# Generic complex polynomial vector fields with real coefficients

Jonathan Godin and Christiane Rousseau  Université de Moncton, Campus de
Shippagan, 218 boul. J.-D.-Gauthier, Shippagan (NB), E8S 1P6, Canada —
Département de mathématiques et de statistique, Université de Montréal, C.P.
6128, Succursale Centre-ville, Montréal (QC), H3C 3J7, Canada.
<EMAIL_ADDRESS>—<EMAIL_ADDRESS>
###### Abstract.

The paper studies the complex 1-dimensional polynomial vector fields with real
coefficients under topological orbital equivalence preserving the separatrices
of the pole at infinity. The number of generic strata is determined, and a
complete parametrization of these strata is given in terms of a modulus formed
by a combinatorial and an analytic part. The bifurcation diagram is described
for the degree 4. A realization theorem is proved for any generic modulus.

##  1\. Introduction

The study of the dynamics of 1-dimensional holomorphic maps involves a mixture
of local and global techniques. Among the local techniques is the study of the
multiple fixed points (also called _parabolic points_) of germs of holomorphic
diffeomorphisms and their unfoldings. For instance, the rightmost real point
of the Mandelbrot set corresponds to a quadratic map with a parabolic point.
For a parabolic fixed point of multiplicity $k+1$, i.e. codimension $k$, it is
natural to embed the corresponding germ of diffeomorphism into a generic
family depending on $k$ parameters. Indeed, the parabolic points organize the
dynamics in their neighborhoods inside the generic families.

Providing a complete description of the dynamics of unfoldings of parabolic
points proved to be a very difficult problem. A simple formal normal form was
known consisting of the time-one map of a rational vector field. But the
change of coordinate to normal form was divergent. For parameter values for
which it is possible to bring the system to the normal form in the
neighbordood of the unfolded simple fixed points, the obstruction to
convergence was explained by the fact that normalizations did not match
globally. But nobody knew how to treat the case where the unfolded simple
fixed points were not linearizable. The idea came from Douady, namely to
normalize on domains with two sectors adherent to two different fixed points.
It was used by Lavaurs and Oudkerk to treat some sectoral regions where the
unfolded simple fixed points were not linearizable ([D94], [L89] and [O99]).
The visionary paper of Douady-Estrada-Sentenac [DES05] made the breakthrough
and opened the way to treat all parameter values by the same method (for a non
exhaustive list see [MRR04], [Ri08], [Ro15]). The shape of the domains of
normalization is controlled by the dynamics of complex 1-dimensional
polynomial vector fields. The paper [DES05] contains a complete study of the
‘‘generic’’ complex polynomial vector fields, meant here to be the
structurally stable ones. It was followed by other works studying the non-
generic cases (see for instance [BD10], [DT16] and [D20]).

Holomorphic dynamics develops in parallel to antiholomorphic dynamics, and in
the latter case parabolic points do also play an important role. The second
iterate of an antiholomorphic parabolic germ is a holomorphic parabolic germ
of the same multiplicity (see for instance [HS14], [IM16] and [GR22]).
However, the natural parameters unfolding an antiholomorphic parabolic germ of
multiplicity $k+1$ are real. An analytic classification of generic unfoldings
of antiholomorphic parabolic germs of multiplicity $k+1$ depending real-
analytically on $k$ parameters can be obtained by extending these germs
antiholomorphically in the parameters (the idea of extending
antiholomorphically was already mentioned by Milnor in [M92]). If
$f_{\varepsilon}$ is such a germ, then the map
$g_{\varepsilon}=f_{\overline{\varepsilon}}\circ f_{\varepsilon}$ is a generic
unfolding of a holomorphic parabolic germ and its modulus is a modulus for
$f_{\varepsilon}$ ([GR23] and [Ro23b]). But the construction of the modulus
appearing there does not respect the real character of the parameters. A
description respecting the real character of the parameters is still needed
and will involve an underlying vector field with real coefficients. This is
the motivation for this study.

The results can also be helpful for any problem where parabolic points
naturally depend on real parameters. This is the case of the Poincaré return
map in the Hopf bifurcation. For the codimension 1 case, it was shown in [A12]
that the orbital classification of families of vector fields unfolding a Hopf
bifurcation follows from the conjugacy of the Poincaré return maps.

The paper describes the complete dynamics of generic polynomial vector fields
$P(z)\frac{\partial}{\partial z}$, where $P$ is a polynomial of degree $k+1$
with real coefficients. In practice, this means providing the phase portrait
of the ODE $\dot{z}=\frac{dz}{dt}=P(z)$ for real time $t$. Modulo an affine
change of coordinate we can always suppose that the polynomial $P$ has the
form

(1)
$P(z)=\begin{cases}z^{k+1}+\varepsilon_{k-1}z^{k-1}+\dots+\varepsilon_{1}z+\varepsilon_{0},&k\
\text{odd},\\\ \pm
z^{k+1}+\varepsilon_{k-1}z^{k-1}+\dots+\varepsilon_{1}z+\varepsilon_{0},&k\
\text{even},\end{cases}$

and we also note it $P_{\varepsilon}$ with
$\varepsilon=(\varepsilon_{k-1},\dots,\varepsilon_{1},\varepsilon_{0})\in\mathbb{R}^{k}$,
to emphasize its coefficients. We only consider the $+$ case. The other case
can be obtained from it by changing $t\mapsto-t$.

######  Remark 1.1.

When complex coefficients are allowed and $k$ is odd, the two cases $\pm$ are
equivalent under a change $z\mapsto\exp\left(\frac{\pi i\ell}{k}\right)z$,
$\ell$ odd.

######  Notation 1.2.

We call $\mathcal{P}_{{\mathbb{C}},k+1}$ (resp.
$\mathcal{P}_{\mathbb{R},k+1}$) the family of polynomial vector fields
$X_{\varepsilon}=P_{\varepsilon}(z)\frac{\partial}{\partial z}$ with complex
(resp. real) coefficients, where

$P_{\varepsilon}(z)=z^{k+1}+\varepsilon_{k-1}z^{k-1}+\dots+\varepsilon_{1}z+\varepsilon_{0}$

and $\varepsilon\in{\mathbb{C}}^{k}$ (resp. $\varepsilon\in\mathbb{R}^{k}$).

######  Remark 1.3.

In the whole paper, we identify $\mathcal{P}_{{\mathbb{C}},k+1}$ (resp.
$\mathcal{P}_{\mathbb{R},k+1}$) with the topological space ${\mathbb{C}}^{k}$
(resp. $\mathbb{R}^{k}$).

In this paper, we determine the equivalence classes of generic complex
1-dimensional polynomial vector fields with real coefficients under
topological orbital equivalence preserving the separatrices of the pole at
infinity: these equivalence classes are called generic _strata_. In Section 2
we recall the known results for $\mathcal{P}_{{\mathbb{C}},k+1}$, and in
Section 3, we derive the corresponding results for
$\mathcal{P}_{\mathbb{R},k+1}$, namely each stratum is characterized by a
combinatorial invariant and parameterized by an analytic invariant. In Section
4.1, we determine the exact number of generic strata in
$\mathcal{P}_{\mathbb{R},k+1}$. In Section 5, we explore the structure of the
family $\mathcal{P}_{\mathbb{R},k+1}$. In particular we show that it contains
a complete unfolding of any non real parabolic point. In Section 6, we
determine the bifurcation diagram of polynomial vector fields in
$\mathcal{P}_{\mathbb{R},k+1}$ of degree 4. Finally, in Section 7, a
realization theorem is proved for any generic combinatorial and analytic
invariant.

##  2\. Preliminaries on polynomial vector fields in
$\mathcal{P}_{{\mathbb{C}},k+1}$

In this section we briefly recall the results of [DES05] on polynomial vector
fields $\mathcal{P}_{{\mathbb{C}},k+1}$.

######  Remark 2.1.

The change

(2)
$(z,\varepsilon_{k-1},\dots,\varepsilon_{1},\varepsilon_{0},t)\mapsto(zr,\varepsilon_{k-1}r^{-(k-2)},\dots,\varepsilon_{1},\varepsilon_{0}r,tr^{k})$

sends a vector field to one with the same phase portrait modulo a zoom. This
allows using scalings when discussing particular situations.

###  2.1. The dynamics of polynomial vector fields
$\mathcal{P}_{{\mathbb{C}},k+1}$

The dynamics is governed by the singular point at $\infty$, which is a pole of
order $k-1$ (see Figure 1(a)). The pole has the behavior of a saddle with $2k$
sepatrices. This means that near the boundary of any sufficiently large disk
containing all the singular points, the trajectories are organized like a
flower (see Figure 1(b)). Note that $\infty$ is reached in finite time.
Moreover, the complexified time is ramified at the pole and covering $k$ times
a neighborhood of $\infty$.

Figure 1. (a) The phase portrait at infinity. (b) The phase portrait near the
boundary of a large disk containing all singular points.

######  Proposition 2.2.

[DES05] Let $X_{\varepsilon}\in\mathcal{P}_{{\mathbb{C}},k+1}$.

  1. (1)

Any separatrix of $\infty$ either lands at a singular point or merges with
another separatrix to form a homoclinic loop.

  2. (2)

Any simple finite singular point is an antisaddle, either a radial node, or a
strong focus, or a center. A singular point is a center if and only if its
eigenvalue is pure imaginary.

