there exists
an integral curve of $X_{x}$, extending $\gamma$ and such that its interval of
definition is $]a-\eta,b[$. In the same way, we prove that for some positive
real number $\eta^{\prime\prime}$, there exists an integral curve of $X_{x}$,
extending $\gamma$ and such that its interval of definition is
$]a,b+\eta^{\prime\prime}[$, whence the corollary. ∎

##  4\. Special integral curves

For $x$ in ${\mathbb{C}}^{n}$, denote by $\gamma_{x}$ the maximal element of
${\mathcal{C}}_{x}$ such that $\gamma_{x}(0)=0$.

###  4.1. Intervals of definition

For $x$ in ${\mathbb{C}}^{n}$, we denote by $I_{x}$ the interval of definition
of $\gamma_{x}$.

######  Lemma 4.1.

Let $x$ be in ${\mathbb{C}}^{n}$.

(i) The interval $I_{x}$ is invariant under the homeomorphism $t\mapsto-t$. In
particular, for some $a(x)$ in ${\mathbb{R}}_{+}^{*}\cup\\{+\infty\\}$,
$I_{x}=]-a(x),a(x)[$.

(ii) For $s$ in ${\mathbb{R}}^{*}$, $\gamma_{sx}(t)=\gamma_{x}(st)$. Moreover,
$I_{sx}=s^{-1}I_{x}$ and $a(sx)=|s|^{-1}a(x)$.

###### Proof.

(i) By the maximality of $\gamma_{x}$ and the unicity of an integral curve
with initial datum it is sufficient to prove that the map $\delta_{x}$,

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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad
t\longmapsto-\gamma_{x}(-t)$

is an integral curve of $X_{x}$ since $0$ is in $I$ and $\gamma_{x}(0)=0$. By
definition, for all $t$ in $-I$,

$\delta_{x}^{\prime}(t)=\gamma_{x}^{\prime}(-t)=X_{x}(\gamma_{x}(-t))=\Phi^{\prime}(\gamma_{x}(-t))^{-1}(x).$

As $\Phi$ is an odd map, $\Phi^{\prime}(y)=\Phi^{\prime}(-y)$ for all $y$ in
${\mathbb{C}}^{n}$. As a result, $\delta_{x}$ is an integral curve of $X_{x}$,
whence the assertion.

(ii) Denote by $\delta$ the map

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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad
t\longmapsto\gamma_{x}(st).$

Then

$\delta^{\prime}(t)=s\gamma^{\prime}_{x}(st)=s\Phi^{\prime}(\gamma_{x}(st))^{-1}(x)=X_{sx}(\delta(t)).$

As a result, $\delta$ is an integral curve of the vector field $X_{sx}$ such
that $\delta(0)=0$. So by the unicity of the integral curve with initial
datum, $s^{-1}I_{x}$ is contained in $I_{sx}$ and $\delta$ is the restriction
of $\gamma_{sx}$ to $s^{-1}I_{x}$. Considering the map

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t\longmapsto\gamma_{sx}(s^{-1}t),$

$sI_{sx}$ is contained in $I_{x}$ by the same argument, whence
$I_{sx}=s^{-1}I_{x}$ and the assertion. ∎

According to Lemma 4.1(i), a map $a$ from ${\mathbb{C}}^{n}$ to
${\mathbb{R}}_{+}^{*}\cup\\{+\infty\\}$ is well defined. Let $\Omega$ be the
subset of elements $(x,t)$ of ${\mathbb{C}}^{n}\times{\mathbb{R}}$ such that
$-a(x)<t<a(x)$ and $\gamma$ the map

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0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(x,t)\longmapsto\gamma(x,t):=\gamma_{x}(t).$

######  Lemma 4.2.

Let $(x,t)$ be in $\Omega$ and $t^{\prime}$ a positive real number such that
$|t|<t^{\prime}<a(x)$. Then for some open neighborhood $\omega$ of $x$ in
${\mathbb{C}}^{n}$, $\omega\times[-t^{\prime},t^{\prime}]$ is contained in
$\Omega$ and the restriction of $\gamma$ to
$\omega\times[-t^{\prime},t^{\prime}]$ is continuous.

###### Proof.

Since $[-t^{\prime},t^{\prime}]$ is a compact interval contained in $I_{x}$
and the map $(y,z)\mapsto X_{z}(y)$ is polynomial, for some open neighborhood
$\omega$ of $x$ in ${\mathbb{C}}^{n}$, $\omega\times[-t^{\prime},t^{\prime}]$
is contained in $\Omega$ and the restriction of $\gamma$ to
$\omega\times[-t^{\prime},t^{\prime}]$ is continuous by [Car67, Théorème
3.6.1]. ∎

######  Corollary 4.3.

(i) The subset $\Omega$ of ${\mathbb{C}}^{n}\times{\mathbb{R}}$ is open and
the map $\gamma$ is continuous.

(ii) The function $a$ is lower semicontinuous.

###### Proof.

(i) By Lemma 4.2, $\Omega$ is the union of open subsets of
${\mathbb{C}}^{n}\times{\mathbb{R}}$ at which the restriction of $\gamma$ is
continuous, whence the assertion.

(ii) By Lemma 4.2, the function $a$ is lower continuous. ∎

###  4.2. A property of the map $\gamma$

Let $\Gamma$ be as in Subsection 3.2 and $\Gamma_{{\mathbb{R}}}$ its
intersection with ${\mathbb{C}}^{n}\times{\mathbb{R}}\times{\mathbb{C}}^{n}$.
Denote by $\psi$ the map

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and $G$ the graph of the map $\gamma$.