  3. (3)

There are no limit cycles.

  4. (4)

Any trajectory is either:

     * •

a fixed point,

     * •

a separatrix of $\infty$ landing at a singular point,

     * •

a homoclinic loop through infinity,

     * •

a periodic orbit of a period annulus,

     * •

a trajectory whose $\alpha$\- and $\omega$\- limits are singular points. In
the case where the $\alpha$\- and $\omega$\- limits are the same singular
point, then this point is a parabolic (multiple) point.

  5. (5)

Any center is surrounded by a period annulus, whose boundary is a union of
homoclinic loops through infinity.

  6. (6)

There is at least one separatrix landing at each radial node or focus.

  7. (7)

If $z_{1},\dots,z_{\ell}$ are the (possibly multiple) singular points of a
vector field $X_{\varepsilon}=P_{\varepsilon}\frac{\partial}{\partial z}$,
then
$\sum_{j=1}^{\ell}\operatorname{Res}\bigl{(}\frac{1}{P_{\varepsilon}},z_{j}\bigr{)}=0$.

  8. (8)

Any homoclinic loop surrounds a group of singular points
$z_{j_{1}},\dots,z_{j_{s}}$ such that
$\sum_{\ell=1}^{s}\operatorname{Res}\bigl{(}\frac{1}{P_{\varepsilon}},z_{j_{\ell}}\bigr{)}\in
i\mathbb{R}^{*}$.

###  2.2. DES-equivalence and structurally stable vector fields

######  Definition 2.3.

  1. (1)

Two vector fields $X_{\varepsilon}$ and $X_{\varepsilon}^{\prime}$ are said to
be _DES-equivalent_ if there exists a homeomorphism of ${\mathbb{C}}$
preserving the separatrices at infinity, which is a topological orbital
equivalence between the vector fields, i.e. it sends the trajectories to
trajectories and preserves the orientation, but not necessarily the
parametrization.

  2. (2)

Each equivalence class (or the corresponding set of parameter values, see
Remark 1.3) is called a _stratum_.

  3. (3)

The strata of full dimension are open sets of the parameter space and called
_generic strata_ , and the vector fields of a generic stratum are called
_generic_ (in the sense of Douady-Estrada-Sentenac). Each vector field
$X_{\varepsilon}$ belonging to a generic stratum is _structurally stable_ ,
i.e. DES-equivalent to all vector fields in a neighborhood of
$X_{\varepsilon}$.

######  Remark 2.4.

Each stratum is a submanifold of the parameter space. Its boundary is a finite
union of strata of lower dimension.

The bifurcation diagram for this equivalence relation divides the parameter
space into open regions (generic strata) of structurally stable vector fields
separated by bifurcation ‘‘surfaces’’, which are unions of strata of lower
dimension. The only bifurcations of real codimension 1 are the homoclinic
loops through infinity. The higher real codimension bifurcations are the
parabolic points (the real codimension is $2s$ for a multiple point of
multiplicity $s+1$) and intersections of lower codimension bifurcations.

######  Proposition 2.5.

A vector field $X_{\varepsilon}\in\mathcal{P}_{{\mathbb{C}},k+1}$ is generic
if and only if it has only simple singular points and no homoclinic loop.

###  2.3. The combinatorial part of the invariant of a structurally stable
vector field

######  Proposition 2.6.

[DES05] Let $X_{\varepsilon}\in\mathcal{P}_{{\mathbb{C}},k+1}$ be a generic
vector field. Two trajectories having the same $\alpha$\- and $\omega$-limits
are called _equivalent_ : this defines an equivalence relation on trajectories
whose $\alpha$\- and $\omega$-limits are singular points. Let us define the
following (non oriented) graph:

  * •

the vertices are the singular points;

  * •

there is an edge between two vertices if and only if there is a trajectory
linking the corresponding singular points. Then the edge is given by the
equivalence classes of trajectories having these two vertices as $\alpha$\-
and $\omega$-limits.

Then this graph is a planar tree graph. On this graph, the only isomorphisms
we allow are the ones provided by orientation preserving homeomorphisms of the
plane. The graph is attached to the $2k$ separatrices in the following way: if
we turn around the graph along a closed curve, then we meet $2k$ singular
points. The separatrices are attached in a one-to-one manner to these $2k$
points in a non-crossing way. (See Figure 2.) This attachement is uniquely
determined by the attachment of a first separatrix to a singular point. The
tree graph of a vector field and its attachment to the separatrices is called
the _combinatorial part_ of the invariant of $X_{\varepsilon}$. All generic
vector fields of a given stratum have the same combinatorial part.

Figure 2. Planar tree graph of a generic vector field and its attachement to
the separatrices, where the vertices are blue, the egdes are brown and the
separatrices are green. The tubular curve around the graph is the path used to
attach the separatrices, described in Proposition 2.6.

######  Theorem 2.7.

[DES05] Let us consider $\mathcal{P}_{{\mathbb{C}},k+1}$ for fixed $k$.

  1. (1)

The generic strata are in bijection with the combinatorial parts and any
combinatorial part is realizable.

  2. (2)

The separatrices at $\infty$ separate a neighborhood of infinity into $2k$
sectoral regions, called _ends_ at $\infty$. There is a unique involution
linking ends two by two by non-crossing curves, which do not intersect the
separatrices. This involution has no fixed point.

  3. (3)

Any non-crossing involution without fixed points between the $2k$ ends is
realizable by a combinatorial invariant.

  4. (4)

The number of generic strata in $\mathcal{P}_{{\mathbb{C}},k+1}$ is $C(k)$,
where

$C(k)=\frac{\binom{2k}{k}}{k+1}$

is the $k$-th Catalan number.

######  Corollary 2.8.

The non crossing involutions without fixed points of Theorem 2.7 are in
bijection with the noncrossing involutions without fixed points inside the
unit circle linking 2 by 2 the marked points

(3) $e_{j}=\exp\left({i\pi{2j-\operatorname{sgn}(j)\over 2k}}\right),\qquad
j\in\\{\pm 1,\ldots,\pm k\\}.$

######  Corollary 2.9.

Alternative ways to describe the combinatorial parts are the following
([DES05] or [D13]):

  1. (1)

The set of combinatorial parts is in bijection with the set of paths in
$\mathbb{N}^{2}$ from $(0,0)$ to $(2k,0)$ with steps $(1,1)$ and $(1,-1)$.
Such a path is called a _Dyck path of semilength $k$_.

  2. (2)

The set of combinatorial parts is in bijection with the set of admissible
brackets for multiplication of $k+1$ numbers, namely the set of sequences
$a_{1},\dots,a_{2k}$, with $a_{i}\in\\{(,)\\}$, $a_{i}=($ for half of the
$a_{i}$, and such that for each $j\in\\{1,\dots,2k\\}$

$\\#\\{i\leq j,a_{i}=(\\}\geq\\#\\{i\leq j,a_{i}=)\\}.$

The attachment of the singular points can be recovered from the combinatorial
invariant. This is described in the following proposition.

######  Proposition 2.10.

[DES05] Let $s_{j}$ be the separatrix of infinity tangent to the direction
$\exp\left(\frac{\pi ij}{2k}\right)$, $j=0,\dots,2k-1$. Let the permutation
$\sigma$ be defined on $\\{0,\ldots,2k-1\\}$ by

$\sigma(j):=\tau(j+1)\quad({\rm mod}\>2k).$

For each $j$, there exists a minimal $r_{j}>0$ such that $\sigma^{\circ
r_{j}}(j)=j$. This partitions $\\{0,\ldots,2k-1\\}$ into $k+1$ subsets of the
form $\\{j,\sigma(j),\dots,\sigma^{r_{j}-1}(j)\\}$. Each subset is the subset
of the indices of the separatrices landing at one singular point. The subsets
are the equivalence classes of the quotient $\\{0,\ldots,2k-1\\}/\sigma$. The
equivalence classes are noted $[j]$.

###  2.4. The analytic part of the invariant of a structurally stable vector
field

Inside a given generic stratum of $\mathcal{P}_{{\mathbb{C}},k+1}$, a vector
field of $X_{\varepsilon}\in\mathcal{P}_{{\mathbb{C}},k+1}$ is characterized
by the _analytic part_ of its invariant, which is defined as follows.

######  Theorem 2.11.

[DES05] Let $X_{\varepsilon}\in\mathcal{P}_{{\mathbb{C}},k+1}$ be a generic
vector field.

  1. (1)

The union of the separatrices, called the _separatrix graph_ , divides
${\mathbb{C}}$ into $k$ open regions, called _zones_ , adherent to exactly two
singular points, one attracting, one repelling. Each zone is the union of all
equivalent trajectories joining its two adherent points.

  2. (2)

In the rectifying coordinate $t=\int\frac{dz}{P_{\varepsilon}(z)}$, each zone
is biholomorphic to a horizontal strip. Images of $\infty$ (which can be
reached in finite time) appear, one on the upper boundary, one on the lower
boundary of the strip. Let $\eta\in{\mathbb{H}}$ be the vector joining the
lower image of $\infty$ to the upper one. Let $\theta\in(0,\pi)$ be the
argument of $\eta$. The image in $z$-space of the line segment $[0,\eta]$ in
$t$-space is a homoclinic loop of
$Y_{\varepsilon}=e^{-i\theta}X_{\varepsilon}$. This loop is included in the
zone. In particular, it has empty intersection with the separatrix graph and
intersects one edge of the planar tree graph. We call $\eta$ the _transversal
time_ of the zone.

  3. (3)

Using the marked points $e^{\prime}_{j}=e^{i\pi{2j-1\over 2k}}$, let
$\eta_{j}=\eta_{(2j-1),\tau(2j-1)}$ for $j\in\\{1,\ldots,k\\}$, where
$\eta_{\ell,\tau(\ell)}$ is the transversal time of the zone with ends at
infinity in the direction $e^{\prime}_{\ell}$ and $e^{\prime}_{\tau(\ell)}$
for odd $\ell$. The _analytic part of the invariant_ is the vector
$\eta=(\eta_{1},\ldots,\eta_{k})\in{\mathbb{H}}^{k}$.

  4. (4)

Any $X_{\varepsilon}\in\mathcal{P}_{{\mathbb{C}},k+1}$ is uniquely determined
by the combinatorial and analytic parts of its invariant.

Any stratum of generic vector fields is analytically parameterized by
${\mathbb{H}}^{k}$, namely the map $\mathbf{\eta}\mapsto\varepsilon$ is
holomorphic. In particular, any couple of a combinatorial part and an analytic
part is realizable by a unique generic vector field in
$\mathcal{P}_{{\mathbb{C}},k+1}$.

##  3\. Generic vector fields in $\mathcal{P}_{\mathbb{R},k+1}$

######  Definition 3.1.

A vector field $X\in\mathcal{P}_{\mathbb{R},k+1}$ is _generic_ if it is
structurally stable.

The following proposition is immediate.

######  Proposition 3.2.

A vector field $X\in\mathcal{P}_{\mathbb{R},k+1}$ is generic if and only if it
has only simple singular points and $\mathbb{R}$ is the only possible
homoclinic loop.

###  3.1. The combinatorial part of the invariant

One combinatorial invariant is presented in [BD10] in the general case of
polynomial vector fields with homoclinic loops and/or parabolic points. We
will not need the full generality, so we present a simplified version.

For a generic $X\in\mathcal{P}_{\mathbb{R},k+1}$, when there are singular
points on the real axis, the strata are completely determined by their
combinatorial type, i.e. a non-crossing involution without fixed points $\tau$
inside the unit circle linking 2 by 2 the marked points $e_{j}$ defined in (3)
so that:

  * •

The links do not cross;

  * •

A marked point $e_{j}$ can only be linked to a marked point $e_{j^{\prime}}$
such that either $j-j^{\prime}$ is odd and $jj^{\prime}>0$, or
$j^{\prime}=-j$;

  * •

If $e_{j}$ is linked to $e_{j^{\prime}}$ with $jj^{\prime}>0$, then $e_{-j}$
is linked to $e_{-j^{\prime}}$.

Hence it suffices to describe the linkings of points $e_{j}$ for $j>0$. We set
${\overline{\tau}}(j):=|\tau(j)|$ for $j>0$, which defines a non crossing
involution of $\\{1,\ldots,k\\}$.

There is a slight difference when there is no singular point on the real axis:
marked points in each hemicycle can only be paired with marked points of the
same hemicycle. But in that case $k$ is odd, and there is an odd number of
marked points. Hence no pairing using all marked points is possible and any
pairing of $k-1$ points linking $e_{1}$ to some $e_{j}$ ($j$ even), is
equivalent to the pairing where $e_{k}$ is linked to $e_{j}$.

######  Definition 3.3 (Combinatorial invariant for generic
$X\in\mathcal{P}_{\mathbb{R},k+1}$).

When there are singular points on the real axis, the combinatorial invariant
is the non crossing involution ${\overline{\tau}}$ of $\\{1,\ldots,k\\}$
described above. When the real axis is a homoclinic loop, the combinatorial
invariant is the non crossing involution ${\overline{\tau}}$ of
$\\{1,\ldots,k-1\\}$, with no fixed point described above.

Recovering the attachment of the separatrices to the singular points is
described in Proposition 2.10 when there is at least one real singular point.
Hence we only need to describe the case when the real axis is a homoclinic
loop.

######  Proposition 3.4.

Let $k$ be odd and consider the case where there are no real singular points.
There are $\frac{k+1}{2}$ singular points and $k-1$ free separatrices in each
of the upper and lower half-planes. Let $s_{j}$ be the separatrix of infinity
tangent to the direction $\exp\left(\frac{\pi ij}{2k}\right)$, $j=1,\dots,k-1$
in the upper half-plane. Let the permutation $\overline{\sigma}$ be defined on
$\\{1,\ldots,k-1\\}$ by

$\overline{\sigma}(j):={\overline{\tau}}(j+1)\quad({\rm mod}\>k-1).$

Each equivalence class of the quotient
$\\{1,\ldots,k-1\\}/{\overline{\sigma}}$ is the subset of the indices of the
separatrices landing at one singular point. The equivalence classes are noted
$[j]$.

Note that $j\mapsto-{\overline{\tau}}(-j)$ allows us to partition
$\\{-1,\ldots,-k+1\\}$ in a similar way for the separatrices in the lower
half-plane. In particular, we have $[-j]=-[j]$.

###  3.2. The analytic part of the invariant for a generic vector field of
$\mathcal{P}_{\mathbb{R},k+1}$

######  Proposition 3.5.

Let $X_{\varepsilon}\in\mathcal{P}_{\mathbb{R},k+1}$ be a generic vector
field, and let $m$ be the number of real singular points.

  1. (1)

The transversal times of zones symmetric with respect to the $\mathbb{R}$-axis
are elements of $i\mathbb{R}^{+}$ (these zones correspond to fixed point of
${\overline{\tau}}$). There are $m-1$ such transversal times.

  2. (2)

The transversal times of pairs of zones symmetric to one another with respect
to the $\mathbb{R}$-axis are of the form
$\eta_{j,{\overline{\tau}}(j)},-\overline{\eta_{j,{\overline{\tau}}(j)}}\in{\mathbb{H}}$
for $j,{\overline{\tau}}(j)\in\\{1,\dots,k\\}$, $j$ odd. In the case $m=0$,
then $j,{\overline{\tau}}(j)\in\\{1,\dots,k-1\\}$.

  3. (3)

When $m=0$, the period of the $\mathbb{R}$-axis is an element of
$\mathbb{R}^{+}$ that we call $\eta_{\mathbb{R}}$.

######  Definition 3.6.

The analytic part of the invariant is the vector $\eta$, whose coordinates are
the transversal times and the period of the $\mathbb{R}$-axis if applicable.
The order of the coordinates is described later in Definition 7.3.

##  4\. The generic strata of $\mathcal{P}_{\mathbb{R},k+1}$

We now specialize to vector fields of $\mathcal{P}_{\mathbb{R},k+1}$. The main
difference with $\mathcal{P}_{{\mathbb{C}},k+1}$ is that the real axis is
always invariant. And when there are no real singular points (an open
condition), it is a homoclinic loop through $\infty$.

Hence there are two types of generic strata in $\mathcal{P}_{\mathbb{R},k+1}$,
depending on the number $m$ of singular points on the real axis:

  * •

Strata of generic vector fields with singular points (of node type) on the
real axis ($m>0$). These vector fields are also generic inside
$\mathcal{P}_{{\mathbb{C}},k+1}$.

  * •

Strata of generic vector fields with $m=0$. Note that this case only occurs
when $k$ is odd. Then the real axis is a homoclinic loop of $\infty$, and such
a generic vector field is not generic in $\mathcal{P}_{{\mathbb{C}},k+1}$. In
each hemicycle there are $\frac{k+1}{2}$ singular points and $k-1$
separatrices. In the generic cases the singular points are linked as a planar
tree graph with trajectories, and the tree is linked to the separatrices. This
can be done in $C(\frac{k-1}{2})$ ways as determined in Proposition 2.6.

###  4.1. Number of generic strata

######  Theorem 4.1.

The number $D(k)$ of generic strata of vector fields in
$\mathcal{P}_{\mathbb{R},k+1}$ is

$D(k)=2\binom{k-1}{\lfloor\frac{k-1}{2}\rfloor}=\begin{cases}\binom{k}{k/2},&k\>\text{even},\\\
2\binom{k-1}{(k-1)/2},&k\>\text{odd}.\end{cases}$

For $k$ odd, the number of generic strata with no real singular points is
$C\left(\frac{k-1}{2}\right)$, the $\frac{k-1}{2}$-th Catalan number.

###### Proof.

The result follows from this recurrence formula that will be proved
geometrically:

$D(k+1)=\begin{cases}2D(k),&k\>\text{even},\\\
2\left(D(k)-C\left(\frac{k-1}{2}\right)\right),&k\>\text{odd}.\end{cases}$

(We will see that for $k$ odd, $C\left(\frac{k-1}{2}\right)$ is the number of
strata with no real singular points.)

Since we consider generic strata, the vector fields have $k+1$ simple singular
points and the real axis is invariant. Let $m$ be the number of singular
points on the real axis. Then $k+1-m=2\ell$ is even and there are $\ell$
singular points in each hemicycle.