######  Lemma 4.4.

Let $\Gamma_{0}$ be the connected component of $\Gamma_{{\mathbb{R}}}$
containing ${\mathbb{C}}^{n}\times\\{0,0\\}$ and $\psi_{0}$ the restriction of
$\psi$ to $\Gamma_{0}$.

(i) The maps $\psi$ and $\psi_{0}$ are local homeomorphisms.

(ii) The subset $\Gamma_{0}$ of
${\mathbb{C}}^{n}\times{\mathbb{R}}\times{\mathbb{C}}^{n}$ is closed and
semialgebraic. Moreover, it contains $G$.

(iii) The subset $G$ of $\Gamma_{0}$ is open.

###### Proof.

(i) Let $\Psi,\Delta$ be as in Subsection 3.2, $\Psi_{{\mathbb{R}}}$ the
restriction of $\Psi$ to $\Gamma_{{\mathbb{R}}}$ and $\Delta_{{\mathbb{R}}}$
the intersection of $\Delta$ and
${\mathbb{C}}^{n}\times{\mathbb{R}}\times{\mathbb{C}}^{n}$. As $\Psi$ is an
étale morphism to $\Delta$, $\Psi_{{\mathbb{R}}}$ is a local homeomorphism
from $\Gamma_{{\mathbb{R}}}$ to $\Delta_{{\mathbb{R}}}$. Denoting by $\pi$ the
map

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0.0pt\raise-1.80833pt\hbox{$\scriptstyle{\pi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 64.47334pt\raise 0.0pt\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}}{\hbox{\kern
34.47334pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 64.47334pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}\times{\mathbb{R}}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(x,t,tx)\longmapsto(x,t),$

$\pi$ is a homeomorphism and
$\psi=\pi\raisebox{0.85358pt}{\scriptsize$\circ$}\Psi_{0}$, whence the
assertion.

(ii) Since $\Gamma_{{\mathbb{R}}}$ is a semialgebraic closed subset of
${\mathbb{C}}^{n}\times{\mathbb{R}}\times{\mathbb{C}}^{n}$, $\Gamma_{0}$ is
semialgebraic by [Cos02, Theorem 2.23]. The set $G$ is contained in
$\Gamma_{{\mathbb{R}}}$ since for all $(x,t)$ in $\Omega$,
$\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma_{x}(t)=tx$ by Lemma 3.2.
As $\Omega$ is a connected subset of ${\mathbb{C}}^{n}\times{\mathbb{R}}$, $G$
is a connected subset of
${\mathbb{C}}^{n}\times{\mathbb{R}}\times{\mathbb{C}}^{n}$ by Corollary
4.3(i). Hence $G$ is contained in $\Gamma_{0}$ since
${\mathbb{C}}^{n}\times\\{(0,0)\\}$ is contained in $G$.

(iii) Let $(x,t,y)$ be in $G$ and $(\omega,\tau)$ an adapted pair to $y$. As
$\gamma$ is continous, $\gamma^{-1}(\omega)\times\omega$ is an open subset of
${\mathbb{C}}^{n}\times{\mathbb{R}}\times{\mathbb{C}}^{n}$. Its intersection
with $\Gamma_{{\mathbb{R}}}$ is the graph of the map

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
17.30779pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\gamma^{-1}(\omega)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
71.30779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
41.30779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 71.30779pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(z,s)\longmapsto\tau(sz).$

Since for all $(z,s)$ in $\gamma^{-1}(\omega)$, $\gamma(z,s)$ is in $\omega$
and $\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(z,s)=sz$,
$\gamma^{-1}(\omega)\times\omega\cap\Gamma_{{\mathbb{R}}}$ is contained in
$G$, whence the assertion. ∎

Let $\overline{G}$ be the closure of $G$ in
${\mathbb{C}}^{n}\times{\mathbb{R}}\times{\mathbb{C}}^{n}$ and
$\widetilde{\Omega}$ its image by $\psi$.

######  Lemma 4.5.

Let $(x,t_{0},y)$ be in $\overline{G}\setminus G$ such that $t_{0}$ is
positive. Then $a(x)$ is smaller than $t_{0}$ and for some positive real
number $\rho$, $\\{x\\}\times[t_{0}-\rho,t_{0}]$ is contained in
$\widetilde{\Omega}$.

###### Proof.

By Corollary 4.3(i), $G$ is closed in $\Omega\times{\mathbb{C}}^{n}$. Hence
$t_{0}\geq a(x)$ since $(x,t_{0},y)$ is in $\overline{G}\setminus G$. Let
$(\omega,\tau)$ be an adapted pair to $y$. For $z$ in $\omega$, denote by
$\lambda_{z}$ an integral curve of the vector field $X_{t_{0}^{-1}\Phi(z)}$
such that $\lambda_{z}(t_{0})=z$. In particular, $\lambda_{y}$ is an integral
curve of the vector field $X_{x}$ such that $\lambda_{y}(t_{0})=y$ and by
Lemma 3.2, for all $z$ in $\omega$ and for all $t$ in the interval of
definition of $\lambda_{z}$,

$\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\lambda_{z}(t)=\Phi(z)+(t-t_{0})t_{0}^{-1}\Phi(z)=tt_{0}^{-1}\Phi(z).$

For some positive real number $\eta$, $[t_{0}-\eta,t_{0}+\eta]$ is contained
in the interval of definition of $\lambda_{y}$. Then, by [Car67, Théorème
3.6.1], we can suppose that for some open neighborhood $\omega^{\prime}$ of
$y$ in $\omega$, $[t_{0}-\eta,t_{0}+\eta]$ is contained in the interval of
definition of $\lambda_{z}$ for all $z$ in $\omega^{\prime}$ since the map
$(y,z)\mapsto X_{t_{0}^{-1}\Phi(z)}(y)$ is polynomial. Moreover, the map

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
42.64375pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\omega^{\prime}\times[t_{0}-\eta,t_{0}+\eta]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
64.19376pt\raise 5.91667pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.91667pt\hbox{$\scriptstyle{\lambda}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 96.64375pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
66.64375pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 96.64375pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(z,t)\longmapsto\lambda_{z}(t)$

is continuous. Hence we can suppose that the image of $\lambda$ is contained
in $\omega$.

For some sequence $(x_{m},t_{m}),\,m=1,2,\ldots$ in $\Omega$,

$\lim_{m\rightarrow+\infty}\gamma(x_{m},t_{m})=y\quad\text{and}\quad\lim_{m\rightarrow+\infty}(x_{m},t_{m})=(x,t_{0}).$

For some positive integer $M$ and for $m$ bigger than $M$,
$\gamma(x_{m},t_{m})$ is in $\omega^{\prime}$ and for $s$ positive real number
smaller than $\eta/2$, $t_{m}-s$ is in $[t_{0}-\eta,t_{0}+\eta]$. Since
$\Phi(\omega^{\prime})$ is a neighborhood of $\Phi(y)$ and
$\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(x_{m},t_{m})=t_{m}x_{m}$,
we can suppose that for $m$ bigger than $M$, $t_{0}x_{m}$ is in
$\Phi(\omega^{\prime})$. Setting $z_{m}:=\tau(t_{0}x_{m})$, $\lambda_{z_{m}}$
is an integral curve of $X_{x_{m}}$ and
$\lambda_{z_{m}}(t_{m})=\gamma(x_{m},t_{m})$ since

$\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\lambda_{z_{m}}(t_{m})=\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(x_{m},t_{m})=t_{m}x_{m}.$

So, by the unicity of the integral curve with initial datum,
$\lambda_{z_{m}}(t)=\gamma_{x_{m}}(t)$ for all $t$ in the intersection of
$[t_{0}-\eta,t_{0}+\eta]$ and $]-a(x_{m}),a(x_{m})[$. As a result, for $s$
positive real number smaller than $\eta/2$,

$\lim_{m\rightarrow+\infty}\gamma(x_{m},t_{m}-s)=\lim_{m\rightarrow+\infty}\lambda_{z_{m}}(t_{m}-s)=\lambda_{y}(t_{0}-s)$

by continuity of $\lambda$. Then, if $t_{0}>a(x)$, for some positive real
number $\rho$, $\rho\leq\eta/2$, $t_{0}-\rho>a(x)$ and
$(x,t_{0}-s,\lambda_{y}(t_{0}-s))$ is in $\overline{G}\setminus G$ for all $s$
in $[0,\rho]$. Otherwise, $t_{0}=a(x)$ and for $s$ positive real number
smaller than $\eta/2$,

$\gamma(x,a(x)-s)=\lim_{m\rightarrow+\infty}\gamma(x_{m},t_{m}-s)=\lim_{m\rightarrow+\infty}\lambda_{z_{m}}(t_{m}-s)=\lambda_{y}(t_{0}-s)$

by continuity of $\gamma$ and $\lambda$. It is not possible since

$\lim_{s\rightarrow 0}\|\gamma(x,a(x)-s)\|=+\infty$

by Corollary 3.5, whence the lemma. ∎

###  4.3. Etale covering

Let $K$ be the field of rational fractions
${\mathbb{C}}({x}_{1}{,\ldots,\hskip 1.79993pt}{x}_{n})$ and $K^{\prime}$ the
subfield of $K$ generated by the coordinates of $\Phi$. Denote by $d$ the
degree of the extension $K$ of $K^{\prime}$.

######  Lemma 4.6.

Let $X$ and $Y$ be two irreducible complex algebraic varieties and $\psi$ a
morphism from $X$ to $Y$. For $i$ nonnegative integer, denote by $Y_{i}$ the
subset of elements $y$ such that $|\psi^{-1}(\\{y\\})|=i$. Then, for all
nonnegative integer $i$, $Y_{i}$ is a Zariski constructible subset of $Y$.