Going from $k$ to $k+1$. Let us introduce the term $k$-stratum to specify the
value of $k$ for a given stratum. Also the marked points depend on $k$, so let
us define

(4) $e_{j,k}=\exp\left({i\pi{2j-\operatorname{sgn}(j)\over 2k}}\right),\qquad
j\in\\{\pm 1,\ldots,\pm k\\}.$

From any generic $k$-stratum with $m\neq 0$, we construct two different
generic $(k+1)$-strata. All the existing generic $(k+1)$-strata are obtained
in this way, and all the $(k+1)$-strata obtained in the construction are
distinct. Let us now describe the construction. Let us start with a generic
$k$-stratum with $m\neq 0$. It has $m$ singular points on the real axis and
symmetric branches (possibly void) starting from the real singular points.
Hence each real singular point is the root of a tree (possibly with one
vertex) in the upper hemicycle, and a root of the symmetric image of the tree
in the lower hemicycle.

Let $z_{m}<z_{m-1}<\dots<z_{1}$ be the real singular points for the
$k$-stratum. The construction consists in adding a singular point $z_{s}$ on
the real axis to the left of $z_{m}$ and two extra separatrices on both sides
of the negative real axis. In practice, this is the $k+1$-(stratum) of
$P(z)+\eta z^{k+2}$ with $\eta>0$ small. Before adding $z_{s}$, the negative
infinite part of the real axis is a separatrix landing in $z_{m}$. After
adding $z_{s}$, this separatrix splits into the two new separatrices which
again land in $z_{m}$ and the negative infinite part of the real axis becomes
a separatrix landing in $z_{s}$. This gives the first new generic
$(k+1)$-stratum (see Figures 3 and 4).

As for the second generic $(k+1)$-stratum, we let $z_{s}$ coallesce with
$z_{m}$ and the two points move together as complex conjugates outside the
real axis. For instance, $z_{m}$ moves in the upper hemicycle, and $z_{s}$ in
the lower one. Then $z_{m}$ brings with it its attached tree of singular
points, which remain attached to their separatrices. The new separatrices
created in the construction are attached to $z_{m}$ and $z_{s}$ respectively.
All the other separatrices keep the same attachment. Two cases now occur. If
$m=1$, then the new vector field has no real singular point and the real axis
is a homoclinic loop (Figure 4). Otherwise, the negative infinite part of the
real axis becomes a separatrix landing at $z_{m-1}$ (Figure 3).

Figure 3. The two $(k+1)$-strata produced from a $k$-stratum when $m>1$.
Figure 4. The two $k+1$-strata produced from a $k$-stratum when $m=1$.

Injectivity of the construction.

When adding the extra singular point $z_{s}$, the old marked points $e_{j,k}$
are moved to $e_{j,k+1}$ and two new marked points are added, namely
$e_{\pm(k+1),k+1}$. If $z_{s}$ is on the real axis the new combinatorial
invariant associates $e_{k+1,k+1}$ with $e_{-(k+1),k+1}$ and the rest of the
linking is the same as before.

If $z_{m}$ and $z_{s}$ are complex conjugates, and $m>1$, then the $k$-pairing
for the $k$-stratum associates some $e_{j,k}$ to $e_{-j,k}$ with a link
cutting the segment $[z_{m},z_{m-1}]$. Then for the $(k+1)$-stratum,
$e_{k+1,k+1}$ will be linked to $e_{j,k+1}$ and the rest of the pairing is
identical. If $z_{m}$ and $z_{s}$ are complex conjugates, and $m=1$, then the
new pairing for the $(k+1)$-stratum is the same as before for all $e_{j,k+1}$,
$j=1,\dots k$, and as discussed above, pairing $e_{k+1,k+1}$ is irrelevant.
(See also Table 1.)

Condition |  $k$-pairing |  $\qquad\qquad\quad(k+1)$-pairing
---|---|---
$z_{s}<z_{m}$ | $\tau_{k}$ | $\begin{cases}\tau_{k+1}(j)=\tau_{k}(j),&1\leq j\leq k\\\ \tau_{k+1}(k+1)=-(k+1),\end{cases}$
$z_{m}=\overline{z}_{s}\in{\mathbb{H}}$, $m>1$ | $\begin{cases}\tau_{k}(j)=-j\\\ \tau_{k}(j^{\prime})\neq-j^{\prime},&j^{\prime}>j\end{cases}$ | $\begin{cases}\tau_{k+1}(j)=k+1\\\ \tau_{k+1}(\ell)=\tau_{k}(\ell),&\ell\neq j,k+1\end{cases}$
$z_{m}=\overline{z}_{s}\in{\mathbb{H}}$, $m=1$ | $\tau_{k}$ | $\begin{cases}\tau_{k+1}(j)=\tau_{k}(j),&1\leq j\leq k\\\ \tau_{k+1}(k+1)\>\text{irrelevant}\end{cases}$
Table 1. Changes in the pairing when passing from a $k$-stratum with pairing
$\tau_{k}$ to a $(k+1)$-stratum with pairing $\tau_{k+1}$.

Any pairing of a $(k+1)$-stratum can be uniquely obtained by this method from
a pairing of a $k$-stratum with at least one real singular point and either
adding $z_{s}$ on the real axis or as a complex conjugate of $z_{m}$. Indeed,
if $m>0$ for the $(k+1)$-stratum, then its tree graph is symmetric with
respect to the real axis. If this tree graph has one end on the leftmost side
of the real axis then it can only come from a $k$-stratum by adding $z_{s}$ on
the real axis. Otherwise, it comes from a $k$-stratum with $z_{m}$ and $z_{s}$
complex conjugates. ∎

######  Corollary 4.2.

For any value of $k$, half of the strata with real singular points have the
leftmost singular point with no attached branches to it in the hemicycles.
Similarly, half of the strata with real singular points have the rightmost
singular point with no branches in the hemicycles.

###### Proof.

Indeed this is the case for $k=1$ and $k=2$. For any higher $k$ and for the
leftmost point, it follows from the construction in the proof of Theorem 4.1.
The case of the rightmost case follows by symmetry using $z\mapsto-z$. ∎

######  Definition 4.3.

A _dispersed Dyck path of length $n$_ is a path in $\mathbb{N}^{2}$ from
$(0,0)$ to $(n,0)$ with steps $(1,1)$, $(1,-1)$ and $(1,0)$ and with no steps
$(1,0)$ at positive height.

The number of dispersed Dyck paths of length $n$ is well known in the
literature. We provide a proof for the sake of completeness.

######  Proposition 4.4.

The number of dispersed Dyck paths of length $n$ is given by
$\binom{n}{\lfloor\frac{n}{2}\rfloor}$.

###### Proof.

Let $N=\begin{cases}0,&n\>\text{even},\\\ -1,&n\>\text{odd}.\end{cases}$. Let
us show that the number of dispersed Dyck paths of length $n$ is equal to the
number of paths of length $n$ from $(0,0)$ to $(n,N)$ with steps $(1,1)$ and
$(1,-1)$. This latter number is obviously equal to the number of choices for
the steps $(1,1)$, namely $\binom{n}{\lfloor\frac{n}{2}\rfloor}$. The
bijection is constructed as follows.

Consider a dispersed Dyck path. Let the steps $(1,0)$ occur between
$(n_{i},0)$ and $(n_{i}+1,0)$, $i=1,\dots,\ell$. A step $(1,0)$ occuring
between $(n_{i},0)$ and $(n_{i}+1,0)$ is replaced by a step $(1,-1)$ (resp.
$(1,1)$) if $i$ is odd (resp. even). Moreover, for $i$ odd, the part
$\gamma_{i}$ of the path between $(n_{i}+1,0)$ and $(n_{i+1},0)$ is replaced
by $-\gamma_{i}-(0,1)$.

The inverse map is constructed as follows. Consider a path of length $n$ from
$(0,0)$ to $(n,N)$ with steps $(1,1)$ and $(1,-1)$. We change each descending
step from level $0$, namely a step from some $(n_{i},0)$ to $(n_{i}+1,-1)$, to
a step from $(n_{i},0)$ to $(n_{i}+1,0)$. Each ascending step to level $0$
from some $(n_{i},-1)$ to $(n_{i}+1,0)$ is changed to a step from $(n_{i},0)$
to $(n_{i}+1,0)$. And each portion $\delta_{i}$ of the path between
$(n_{i}+1,-1)$ and $(n_{j},-1)$, $n_{j}>n_{i}$ located below the line $y=-1$
is changed to $-\delta_{i}-(0,1)$. ∎

######  Corollary 4.5.

  1. (1)

For $k$ even, the generic strata in $\mathcal{P}_{\mathbb{R},k+1}$ are in
bijection with the set of dispersed Dyck paths of length $k$, whose number is
$Dd(k)=\binom{k}{k/2}$.

  2. (2)

For $k$ odd, the strata generic in both $\mathcal{P}_{\mathbb{R},k+1}$ and
$\mathcal{P}_{{\mathbb{C}},k+1}$ (i.e. at least one real fixed point) are in
bijection with the set of dispersed Dyck paths of length $k$, whose number is
$Dd(k)=\binom{k}{\lfloor k/2\rfloor}$. The number of generic strata with no
real singular points is $C\left(\frac{k-1}{2}\right)$. Hence

$D(k)=Dd(k)+C\left(\frac{k-1}{2}\right)=2\binom{k-1}{(k-1)/2}.$

###### Proof.

Recall that for even $k$ or for odd $k$ with at least one real singular point,
any generic vector field of $\mathcal{P}_{\mathbb{R},k+1}$ is generic in
$\mathcal{P}_{{\mathbb{C}},k+1}$. Because of the symmetry, to describe a non-
crossing involution $\tau$ between the marked points, it suffices to describe
the images $\tau(j)$ for $j=1,\dots,k$ as steps of a dispersed Dyck path:

  * •

The step is $(1,1)$ if $\tau(j)=j^{\prime}$ with $j^{\prime}>j$;

  * •

The step is $(1,-1)$ if $\tau(j)=j^{\prime}$ with $j^{\prime}<j$;

  * •

The step is $(1,0)$ if $\tau(j)=-j$.