###### Proof.

Let $Y^{\prime}_{i}$ be the subset of elements $y$ of $Y$ such that
$|\psi^{-1}(\\{y\\})|\geq i$. Denote by $Z_{i}$ the subset of elements
$(y,{x}_{1}{,\ldots,\hskip 1.79993pt}{x}_{i})$ of $Y\times X^{i}$ such that
$y={\psi}(x_{1}){=\cdots=\hskip 1.79993pt}{\psi}(x_{i})$. Then $Z_{i}$ is a
Zariski closed subset of $Y\times X^{i}$. Let $Z^{\prime}_{i}$ be the Zariski
open subset of elements $(y,{x}_{1}{,\ldots,\hskip 1.79993pt}{x}_{i})$ of
$Z_{i}$ such that $x_{j}\neq x_{k}$ for $j\neq k$. Then $Y^{\prime}_{i}$ is a
Zariski constructible subset of $Y$ as the image of $Z^{\prime}_{i}$ by the
projection

$(y,{x}_{1}{,\ldots,\hskip 1.79993pt}{x}_{i})\longmapsto y.$

By definition, $Y_{i}=Y^{\prime}_{i}\setminus Y^{\prime}_{i+1}$. Hence $Y_{i}$
is a Zariski constructible subset of $Y$. ∎

######  Corollary 4.7.

Suppose that $\psi$ is a quasi-finite dominant morphism. Let $j$ be the degree
of the field extension ${\mathbb{C}}(Y)\subset{\mathbb{C}}(X)$. Then $Y_{j}$
contains a dense Zariski open subset of $Y$.

###### Proof.

As $j$ is the degree of the separable field extension
${\mathbb{C}}(Y)\subset{\mathbb{C}}(X)$, $Y_{j}$ contains the generic point of
$Y$. So, by Lemma 4.6, $Y_{j}$ contains a dense Zariski open subset of $Y$. ∎

######  Proposition 4.8.

Let $U$ be the subset of elements $x$ of ${\mathbb{C}}^{n}$ such that
$|\Phi^{-1}(\\{x\\})|=d$. Then $U$ is a dense Zariski open subset of
${\mathbb{C}}^{n}$ and the restriction of $\Phi$ to $\Phi^{-1}(U)$ is a
covering map of degree $d$. In particular, it is a finite morphism.

###### Proof.

As $\Phi$ is étale, $\Phi$ is a quasi finite morphism. So, by Corollary 4.7
and Proposition 2.4(iii), $U$ is a Zariski dense open subset of
${\mathbb{C}}^{n}$. Let $x$ be in $U$ and ${x}_{1}{,\ldots,\hskip
1.79993pt}{x}_{d}$ the elements of $\Phi^{-1}(\\{x\\})$. For $i=1,\ldots,d$,
denote by $(\omega_{i},\tau_{i})$ an adapted pair to $x_{i}$. Since
${x}_{1}{,\ldots,\hskip 1.79993pt}{x}_{d}$ are pairwise different, we can
choose ${\omega}_{1}{,\ldots,\hskip 1.79993pt}{\omega}_{d}$ pairwise disjoint.
Let $\omega$ be the intersection of ${\Phi}(\omega_{1}){,\ldots,\hskip
1.79993pt}{\Phi}(\omega_{d})$. Then $\omega$ is an open neighborhood of $x$
and $\Phi^{-1}(\omega)$ is the disjoint union of the open subsets
$\omega_{1}\cap\Phi^{-1}(\omega),\ldots,\omega_{d}\cap\Phi^{-1}(\omega)$. As a
result, the restriction of $\Phi$ to $\Phi^{-1}(U)$ is an étale covering of
degree $d$ of $U$, whence the proposition. ∎

###  4.4. A property of $\Gamma_{0}$

Let $U$ be the Zariski open subset of ${\mathbb{C}}^{n}$ as in Proposition 4.8
and $Y$ its complement in ${\mathbb{C}}^{n}$.

######  Lemma 4.9.

Let $Z$ be the closure in ${\mathbb{C}}^{n}$ of the union of
$tY,\,t\in{\mathbb{R}}^{*}_{+}$ and $O:={\mathbb{C}}^{n}\setminus Z$.

(i) The subsets $Z$ and $O$ of ${\mathbb{C}}^{n}$ are semialgebraic. In
particular, the connected components of $O$ and $Z$ are finitely many and
semialgebraic.

(ii) The open subset $O$ of ${\mathbb{C}}^{n}$ is dense in ${\mathbb{C}}^{n}$.

(iii) The connected components of $O$ and $Z$ are invariant under
${\mathrm{H}}_{n}$.

###### Proof.

(i) Let $Z_{*}$ be the union of $tY,\,t\in{\mathbb{R}}^{*}_{+}$. As the map

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
20.11348pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{R}}^{*}_{+}\times{\mathbb{C}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
74.11348pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
44.11348pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 74.11348pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(t,x)\longmapsto
tx$

is semialgebraic and $Y$ is an algebraic subvariety of ${\mathbb{C}}^{n}$,
$Z_{*}$ is a semialgebraic subset of ${\mathbb{C}}^{n}$. Hence $Z$ and $O$ are
semialgebraic subsets of ${\mathbb{C}}^{n}$, whence the assertion by [Cos02,
Theorem 2.23].

(ii) By definition, $Y$ is an algebraic subvariety of dimension smaller than
$n$ of ${\mathbb{C}}^{n}$. Hence its dimension as a semialgebraic subset of
${\mathbb{C}}^{n}$ is at most $2n-2$ by Lemma 2.2(iv). As a result, $Z_{*}$
and $Z$ have dimension at most $2n-1$ by Lemma 2.2, (i) and (v). Hence $O$ is
a dense open subset of ${\mathbb{C}}^{n}$ by Lemma 2.2(iii).