∎

###  4.2. Description of the strata

In this section, we count the number of generic strata with another approach,
linked to their geometry.

######  Theorem 4.6.

Let $D(k,m)$ be the number of generic strata with exactly $m$ real singular
points. Then

(5)
$D(k,m)=\begin{cases}\displaystyle\sum\limits_{\begin{subarray}{c}n_{1}+\cdots+n_{m}=\frac{k-m+1}{2}\\\
n_{j}\geq 0\end{subarray}}C(n_{1})\cdots C(n_{m}),&m>0,\\\\[22.76219pt]
C\bigl{(}\frac{k-1}{2}\bigr{)},&m=0,\end{cases}$

where $C(n)$ is the $n$-th Catalan number. The number $D(k)$ of generic strata
is

$D(k)=\begin{cases}\displaystyle\sum_{m\in\\{1,3,\dots,k+1\\}}D(k,m),&k\
\text{even},\\\ \displaystyle\sum_{m\in\\{0,2,\dots,k+1\\}}D(k,m),&k\
\text{odd}.\end{cases}$

This yields the following combinatorial formulas

$D(k)=\begin{cases}\displaystyle\sum_{m\in\\{1,3,\dots,k+1\\}}\sum_{\begin{subarray}{c}n_{1}+\cdots+n_{m}=\frac{k-m+1}{2}\\\
n_{j}\geq 0\end{subarray}}C(n_{1})\cdots C(n_{m})=\binom{k}{k/2},&k\
\text{even},\\\
\displaystyle\sum_{m\in\\{2,4,\dots,k+1\\}}\sum_{\begin{subarray}{c}n_{1}+\cdots+n_{m}=\frac{k-m+1}{2}\\\
n_{j}\geq 0\end{subarray}}C(n_{1})\cdots\textstyle
C(n_{m})+C\\!\left(\frac{k-1}{2}\right)\\\ \hskip
190.00029pt=2\binom{k-1}{(k-1)/2},&k\ \text{odd}.\end{cases}$

###### Proof.

Let $m$ be the number of real singular points, which is odd (resp. even) when
$k$ is even (resp. odd). In all cases, $k-m+1$ is even, and there are
${k-m+1\over 2}$ singular points in each of the upper and lower half-planes.
As previously, each real singular point is the root of two symmetric trees,
one in the closed upper half-plane, called _upper tree_ , and its mirror image
in the closed lower half-plane. Because of the symmetry, we only discuss the
upper half-plane.

Let us first discuss $m>0$, and let $z_{m}<\ldots<z_{1}$ be the real singular
points and $n_{j}+1$ be the number of vertices of the upper tree rooted at
$z_{j}$, so $n_{j}$ is the number of edges of the upper tree and also the
number of non real vertices. Then we have the relation

$\sum_{j=1}^{m}n_{j}={k-m+1\over 2}.$

For $n_{1},\ldots,n_{m}$ fixed, then $C(n_{1})\cdots C(n_{m})$ is the number
of possible ‘‘ordered’’ forests of $m$ trees with the $j$-th tree having
$n_{j}$ edges, where $C(n)=\frac{1}{n+1}{2n\choose n}$ is the $n$-th Catalan
number, which counts the number of rooted trees with $n$ edges. Therefore, the
number of forests with $m$ rooted trees with a total number of
$\frac{k-m+1}{2}$ edges is $D(k,m)$ given in (5).

In the case $m=0$, there are $\frac{k+1}{2}$ singular points ($\frac{k-1}{2}$
edges) and $k-1$ separatrices in the upper half-plane that form a tree.
Quotienting the real axis to a regular point, this is completely similar to
the DES case for $k^{\prime}=\frac{k-1}{2}$ and the number of configurations
of a tree attached to the separatrices is $D(k,0)=\textstyle
C\bigl{(}k^{\prime}\bigr{)}=\textstyle C\bigl{(}\frac{k-1}{2}\bigr{)}.$ ∎

##  5\. Structure of the family $\mathcal{P}_{\mathbb{R},k+1}$

The bifurcations occuring in the family $\mathcal{P}_{\mathbb{R},k+1}$ are of
the following types:

  1. (1)

Bifurcations of homoclinic loops through $\infty$. The homoclinic loop along
the real axis occurs on an open set in parameter space while the other
bifurcations of homoclinic loops occur on algebraic sets of real codimension
1.

  2. (2)

Bifurcation of parabolic singular points. The bifurcation of multiple singular
points of multiplicity $n+1$ occurs in a set of real codimension $n$ when the
singular point is on the real axis and of complex codimension $n$ otherwise.
We will show below that full unfoldings exist inside the family
$\mathcal{P}_{\mathbb{R},k+1}$.

  3. (3)

Intersection of the former types.

The family $\mathcal{P}_{\mathbb{R},k+1}$ of polynomial vector fields with
real coefficients $P_{\varepsilon}(z)\frac{\partial}{\partial z}$ for

$P_{\varepsilon}(z)=z^{k+1}+\varepsilon_{k-1}z^{k-1}+\dots+\varepsilon_{1}z+\varepsilon_{0}$

with $\varepsilon\in\mathbb{R}^{k}$ is as complete as possible under the
constraint that the real axis is invariant.

######  Theorem 5.1.

Let $k\geq 2$ and $z_{0}$ be a non real singular point of
$P_{\varepsilon^{*}}(z)\frac{\partial}{\partial z}$ for a particular value
$\varepsilon^{*}$ of $\varepsilon$. Then a full unfolding exists in the family
$\mathcal{P}_{\mathbb{R},k+1}$.

###### Proof.

Let $s$ be the multiplicity of $z_{0}$: $z_{0}$ is simple if $s=1$ and
multiple otherwise. We can of course suppose that $z_{0}=a+i$, with
$a\in\mathbb{R}$. Then
$P_{\varepsilon^{*}}(z)=(z-a-i)^{s}(z-a+i)^{s}Q_{\varepsilon^{*}}(z)$, where
${\rm deg}(Q_{\varepsilon^{*}})=\ell=k+1-2s$ and $Q_{\varepsilon^{*}}$ is a
polynomial with real coefficients.

If $\ell>0$, let us consider the following unfolding

$P_{\varepsilon(\nu)}(z)=R_{\varepsilon(\nu)}(z)\overline{R}_{\varepsilon(\nu)}(z)Q_{\varepsilon(\nu)}(z),$

with

$\begin{cases}R_{\varepsilon(\nu)}(z)=(z-a-i)^{s}+\sum_{j=0}^{s-1}(\delta_{j}+i\eta_{j})(z-a-i)^{j},\\\\[6.0pt]
\overline{R}_{\varepsilon(\nu)}(z)=(z-a+i)^{s}+\sum_{j=0}^{s-1}(\delta_{j}-i\eta_{j})(z-a+i)^{j},\\\\[6.0pt]
Q_{\varepsilon(\nu)}(z)=Q_{\varepsilon^{*}}(z)-2\delta_{s-1}z^{\ell-1}+\sum_{j=0}^{l-2}\mu_{j}z^{j},\end{cases}$

where
$\nu=(\delta_{0},\eta_{0},\dots,\delta_{s-1},\eta_{s-1},\mu_{0},\dots\mu_{\ell-2})\in\mathbb{R}^{k}$.
Let $\varepsilon(\nu)=\varepsilon^{*}+\nu$. Then
$P_{\varepsilon(\nu)}\in\mathcal{P}_{\mathbb{R},k+1}$ and, from its form,
$P_{\varepsilon(\nu)}$ is a complete unfolding of $P_{\varepsilon^{*}}$ in the
neighborhood of $a+i$, since
$\overline{R}_{\varepsilon(\nu)}(z)Q_{\varepsilon(\nu)}(z)=C+O(\nu)+O(z-a-i)$,
i.e. $R_{\varepsilon(\nu)}(z)$ is the Weierstrass polynomial associated to
$a+i$. By symmetry, $P_{\varepsilon(\nu)}$ is a complete unfolding of
$P_{\varepsilon^{*}}$ in the neighborhood of $a-i$. Using induction on the
degree of $P_{\varepsilon^{*}}$, we can also conclude that
$P_{\varepsilon(\nu)}$ is a complete unfolding of $P_{\varepsilon^{*}}$ in the
neighborhood of each zero of $Q_{\varepsilon^{*}}$.

Let us now consider the case $\ell=0$. In this case, necessarily $a=0$, and we
consider the unfolding
$P_{\varepsilon(\nu)}(z)=R_{\varepsilon(\nu)}(z)\overline{R}_{\varepsilon(\nu)}(z)$
with

$\begin{cases}R_{\varepsilon(\nu)}(z)=(z-i)^{s}+\sum_{j=0}^{s-2}(\delta_{j}+i\eta_{j})(z-i)^{j}+i\eta_{s-1}(z-i)^{s-1},\\\
\overline{R}_{\varepsilon(\nu)}(z)=(z+i)^{s}+\sum_{j=0}^{s-2}(\delta_{j}-i\eta_{j})(z+i)^{j}-i\eta_{s-1}(z-i)^{s-1},\end{cases}$

where
$\nu=(\delta_{0},\eta_{0},\dots,\delta_{s-2},\eta_{s-2},\eta_{s-1})\in\mathbb{R}^{k}$.
Then $P_{\varepsilon(\nu)}$ is a complete unfolding of $P_{\varepsilon^{*}}$
in the neighborhood of $\pm i$. ∎

######  Remark 5.2.

Consider $P_{\varepsilon}$ generic. Since the eigenvalues of real singular
points are real and the eigenvalues of complex conjugate points are complex
conjugate, then $k+1$ real parameters would be needed if the eigenvalues were
to be independent. But the eigenvalues of the singular points
$z_{1},\dots,z_{k+1}$ of $P_{\varepsilon}$ satisfy
$\sum_{j=1}^{k+1}\frac{1}{P_{\varepsilon}^{\prime}(z_{j})}=0$. Since this sum
is real, this is a condition of real codimension 1. This explains why a
generic $P_{\varepsilon}$ depends on $k$ real parameters.

##  6\. The case $k=3$

######  Theorem 6.1.

The bifurcation diagram of the vector field
$(z^{4}+\varepsilon_{2}z^{2}+\varepsilon_{1}z+\varepsilon_{0})\frac{\partial}{\partial
z}$ is given in Figure 5. The bifurcation diagram has a conic structure using
(2). Its intersection with a sphere minus a point in the regular stratum with
no real singular point is given in Figure 6.

Figure 5. Bifurcation diagram of
$(z^{4}+\varepsilon_{2}z^{2}+\varepsilon_{1}z+\varepsilon_{0})\frac{\partial}{\partial
z}$. Figure 6. The intersection of the bifurcation diagram of
$(z^{4}+\varepsilon_{2}z^{2}+\varepsilon_{1}z+\varepsilon_{0})\frac{\partial}{\partial
z}$ with a sphere minus a point.

###### Proof.

The codimension 1 bifurcations are of two types: real parabolic points of
multiplicity 2 and pairs of homoclinic loops symmetric with respect to the
real axis. The higher order bifurcations occur at the multiple points, located
either at the boundary of the codimension 1 bifurcation surfaces, or at the
intersection of the codimension 1 bifurcation surfaces. Note that a complex
double parabolic point has real codimension 2.

Bifurcations of parabolic points. They occur when the discriminant
$\Delta=256\varepsilon_{0}^{3}-27\varepsilon_{1}^{4}+144\varepsilon_{0}\varepsilon_{1}^{2}\varepsilon_{2}-128\varepsilon_{0}^{2}\varepsilon_{2}^{2}-4\varepsilon_{1}^{2}\varepsilon_{2}^{3}+16\varepsilon_{0}\varepsilon_{2}^{4}$
vanishes. The locus where $\Delta=0$ corresponding to multiple real roots is
well known: it is the swallow tail. There is also the locus of nonreal pairs
of parabolic points $(z_{0},\overline{z}_{0})$. Because their sum vanishes,
then $z_{0}=ia$ for some $a\in\mathbb{R}^{*}$. This occurs along the curve
$\varepsilon_{1}=0$, $\varepsilon_{2}^{2}-4\varepsilon_{0}=0$ and
$\varepsilon_{2}>0$. (Note that the other part of the curve
$\varepsilon_{1}=0$, $\varepsilon_{2}^{2}-4\varepsilon_{0}=0$,
$\varepsilon_{2}<0$, corresponds to a pair of real parabolic points: this is
the self-intersection curve of the swallow tail.)

Bifurcations of homoclinic loops. Since the real axis is invariant, it is a
homoclinic loop as soon as there are no real singular point. This occurs in an
open region in parameter space, and it is not a bifurcation. Otherwise,
homoclinic loops occur in symmetric pairs when one singular point has a pure
imaginary eigenvalue. There are two cases to consider: $m=0$ and $m=2$.

When $m=2$, then $P$ has the form $P(z)=(z-2a)(z-2b)((z+a+b)^{2}+c^{2})$ with
$c\in\mathbb{R}^{*}$. $P^{\prime}(-a-b\pm ic)$ is pure imaginary only if
$(a+b)c^{2}=0$. Hence $a+b=0$, $P(z)=(z-2a)(z+2a)(z^{2}+c^{2})$ and
$P^{\prime}(\pm ic)=\mp 2ic(4a^{2}+c^{2})\neq 0$. This occurs on
$\varepsilon_{0}<0$, $\varepsilon_{1}=0$. (Note that necessarily
$\varepsilon_{0}<0$ when $\varepsilon_{1}=0$ and $m=2$.) When
$\varepsilon_{0}=0$, this is the particular case of a parabolic real point at
$z=0$.

When $m=0$, then $P$ has the form $P(z)=((z-a)^{2}+b^{2})((z+a)^{2}+c^{2})$
for $b,c\in\mathbb{R}^{*}$. Then $P^{\prime}(a+ib)\in i\mathbb{R}^{*}$ if and
only if $a=0$ and $b\neq 0,\pm c$. This occurs for $\varepsilon_{1}=0$,
$\varepsilon_{2}^{2}-4\varepsilon_{0}>0$ and
$\varepsilon_{0},\varepsilon_{2}>0$. This surface is bounded by
$\varepsilon_{1}=\varepsilon_{0}=0$, $\varepsilon_{2}>0$ where a real
parabolic point occurs at $z=0$ and homoclinic loops around $\pm
i\sqrt{\varepsilon_{2}}$, and $\varepsilon_{1}=0$,
$\varepsilon_{2}^{2}-4\varepsilon_{0}=0$, $\varepsilon_{2}>0$ where a pair of
symmetric purely imaginary parabolic points occurs (and no homoclinic loops).

∎

##  7\. Realization

We briefly discussed the invariants of generic vector fields of
$\mathcal{P}_{\mathbb{R},k+1}$ in Section 3. In this section, we precisely
define their modulus of classification and we prove that any modulus is
realizable as a generic vector field of $\mathcal{P}_{\mathbb{R},k+1}$.

###  7.1. Properties of the combinatorial and analytic invariants

For a generic vector field of $\mathcal{P}_{\mathbb{R},k+1}$, recall that the
combinatorial invariant is a non crossing involution (see Definition 3.3) and
the analytic invariants are the transversal times and potentially the period
of a homoclinic loop along $\mathbb{R}$. From Propositions 2.10 and 3.4 we can
recover how the separatrices attach to the singular points from the
combinatorial data. For the rest of the section, we use the notation $z_{[j]}$
for a singular point, where $[j]$ is an equivalence class of separatrices
landing at the same singular point. The combinatorial data tells us how many
singular points are real, as seen in Lemma 7.1 below. Indeed,
$\overline{z_{[j]}}=z_{[-j]}$, hence $z_{[j]}$ is real if and only if
$[j]=[-j]$.

Recall that the separatrix graph of $X$ divides ${\mathbb{C}}$ into simply
connected zones. We denote a zone with two ends in the direction $e_{j},e_{k}$
by $Z_{j,k}$ (or $Z_{k,j}$ interchangeably), and the transversal time in
$Z_{j,k}$ by $\eta_{j,\tau(j)}$.

The following lemma explains how to recover some features of the dynamics,
including the presence of a homoclinic loop along the real axis and the number
of real singular points from the combinatorial data.

######  Lemma 7.1.

Let $X=P\frac{\partial}{dz}\in\mathcal{P}_{\mathbb{R},k+1}$ be a generic
vector field. Let ${\overline{\tau}}$ be the combinatorial invariant of $X$:
its domain $\operatorname{dom}({\overline{\tau}})$ is a set of cardinality
$k-1$ (resp. $k$) if $\mathbb{R}$ is a homoclinic loop (resp. otherwise). Let
$\ell=k-\\#\operatorname{dom}({\overline{\tau}})$. (Then $\ell=1$ if
$\mathbb{R}$ is a homoclinic loop, and $\ell=0$ otherwise.)

  1. (1)

${\overline{\tau}}$ is equivalent to the non crossing involution with no fixed
points, $\tau$, of $\\{\pm 1,\ldots,\pm(k-\ell)\\}$ that pairs the ends of
zones of $X$.

  2. (2)

The number $m\in\mathbb{N}$ of real singular points of $X$ is given by

$\displaystyle m+\ell-1$ $\displaystyle=\text{number of fixed points of
${\overline{\tau}}$,}$ $\displaystyle=\text{number of pairs
$\left(j,\tau(j)\right)$ with $j>0$ and $\tau(j)<0$,}$
$\displaystyle=\text{number of zones whose interior intersects $\mathbb{R}$.}$

  3. (3)

The transversal times satisfy
$\eta_{j,\tau(j)}=-\overline{\eta_{-j,\tau(-j)}}$. In particulier, there are
$m+\ell-1$ transversal times in $i\mathbb{R}_{+}$.

###### Proof.

For (1), $\tau$ and $j\mapsto-j$ commute since $P$ commutes with the complex
conjugation. Then, we have

$\tau(j)=\begin{cases}{\overline{\tau}}(j),&{\overline{\tau}}(j)\neq j,\\\
-j,&{\overline{\tau}}(j)=j,\end{cases}\hbox to0.0pt{\qquad for $j>0$,\hss}$

and $\tau(-j)=-\tau(j)$ for $j<0$.