(iii) As $O$ and $Z$ are invariant under ${\mathrm{H}}_{n}$,
${\mathrm{H}}_{n}$ permutes the connected components of $O$ and $Z$. Then, by
(ii), the connected components of $O$ and $Z$ are invariant under
${\mathbb{R}}^{*}_{+}$ since ${\mathbb{R}}^{*}_{+}$ has no subgroup of finite
index different from ${\mathbb{R}}^{*}_{+}$. ∎

Let $\Gamma_{O}$ be the intersection of $\Gamma_{{\mathbb{R}}}$ and
$O\times{\mathbb{R}}_{+}^{*}\times{\mathbb{C}}^{n}$ and $\psi_{O}$ the
restriction of $\psi$ to $\Gamma_{O}$.

######  Lemma 4.10.

The map $\psi_{O}$ is a proper map to $O\times{\mathbb{R}}^{*}_{+}$.

###### Proof.

Denote by $\Delta_{O}$ the graph of the map

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
19.84pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{O\times{\mathbb{R}}^{*}_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
73.84pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
43.84pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 73.84pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(x,t)\longmapsto
tx,$

and $\Psi_{O}$ the map

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
9.40627pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Gamma_{O}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
28.24226pt\raise 6.69pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.05pt\hbox{$\scriptstyle{\Psi_{O}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 63.40627pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.40627pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 63.40627pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Delta_{O}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(x,t,y)\longmapsto(x,t,\Phi(y)).$

Then $\psi_{O}=\pi\raisebox{0.85358pt}{\scriptsize$\circ$}\Psi_{O}$ with $\pi$
as in the proof of Lemma 4.4(i). By definition $Z$ and $O$ are invariant under
${\mathrm{H}}_{n}$. Hence $\Delta_{O}$ is contained in
$O\times{\mathbb{R}}^{*}_{+}\times O$. By Proposition 4.8, the restriction of
$\Phi$ to $\Phi^{-1}(O)$ is a proper map from $\Phi^{-1}(O)$ to $O$. Hence
$\Psi_{O}$ is a proper map from $\Gamma_{O}$ to $\Delta_{O}$ since
$\Psi(x,t,y)=(x,t,\Phi(y))$ and $\Phi(y)=tx$ for all $(x,t,y)$ in
$\Gamma_{O}$. As a result, $\psi_{O}$ is a proper map from $\Gamma_{O}$ to
$O\times{\mathbb{R}}^{*}_{+}$ since the restriction of $\pi$ to $\Delta_{O}$
is a homeomorphism onto $O\times{\mathbb{R}}^{*}_{+}$. ∎

######  Corollary 4.11.

The open subset $O$ of ${\mathbb{C}}^{n}$ is contained in
$a^{-1}(\\{+\infty\\})$.

###### Proof.

Suppose that it is not true. A contradiction is expected. As $\Omega\cap
O\times{\mathbb{R}}^{*}_{+}$ is not closed in $O\times{\mathbb{R}}^{*}_{+}$,
$G\cap\Gamma_{O}$ is not closed in $\Gamma_{O}$ by Lemma 4.10. For $x$ in $O$,
set:

$I_{x}:=\\{t\in[a(x),+\infty[\,|\,\\{(x,t)\\}\times{\mathbb{C}}^{n}\cap\overline{G}\setminus
G\neq\emptyset\\}.$

As $G\cap\Gamma_{O}$ is not closed in $\Gamma_{O}$, for some $x$ in $O$,
$I_{x}$ is nonempty. By Lemma 4.4(iii), $\overline{G}\setminus G$ is a closed
subset of $\Gamma_{0}$. So, by Lemma 4.10, $\psi_{O}((\overline{G}\setminus
G)\cap\Gamma_{O})$ is a closed subset of $O\times{\mathbb{R}}^{*}_{+}$. As a
result, by Lemma 4.5, $I_{x}$ is a closed subset of ${\mathbb{R}}$ contained
in $]a(x),+\infty[$. Denoting by $t_{0}$ the smallest element of $I_{x}$,
$]a(x),t_{0}[\cap I_{x}$ is nonempty again by Lemma 4.5, whence the
contradiction. ∎

Let $U$ be as in Proposition 4.8.

######  Proposition 4.12.

(i) The sets $\overline{G}$ and $\Gamma_{0}$ are equal.

(ii) Set $\widetilde{\Omega}:=\psi(\overline{G})$. Then $\widetilde{\Omega}$
is an open subset of ${\mathbb{C}}^{n}\times{\mathbb{R}}$ and $\Gamma_{0}$ is
the graph of a continuous semialgebraic map $\widetilde{\gamma}$ from
$\widetilde{\Omega}$ to ${\mathbb{C}}^{n}$.

(iii) For $(x,t)$ in ${\mathbb{C}}^{n}\times{\mathbb{R}}$, if $tx$ is in $U$
then $(x,t)$ is in $\widetilde{\Omega}$.

###### Proof.

(i) Let $(x,t,y)$ be in $\overline{G}$ and $(\omega,\tau)$ an adapted pair to
$y$ such that $\omega$ is connected. Denote by $\delta$ the connected
component of $(x,t)$ in the inverse image of $\Phi(\omega)$ by the map
$(z,s)\mapsto sz$. Then $\delta$ is a connected neighborhood of $(x,t)$. By
Corollary 4.11 and Lemma 4.9(ii), $O\times{\mathbb{R}}$ is a dense subset of
$\Omega$ so that $O\times{\mathbb{R}}\cap\delta$ is dense in $\delta$. As
$(x,t,y)$ is in $\overline{G}$, the subset of elements $(z,s)$ of this
intersection such that $(z,s,\tau(sz))$ is in $\overline{G}$ is closed and
nonempty. Hence $(z,s,\tau(sz))$ is in $\overline{G}$ for all $(z,s)$ in
$\delta$ since $\gamma(z,s)=\tau(sz)$ for all $(z,s)$ in $\delta\cap
O\times{\mathbb{R}}$. Moreover, for all $(z,s,u)$ in the intersection of
$\delta\times\omega$ and $\Gamma_{{\mathbb{R}}}$, $u=\tau(sz)$. As a result,
$\overline{G}$ is an open subset of $\Gamma_{0}$, whence
$\overline{G}=\Gamma_{0}$ since $\Gamma_{0}$ is connected by definition.