For (2), if $\ell=1$, then ${\overline{\tau}}$ has no fixed points, so $m=0$
is the number of fixed points of ${\overline{\tau}}$. For the case $\ell=0$,
let $n$ be the number of zones with an interior intersecting the real axis.
This divides the real axis in $n$ finite intervals, each of which is an orbit
of $P\frac{\partial}{\partial z}$ and, in addition, the two separatrices,
along $\mathbb{R}_{+}$ and $\mathbb{R}_{-}$ on the boundary of some zones.
This partitions $\mathbb{R}$ into $n+2$ orbits and $n+1$ singular points, one
between each orbit. We conclude that $m=n+1$.

(3) is a direct computation. ∎

######  Notation 7.2.

We will use the notation
$\operatorname{Fix}({\overline{\tau}}):=\\{j\mid{\overline{\tau}}(j)=j\\}$.

For $m=0$, we have ${k-1\over 2}$ zones in the upper half-plane. We can define
a transversal time for each of them. We also define the period of the
$\mathbb{R}$-axis

$\eta_{\mathbb{R}}=\int_{\mathbb{R}}\frac{{\rm d}z}{P(z)},$

which is an invariant. Note that $\eta_{\mathbb{R}}>0$ since we only consider
the case where the leading coefficient of $P$ is positive.

######  Definition 7.3 (Modulus of classification of a generic vector field).

Let $P\frac{\partial}{\partial z}$ be a generic vector field of
$\mathcal{P}_{\mathbb{R},k+1}$. Let $m$ be the number of real roots of $P$,
and $\ell$ be the number of homoclinic loops (either 0 or 1). The modulus of
invariants of $P$ is a triple $({\overline{\tau}},\eta)$, where
${\overline{\tau}}$ is a non crossing involution of $\\{1,\ldots,k-\ell\\}$
with $m-1+\ell$ fixed points, and $\eta\in{\mathbb{H}}^{{k-m+1\over
2}-\ell}\times(i\mathbb{R}^{+})^{m-1+\ell}\times(\mathbb{R}^{+})^{\ell}$ is
the vector whose coordinates are the transversal times and period of
$\mathbb{R}$, ordered as follows (see Figure 7):

  1. (1)

let $0<q_{1}<\cdots<q_{(k-m+1)/2-\ell}$ be the odd numbers in
$\\{1,\dots,k-\ell\\}$ which are not fixed points of ${\overline{\tau}}$; then
we set $\eta_{j}$ as the transversal time for the zone
$Z_{q_{j},{\overline{\tau}}(q_{j})}$;

  2. (2)

next, we consider $0<r_{1}<\cdots<r_{m-1}$ in
$\operatorname{Fix}({\overline{\tau}})$, and we set $\eta_{j+(k-m+1)/2-\ell}$
as the transversal time in $Z_{r_{j},-r_{j}}$;

  3. (3)

lastly, if $\ell=1$, the last coordinate is
$\eta_{\mathbb{R}}=\int_{\mathbb{R}}\frac{dz}{P(z)}\in\mathbb{R}^{+}$.

(a) Zones with $k=7$, $m=4$ and $\ell=0$. (b) Zones with $k=7$, $m=0$ and
$\ell=1$.

Figure 7. Zones with marked ends
$e_{j}=\exp\left(i\pi{2j-\operatorname{sgn}(j)\over 2k}\right)$. Transversal
times of zones in symmetric pairs are ordered first, then of zones crossing
the real line (if applicable), then the period of the homoclinic loop (if
applicable).

###  7.2. Realisation

This is the main result of the section.

######  Theorem 7.4 (Realisation and moduli space).

Let $\mathcal{S}_{k}$ be the set of non crossing involutions of
$\\{1,\ldots,k\\}$, and $\mathcal{S}_{k-1}^{*}$ be the set of the non crossing
involutions of $\\{1,\dots,k-1\\}$ without fixed point. (Note that
$\mathcal{S}_{k-1}^{*}$ is empty when $k$ is even.) For
${\overline{\tau}}\in\mathcal{S}_{k}\cup\mathcal{S}_{k-1}^{*}$, let
$\ell=k-\\#\operatorname{dom}({\overline{\tau}})$ and
$m=\\#\operatorname{Fix}({\overline{\tau}})+1-\ell$. The moduli space
$\mathcal{M}_{k}$ of the generic vector fields in
$\mathcal{P}_{\mathbb{R},k+1}$ is

$\mathcal{M}_{k}=\bigcup_{{\overline{\tau}}\in\mathcal{S}_{k}\cup\mathcal{S}_{k-1}^{*}}\\{{\overline{\tau}}\\}\times{\mathbb{H}}^{{k-m+1\over
2}-\ell}\times(i\mathbb{R}^{+})^{m-1+\ell}\times(\mathbb{R}^{+})^{\ell}.$

For each $({\overline{\tau}},\eta)\in\mathcal{M}_{k}$, there exists a unique
monic centered generic polynomial $P$ of degree $k+1$ with real coefficients
such that $P\frac{\partial}{\partial z}$ has $({\overline{\tau}},\eta)$ for
its modulus, $m$ real singular points and $\ell$ homoclinic loop.

######  Remarks 7.5.

  1. (1)

The theorem is stated for polynomials with a leading coefficient of $+1$. When
$k$ is odd, the polynomials
$P(z)=z^{k+1}+\varepsilon_{k-1}z^{k-1}+\cdots+\varepsilon_{0}$ and
$Q(z)=-z^{k+1} -\varepsilon_{k-1}z^{k-1}-\cdots-\varepsilon_{0}$ have the same
invariants $(\overline{\tau},\eta)$, but they are not equivalent. Indeed, they
have the same orbits, but with reversed orientation. We could add a sign to
the modulus for $k$ odd, but since we are only concerned with polynomials with
a positive leading coefficient, we omit this addition.

  2. (2)

In all cases, the total number of real parameters is $k$ as expected since a
generic stratum is an open set of $\mathbb{R}^{k}$.

The proof has two main steps:

  * •

1\. the construction of a Riemann surface $M$ conformally equivalent to a
sphere punctured in $k+1$ points, on which the vector field
$1\frac{\partial}{\partial Z}$ is well-defined and with a distinguished point
at infinity; the polynomial vector field is then obtained as the pushforward
of $1\frac{\partial}{\partial z}$ on ${\mathbb{C}}$ punctured in $k+1$ points
and holomorphically extended with singular points at the punctures;

  * •

2\. defining a complex conjugation on $M$ to prove that the polynomial vector
field has real coefficients.

The construction of the Riemann surface is done in [DES05] in the DES-generic
case, and in [BD10] when there are homoclonic loops. We will briefly go over
the parts of the construction needed for the proof of our theorem so that the
paper be self-contained.

#### Construction of a rectifying Riemann surface.

Let $\ell=k-\\#\operatorname{dom}({\overline{\tau}})$ and
$m=\\#\operatorname{Fix}({\overline{\tau}})+1-\ell$.

We consider $k-\ell$ horizontal strips grouped in two types. (See Figure 8(b)
for $\ell=0$ and Figure 9(b) for $\ell=1$.)

  1. (1)

For each $j\in\operatorname{Fix}({\overline{\tau}})$, set

$B_{j,\tau(j)}:=\bigl{\\{}z\in{\mathbb{C}}:\textstyle|\Im
z|<\frac{\Im\eta_{j,\tau(j)}}{2}\bigr{\\}}$

and $B_{-j,\tau(-j)}:=B_{j,\tau(j)}$. We define the points
$E_{j}:=(-1)^{j}\frac{\eta_{j,\tau(j)}}{2}$ and $E_{\tau(j)}=-E_{j}$ on the
boundary of $B_{j,\tau(j)}$.

  2. (2)

For each
$j\in\\{1,\ldots,k-\ell\\}\setminus\operatorname{Fix}({\overline{\tau}})$, set

$B_{j,\tau(j)}=\bigl{\\{}z\in{\mathbb{C}}:\textstyle 1<\Im
z<\Im\eta_{j,\tau(j)}+1\bigr{\\}}$

and $B_{-j,\tau(-j)}:=\overline{B_{j,\tau(j)}}$ (the complex conjugate of
$B_{j,\tau(j)}$). We define the point $E_{j}=i$ for $j$ odd or
$E_{j}=\eta_{j,\tau(j)}+i$ for $j$ even, then we define
$E_{\tau(j)}=E_{j}+(-1)^{j+1}\eta_{j,\tau(j)}$. Lastly, set
$E_{-j}=\overline{E_{j}}$ and $E_{\tau(-j)}=\overline{E_{\tau(j)}}$. In the
case $\ell=1$, we set $E_{k}=E_{1}-\eta_{\mathbb{R}}$ in $B_{1,\tau(1)}$ and
$E_{-k}=\overline{E_{k}}$ in $B_{-1,\tau(-1)}$.

We then have $m-1+\ell$ symmetric strips and $\frac{k-m+1}{2}-\ell$ pairs of
symmetric strips, for a total of $k-\ell$ horizontal strips, and $2k$ marked
points on the boundaries of the strips. On each strip, the vector field
$1\frac{\partial}{\partial Z}$ is well-defined. The first type of strips will
correspond to zones intersecting the real axis and the real line in the strip
will be mapped on an interval of the real axis. The other type of strips will
correspond to pairs of zones symmetric with respect to the real axis.