(ii) By (i), $\widetilde{\Omega}=\psi(\Gamma_{0})$. Hence $\widetilde{\Omega}$
is a semialgebraic open subset of ${\mathbb{C}}^{n}\times{\mathbb{R}}$ since
$\psi$ is a semialgebraic local homeomorphism by Lemma 4.4(i). For $i$
nonnegative integer, set

$\widetilde{\Omega}_{i}:=\\{(x,t)\in\widetilde{\Omega}\,|\,|\psi_{0}^{-1}(\\{(x,t)\\}|=i\\}.$

By Proposition 2.4(iii), for some positive integer $j$,
$\widetilde{\Omega}_{i}$ is empty for $i>j$ and $\widetilde{\Omega}_{j}$ is a
nonempty open subset of $\widetilde{\Omega}$ since $\psi_{0}$ is a local
homeomorphism. By (i), $\psi_{0}^{-1}(O\times{\mathbb{R}})$ is contained in
$G$ by continuity of $\gamma$. Then $O\times{\mathbb{R}}$ is contained in
$\widetilde{\Omega}_{1}$ and $j=1$ since $O\times{\mathbb{R}}$ is dense in
$\widetilde{\Omega}$. As a result, $\Gamma_{0}$ is the graph of an extension
$\widetilde{\gamma}$ of $\gamma$ to $\widetilde{\Omega}$.

Let $(x,t,y)$ be in $\Gamma_{0}$ and $\omega,\tau,\delta$ as in the proof of
(i). Then $\tilde{\gamma}(x,t)=y$ and the restriction of $\tilde{\gamma}$ to
$\delta$ is the map

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.66666pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
59.66666pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.66666pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(x,t)\longmapsto\tau(tx).$

Hence the restriction of $\widetilde{\gamma}$ to $\delta$ is continuous,
whence the assertion since $\Gamma_{0}$ is semialgebraic.

(iii) Suppose that $(x,t)$ is not in $\widetilde{\Omega}$ and $tx$ is in $U$.
A contradiction is expected. By Corollary 4.11, Lemma 4.9(ii) and (i),
$\widetilde{\Omega}$ is a dense open semialgebraic subset of
${\mathbb{C}}^{n}\times{\mathbb{R}}$. So, by Proposition 2.3, there exists a
continuous semialgebraic arc $\theta$,

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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37.02835pt\raise 5.91667pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 69.00002pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
39.00002pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 69.00002pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}\times{\mathbb{R}}^{*}_{+}}$}}}}}}}\ignorespaces}}}}\ignorespaces\qquad
s\longmapsto\theta(s)$

such that $\theta(0)=(x,t)$ and $\theta(s)\in\widetilde{\Omega}$ for all $s$
in $]0,1]$. As $tx$ is in $U$, for some open neighborhood $\omega$ of $tx$ in
${\mathbb{C}}^{n}$, the restriction of $\Phi$ to $\Phi^{-1}(\omega)$ is a
proper map by Proposition 4.8. By continuity of $\widetilde{\gamma}$, for some
positive number $\varepsilon$,
$\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\widetilde{\gamma}\raisebox{0.85358pt}{\scriptsize$\circ$}\theta([0,\varepsilon])$
is a compact subset of $\omega$. Hence the map
$s\mapsto\|\widetilde{\gamma}\raisebox{0.85358pt}{\scriptsize$\circ$}\theta(s)\|$
is bounded. So, by Lemma 2.5(i), for some $y$ in ${\mathbb{C}}^{n}$,

$\lim_{s\rightarrow
0}\widetilde{\gamma}\raisebox{0.85358pt}{\scriptsize$\circ$}\theta(s))=y.$

As a result, $(x,t,y)$ is in $\Gamma_{0}$ and $(x,t)$ is in
$\widetilde{\Omega}$, whence the contradiction. ∎

##  5\. Polynomial automorphism

Let $\Gamma$ be as in Subsection 3.2 and $\Gamma_{*}$ the connected component
of $\Gamma$ containing ${\mathbb{C}}^{n}\times\\{(0,0)\\}$. Denote by
$\psi_{*}$ the map

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8.43001pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{$\textstyle{\Gamma_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
28.49417pt\raise 6.73334pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.1pt\hbox{$\scriptstyle{\psi_{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 62.43001pt\raise 0.0pt\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}}{\hbox{\kern
32.43001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 62.43001pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}\times{\mathbb{C}}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(x,t,y)\longmapsto(x,t).$

Let $U$ be as in Proposition 4.8, $\Omega_{*}$ the subset of elements $(x,t)$
of ${\mathbb{C}}^{n}\times{\mathbb{C}}$ such that $tx$ is in $U$ and
$\gamma_{*}$ the map

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
9.01335pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{$\textstyle{\Omega_{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
29.63889pt\raise 5.625pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.99167pt\hbox{$\scriptstyle{\gamma_{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 63.01335pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.01335pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 63.01335pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(x,t)\longmapsto\widetilde{\gamma}(tx,1)$

with $\widetilde{\gamma}$ as in Proposition 4.12(ii). By Proposition 4.12,(ii)
and (iii), $\gamma_{*}$ is well defined and continuous.

######  Lemma 5.1.

Let $G_{*}$ be the graph of $\gamma_{*}$.

(i) The map $\psi_{*}$ is a local homeomorphism.

(ii) The subset $\Gamma_{*}$ of
${\mathbb{C}}^{n}\times{\mathbb{C}}\times{\mathbb{C}}^{n}$ is an irreducible
component of $\Gamma$. Moreover, it contains $G_{*}$.

(iii) The subset $G_{*}$ of $\Gamma_{*}$ is open.

###### Proof.

(i) Let $\Psi,\Delta$ be as in Subsection 3.2 and $\Psi_{*}$ the restriction
of $\Psi$ to $\Gamma_{*}$. As $\Phi$ is étale, $\Psi$ is an étale morphism to
$\Delta$. Hence $\Psi_{*}$ is a local homeomorphism from $\Gamma_{*}$ to
$\Delta$. Denoting by $\pi_{*}$ the map

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28.40591pt\raise 5.36667pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.25pt\hbox{$\scriptstyle{\pi_{*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 62.00002pt\raise 0.0pt\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}}{\hbox{\kern
32.00002pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 62.00002pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}\times{\mathbb{C}}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(x,t,tx)\longmapsto(x,t),$

$\pi_{*}$ is a homeomorphism and
$\psi_{*}=\pi_{*}\raisebox{0.85358pt}{\scriptsize$\circ$}\Psi_{*}$, whence the
assertion.

(ii) The subset $\Gamma_{*}$ of
${\mathbb{C}}^{n}\times{\mathbb{C}}\times{\mathbb{C}}^{n}$ is an irreducible
component of $\Gamma$ by Lemma 3.3(i). By definition, $\Omega_{*}$ is a
Zariski open subset of ${\mathbb{C}}^{n}\times{\mathbb{C}}$. Hence it is
connected. As a result, $G_{*}$ is a connected subset of $\Gamma$ since
$\gamma_{*}$ is continuous. Moreover, for all $x$ in $U$, $\gamma_{*}(x,1)$ in
$G_{*}\cap\Gamma_{0}$ by Proposition 4.12(iii), whence the assertion since
$\Gamma_{0}$ is contained in $\Gamma_{*}$.

(iii) Let $(x,t,y)$ be in $G_{*}$ and $(\omega,\tau)$ an adapted pair to $y$.
As $\gamma_{*}$ is continuous, $\gamma_{*}^{-1}(\omega)\times\omega$ is an
open subset of ${\mathbb{C}}^{n}\times{\mathbb{C}}\times{\mathbb{C}}^{n}$. Its
intersection with $\Gamma$ is the graph of the map

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40.18779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 70.18779pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\qquad(z,s)\longmapsto\tau(sz).$

Since for all $(z,s)$ in $\gamma_{*}^{-1}(\omega)$, $\gamma_{*}(z,s)$ is in
$\omega$ and $\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\gamma(z,s)=sz$,
$\gamma_{*}^{-1}(\omega)\times\omega\cap\Gamma$ is contained in $G_{*}$,
whence the assertion. ∎

Let $\overline{G_{*}}$ be the closure of $G_{*}$ in
${\mathbb{C}}^{n}\times{\mathbb{C}}\times{\mathbb{C}}^{n}$ and
$\widetilde{\Omega_{*}}$ its image by $\psi_{*}$.

######  Proposition 5.2.

(i) The sets $\overline{G_{*}}$ and $\Gamma_{*}$ are equal.

(ii) The set $\widetilde{\Omega_{*}}:=\psi_{*}(\overline{G_{*}})$ is a Zariski
open subset of ${\mathbb{C}}^{n}\times{\mathbb{C}}$ and $\Gamma_{*}$ is the
graph a regular map $\widetilde{\gamma_{*}}$ from $\widetilde{\Omega_{*}}$ to
${\mathbb{C}}^{n}$.

###### Proof.

(i) Let $(x,t,y)$ be in $\overline{G_{*}}$ and $(\omega,\tau)$ an adapted pair
to $y$ such that $\omega$ is connected. Denote by $\delta$ the connected
component of $(x,t)$ in the inverse image of $\Phi(\omega)$ by the map
$(z,s)\mapsto sz$. Then $\delta$ is a connected neighborhood of $(x,t)$. Since
$\Omega_{*}$ is a Zariski open subset of ${\mathbb{C}}^{n}\times{\mathbb{C}}$,
$\Omega_{*}\cap\delta$ is dense in $\delta$. As $(x,t,y)$ is in
$\overline{G_{*}}$, the subset of elements $(z,s)$ of this intersection such
that $(z,s,\tau(sz))$ is in $\overline{G_{*}}$ is closed and nonempty. Hence
$(z,s,\tau(sz))$ is in $\overline{G_{*}}$ for all $(z,s)$ in $\delta$ since
$\gamma_{*}(z,s)=\tau(sz)$ for all $(z,s)$ in $\delta\cap\Omega_{*}$.
Moreover, for all $(z,s,u)$ in the intersection of $\delta\times\omega$ and
$\Gamma_{*}$, $u=\tau(sz)$. As a result, $\overline{G_{*}}$ is an open subset
of $\Gamma_{*}$, whence $\overline{G_{*}}=\Gamma_{*}$ since $\Gamma_{*}$ is
connected by definition.