Let us consider a zone not containing the real axis as a homoclinic loop in
its boundary. For $j>0$ and odd, the marked point $E_{j}$ divides the lower
boundary of the strip in half-lines $S_{j-1}$ on the left and $S_{j}$ on the
right, and the marked point $E_{\tau(j)}$ divides the upper boundary of the
strip in half-lines $S_{\tau(j)}$ on the left and $S_{\tau(j)-1}$ on the right
if $\tau(j)>0$ (resp. half-lines $S_{\tau(j)+1}$ on the left and $S_{\tau(j)}$
on the right if $\tau(j)<0$) as in Figures 8(b) and 9(b). For $j$ even and
${\overline{\tau}}(j)=j$, the marked point $E_{j}$ divides the upper boundary
of the strip in half-lines $S_{j}$ on the left and $S_{j-1}$ on the right, and
the marked point $E_{\tau(j)}=E_{-j}$ divides the lower boundary of the strip
in half-lines $S_{-j}$ on the left and $S_{-j+1}$ on the right (Figure 8). For
$j<0$, we define $S_{j}:=\overline{S_{-j}}$. (The ordering of the $S_{j}$’s is
completely determined by $\tau$, but since $\tau(-j)=-\tau(j)$, this is
equivalent.)

We obtain a Riemann surface $M$ by identifying the strips along a half-line
$S_{\ell}$ on some $B_{j,\tau(j)}$ with a half-line $S_{\ell}$ on some
$B_{n,\tau(n)}$. Recall from Propositions 2.10 and 3.4 that the combinatorial
invariant partitions $\\{\pm 1,\ldots,\pm(k-\ell)\\}$ in classes $[j]$. Now,
if we glue every half strip along the separatrices $S_{j^{\prime}}$ on
$S_{j^{\prime}}$ for every $j^{\prime}\in[j]$, we obtain a half infinite
cylinder conformally equivalent to an open set $U$ with the topological type
of a punctured disk. This allows us to add a point $Z_{[j]}$ at the unbounded
end of the cylinder. Since there are $k+1$ equivalence classes, there are
$k+1$ half cylinders, so that we can add $k+1$ points to $M$ in this way.

Lastly, we identify $\\{E_{j}\\}_{j}$ to a single point $E$ and we define a
neighbourhood $V(E)$ of $E$ by taking a small half disk around each $E_{j}$
inside $B_{j,\tau(j)}\cup\partial B_{j,\tau(j)}$ and by gluing them together
along segments of separatrices. This is a $k$-covering of $E$ and the local
chart $h\colon\bigl{(}V(E),E\bigr{)}\to(\hat{\mathbb{C}},\infty)$, given by
$\zeta={C\over(Z-E)^{1/k}}$ sends $1\frac{\partial}{\partial Z}$ to
$\zeta^{k+1}\frac{\partial}{\partial\zeta}$, for some $C$ satisfying
$kC^{k}=-1$. There are $k$ such charts: we choose the one for which the image
of $S_{0}$ is oriented along the positive real axis and we call it $h_{E}$.

Let $\widetilde{M}=M\cup\\{Z_{[j]}\\}_{j}\cup\\{E\\}$ be the Riemann surface
thus obtained. It it proved in [DES05] ($m>0$) or in [BD10] ($m=0$) that
$\widetilde{M}$ is homeomorphic to a sphere, and, following from Riemann’s
Uniformization Theorem, isomorphic to the Riemann sphere.

(a) Zones in the $z$ coordinate (for $k=7$). The separatrix graph is in gray.
The marked points $e_{j}$ represent the infinity in the direction
$\exp\\!\left(\pi i\frac{2j-1}{2k}\right)$.  (b) Horizontal strips in the
rectifying coordinate $t$. The coordinate is ramified at $z_{[j]}$ and
$z_{[j]}$ corresponds to $Z_{[j]}$ ‘‘at infinity’’. The marked point $e_{j}$
(infinity in the $z$ coordinate) is mapped on $E_{j}$ on the boundary of a
strip. The dashed lines are the images of the transversal curves.

Figure 8. Representation of the zones and the strips in codimension $7$ with
at least one singular point on the real axis. A zone of a color is mapped on
the strip of the same color.

(a) Strips in the $z$ coordinate. (b) Strips in the rectifying coordinate $Z$.

Figure 9. Representation of the strips in codimension $7$ with no singular
point on the real axis. A zone of a color is mapped on the strip of the same
color. The singular points $z_{[j]}$ are mapped on the $Z_{[j]}$ ‘‘at
infinity’’.

#### Existence of the monic polynomial vector field

There exists a unique isomorphism $H\colon\widetilde{M}\to\hat{\mathbb{C}}$
such that $H$ in the chart $\bigl{(}V(E),h_{E}\bigr{)}$ has the form
$\zeta\mapsto\zeta+o\bigl{(}{1\over\zeta}\bigr{)}$. The vector field
$1\frac{\partial}{\partial Z}$ is well defined on $M$, so we define the
pushforward $P\frac{\partial}{\partial
z}:=H_{\ast}\bigl{(}1\frac{\partial}{\partial Z}\bigr{)}$. Since $P$ is
bounded near each $z_{[j]}$, it can be extended over each $z_{[j]}$ to a
finite value, which we need to prove to be zero. A chart on the half-cylinder
associated to $z_{[j]}$ compatible with the glueing of points $E_{n}$ is given
by $\xi=\exp(\lambda_{j}Z)$ where $\lambda_{j}$ is the sum of the transversal
times of the infinite half-strips glued to form the half-cylinder. The
pushforward in the $\xi$-coordinate shows that the vector field has a simple
zero at $z_{[j]}$, a property invariant by change of coordinate. Hence,
$P(z_{[j]})=0$. In a neighbourhood of infinity, we have $(h_{E}\circ
H^{-1})_{*}\bigl{(}P\frac{\partial}{\partial
z}\bigr{)}(\zeta)=\zeta^{k+1}\frac{\partial}{\partial\zeta}$. It follows that
$P$ is a monic polynomial of degree $k+1$.

######  Remark 7.6.

We choose $h$ so that it maps $S_{0}$ to $s_{0}$ tangent to $\mathbb{R}_{+}$
at infinity. With this choice, it then follows that the polynomial obtained
has a positive leading coefficient because of the way we attached the
separatrices on the Riemann surface.

#### Proof of Theorem 7.4

We define a complex conjugation $\Sigma$ of $\widetilde{M}$. In the strips, we
set $\Sigma_{j,\tau(j)}\colon B_{j,\tau(j)}\to B_{-j,\tau(-j)}$ defined by
$\Sigma_{j,\tau(j)}(Z)=\overline{Z}$, which is well-defined since
$\eta_{j,\tau(j)}=-\overline{\eta_{-j,\tau(-j)}}$. These maps extend to the
separatrices on the boundaries of the strips, and because
$\overline{S_{-j}}=S_{j}$ (a consequence of $\tau(-j)=-\tau(j)$), this defines
a global map $\Sigma$ on $M$. Moreover, it maps a half infinite cylinder of
$Z_{[j]}$ on a half infinite cylinder of $Z_{[-j]}$, so that we can extend
$\Sigma$ over the $Z_{[j]}$’s. Lastly, it extends on $\\{E_{j}\\}_{j}$ by
$\Sigma(E_{j})=E_{-j}$ and maps a neighbourhood of $E$ on itself. Hence,
$\Sigma$ is well-defined on $\widetilde{M}$.

Now, let $\sigma=h\circ\Sigma\circ h^{-1}$. It is an antiholomorphic
involution of $\hat{\mathbb{C}}$ that maps $\infty$ on itself and $s_{0}$ on
itself. Then $\sigma(z)=\overline{z}+ia$, for $a\in\mathbb{R}$. The
translation $z\mapsto{ia\over 2}$ kills $a$, so we may as well suppose that
$a=0$. Then $\sigma=id$ only on $\mathbb{R}$.

  1. (1)

On a strip such that $B_{-j,\tau(-j)}=\overline{B_{j,\tau(j)}}$, $\Sigma=id$
on the real axis. This means that this orbit of $1\frac{\partial}{\partial Z}$
is mapped by $h$ on an interval of $\mathbb{R}$, and the $\alpha$\- and
$\omega$-limits are real singular points. There are $m-1+\ell$ such strips.

  2. (2)

On the other strips, there are no invariant orbits under $\Sigma$, except
possibly separatrices.

     * •

When $m=0$, then $S_{0}=S_{k}=S_{-k}$ is the only invariant orbit under
$\Sigma$, on which $\Sigma=id$, so it is mapped on $\mathbb{R}$ by $h$.

     * •

When $m>1$, then $S_{0}$ and $S_{k}$ are the only two invariant orbits under
$\Sigma$. They are both mapped on an half-line of $\mathbb{R}$ with the
landing points $z_{[0]},z_{[k]}\in\mathbb{R}$ respectively. (If $m=1$, then
$z_{[0]}=z_{[k]}$.)

In all cases, $\mathbb{R}$ is invariant under $P\frac{\partial}{\partial z}$,
hence $P$ has real coefficients. With a real translation, it is now possible
to center $P$. ∎

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