(ii) By (i), $\widetilde{\Omega_{*}}=\psi_{*}(\Gamma_{*})$. Hence
$\widetilde{\Omega_{*}}$ is a Zariski open subset of
${\mathbb{C}}^{n}\times{\mathbb{C}}$ since $\psi_{*}$ is a morphism of
algebraic varieties and a local homeomorphism by Lemma 5.1(i). For $i$
nonnegative integer, set

$\widetilde{\Omega_{*}}_{i}:=\\{(x,t)\in\widetilde{\Omega_{*}}\,|\,|\psi_{*}^{-1}(\\{(x,t)\\}|=i\\}.$

By Proposition 2.4(iii), for some positive integer $j$,
$\widetilde{\Omega_{*}}_{i}$ is empty for $i>j$ and
$\widetilde{\Omega_{*}}_{j}$ is a nonempty open subset of
$\widetilde{\Omega_{*}}$ since $\psi_{*}$ is a semialgebraic local
homeomorphism. By (i), $\psi_{*}^{-1}(\Omega_{*})$ is contained in $G_{*}$
since $\gamma_{*}$ is continuous. Then $\Omega_{*}$ is contained in
$\widetilde{\Omega_{*}}_{1}$ and $j=1$ since $\Omega_{*}$ is dense in
$\widetilde{\Omega_{*}}$. As a result, $\Gamma_{*}$ is the graph of an
extension $\widetilde{\gamma_{*}}$ of $\gamma_{*}$ to
$\widetilde{\Omega_{*}}$. In particular, $\psi_{*}$ is a birational morphism
since ${\mathbb{C}}$ is a separable field. Then, by [Mu88, §9], $\psi_{*}$ is
an isomorphism of the algebraic variety $\Gamma_{*}$ onto
$\widetilde{\Omega_{*}}$ and $\widetilde{\gamma_{*}}$ is a regular map. ∎

######  Corollary 5.3.

Let $\varphi$ be the map from $U$ to ${\mathbb{C}}^{n}$ such that
$\varphi(x)=\gamma_{*}(x,1)$ for all $x$ in $U$. Then $\varphi$ is a regular
map such that $\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\varphi(x)=x$ for
all $x$ in $U$.

###### Proof.

By Proposition 5.2(ii), $\widetilde{\gamma_{*}}$ is a regular map from
$\widetilde{\Omega_{*}}$ to ${\mathbb{C}}^{n}$. Hence $\varphi$ is a regular
map from $U$ to ${\mathbb{C}}^{n}$ since the map $x\mapsto(x,1)$ from $U$ to
$\widetilde{\Omega_{*}}$ is regular. By definition, for all $x$ in $U$,

$\Psi(x,1,\widetilde{\gamma_{*}}(x,1))=(x,1,\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\widetilde{\gamma_{*}}(x,1))=(x,1,x),$

whence the assertion. ∎

######  Theorem 5.4.

The polynomial endomorphism $\Phi$ is a polynomial automorphism.

###### Proof.

By Corollary 5.3, $\varphi(U)$ is a Zariski constructible subset of
$\Phi^{-1}(U)$. As $\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\varphi(U)=U$,
$\varphi(U)$ has dimension $n$. In particular, $\varphi(U)$ is dense in
$\Phi^{-1}(U)$. Again by Corollary 5.3(ii), for all $x$ in $U$,
$\varphi\raisebox{0.85358pt}{\scriptsize$\circ$}\Phi\raisebox{0.85358pt}{\scriptsize$\circ$}\varphi(x)=\varphi(x)$.
Hence $\varphi\raisebox{0.85358pt}{\scriptsize$\circ$}\Phi(y)=y$ for all $y$
in $\Phi^{-1}(U)$. Then, for $x$ in $U$ and $y$ in $\Phi^{-1}(\\{x\\})$,
$y=\varphi(x)$, whence $|\Phi^{-1}(\\{x\\})|=1$. So, by Proposition 4.8,
$\Phi$ is a birational morphism since ${\mathbb{C}}$ is a separable field. As
a result, $\Phi$ is a polynomial automorphism by [Bass 82, Theorem 2.1]. ∎

##  6\. Conclusion

By Theorem 5.4, an odd étale polynomial endomorphism of ${\mathbb{C}}^{n}$ is
a polynomial automorphism. So, by [Dru83, Theorem 3], any étale polynomial
endomorphism of ${\mathbb{C}}^{n}$ is a polynomial automorphism. Then, by
[Bass 82, Section I], we have the following theorem:

######  Theorem 6.1.

Let ${\Bbbk}$ be an integral ring of characteristic $0$ and $n$ a positive
integer. Then any polynomial endomorphism of ${\Bbbk}^{n}$ whose Jacobian
matrix has a constant invertible determinant is a polynomial automorphism.

From Theorem 6.1 and [Ka, Theorem 1], we deduce the following theorem:

######  Theorem 6.2.

Let ${\Bbbk}$ be a field of characteristic zero and $n$ a positive integer.
Denote by $A_{n,{\Bbbk}}$ the n-th Weyl algebra over ${\Bbbk}$. Then any
endomorphism of $A_{n,{\Bbbk}}$ is an automorphism.

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