Introduction

As far as we know, the simplest class of non-injective polynomial local diffeomorphisms of
^{1994}). The existence of these maps disproves the *real Jacobian conjecture*, that a polynomial local diffeomorphism of

One of the most known conditions for a local diffeomorphism to be a global one is that it is proper. The asymptotic variety of a map of
^{2000}) and Campbell (^{arXiv:math/9812032 in 1998, 2011}) calculated the asymptotic variety of two Pinchuk maps in details. Our aim in this paper is to do a similar work, i.e., to describe a Pinchuk map, but now from a different point of view.

Let

where the dot denotes derivative with respect to the time

**Lemma 1.**
*The singular points of system (1) are the zeros of
* (1).

The following is a generalization of the characterization of global invertibility of polynomial maps given by Sabatini (^{1998}). This version is due to Braun and Llibre (arXiv:1706.02643 in 2017).

**Theorem 2.**
*Let
*

In the special case that
^{Białynicki-Birula and Rosenlicht 1962}). Hence, in this case,
^{2016}).

Since the phase portrait on the Poincaré sphere of a Hamiltonian polynomial vector field having a global center is simple, i.e., at the infinite either it does not have singular points, or the infinite singular points are formed by two degenerate hyperbolic sectors (for Hamiltonian vector fields, the infinity contains only isolated singular points), it is interesting to know how complex can be the phase portrait of a non-global center of a Hamiltonian system (1).

In this paper we provide the qualitative global phase portrait of the Hamiltonian system (1) when

**Theorem 3.**
*Let
*

To prove Theorem 3, we first study the infinite singular points of system (1). These infinite singular points are very degenerate, and we apply homogeneous and quasi-homogeneous blow ups to study them. Then, we complete the proof of Theorem 3 by proving that the separatrix configuration of system (1) is qualitatively the one presented in Fig. 1.

We think that a good understanding of what fails in the real Jacobian conjecture could be interesting to investigate a related problem, the *Jacobian conjecture* in
^{1982}) and Van Den Essen (^{2000}).

Injectivity, centers and a Pinchuk map

We begin with the proof of Lemma 1.

*Proof of Lemma 1.* Let

which is true if and only in

The point

Now, we select the map
*Pinchuk map* is a non-injective polynomial map with nowhere zero Jacobian determinant of the form

According to Campbell (2011), the points

which is a parametrization of the asymptotic variety of

We consider the map

Observe that

From now on, we restrict our attention to the specific Pinchuk map (2).

We first calculate the coordinates of the point

We observe that

Repeating a similar reasoning now looking

Hence

In the following section we shall investigate the infinite of system (1).

The infinite of system

In this section, we will use results and notations on the *Poincaré compactification* of polynomial vector fields of

For details on this technique we refer the reader to Chapter 5 of (^{Dumortier et al. 2006}) or to (^{González Velasco 1969}).

We call a singular point of a vector field *linearly zero* when the linear part of the vector field at this point is identically zero.

We begin by proving a general fact about the infinite singular points of Hamiltonian systems of the form (1). Writing

Assuming

**Lemma 4.**
*The infinite singular points of the Hamiltonian system (1) are linearly zero.*

Now, we return to the Pinchuk map

We will use the quasi-homogeneous directional blow up technique to desingularize each of these infinite singular points. An exposition about blow-ups can be found in (^{Álvarez et al. 2011}), see also Chapter 3 of (Dumortier et al. 2006). We now recall the directional blow up transformations.

By the *quasi-homogeneous blow up in the positive* (resp. *negative*) *
*, or simply

*(resp.*( α , β )
-blow up in the positive

*negative*)

*, we mean the transformation which carries the variables*x
-direction

respectively. Similarly, by the *quasi-homogeneous blow up in the positive* (resp. *negative*) *
*, or simply

*(resp.*( α , β )
-blow up in the positive

*negative*)

*, we mean the transformations*y
-direction

respectively.

Clearly if

with

with

After the blow up in the

The weights
*Newton polygon* of

The application of

To make the exposition clearer, we shall apply the most part of the blow ups in the
*
*

In the next two subsections we will desingularize the origin of the charts

Since the Hamiltonian system (1) with the polynomials
*Mathematica*. We do not show in each step the whole expressions of the systems

The origin of the chart

We write the compactification of system (1) in the chart

We first apply a

The Newton polygon of system

and

The only singular point of

The reader can follow a schema of each step of the calculations in Fig. 2. We just need to analyze the origin of the systems

We now analyze this linearly zero singularity. We first do a translation bringing this point to the origin, obtaining the new system

Over the line

The discriminant of this quartic equation is negative, thus, it has two real solutions. Those are approximately

Now, we study the linearly zero point

By using Sturm’s theorem (see for instance (^{Isaacson and Keller 1994}); in the software Mathematica, the Sturm theorem is programed by the instruction *CountRoots*) we see that the polynomial of degree

We now begin the process of blowing down.

It is simple to conclude that the phase portrait of the system

Consequently, by considering also the information close to the other two singular points in the line

By translating

The origin of chart

As in the calculations made above, we write the compactified vector field in the chart

The polynomials

It is clear that at

We also apply

We do not need to analyze the other singular points over the lines

and so

Thus, at

By using Sturm’s theorem, we conclude that this equation has no real solution.

Now, we desingularize the point

Clearly,

which is approximately

Now, we apply a translation to bring the point

The Newton polygon of this system has just one compact edge contained in the line

with

Then, after applying

with

and

with

The origin of system

with

The point

The Newton polygon of this system has only one compact edge contained in the straight line

As above, we apply

with

with

System

with

The point

As before the Newton polygon of this system has only one compact edge contained in the straight line

with

Now, over the line

with

It is simple to conclude that

Since the behavior near each appearing singular points in each step above is very simple, the blow down of each step is also very simple: following the arrays in Fig. 5, it is easy to conclude that the origin of

The global phase portrait

We begin with a background on separatrices and canonical regions of the Poincaré compactification
*topologically equivalent* if there exists a homeomorphism

Following Markus (^{1954}), we say that the flow
*parallel* if it is topologically equivalent to one of the following flows: (i) the flow defined in
*stripannular* and *spiral* (or *radial*), respectively.

We denote by
*separatrix* of

If
*canonical region* of
^{1975}) proved that each canonical region of a vector field

To the union of the separatrices of
*separatrix configuration* of
*topologically equivalent* if there exists an orientation preserving homeomorphism from
^{Neumann and O’Brien 1976}, ^{Peixoto 1973}, Dumortier et al. 2006) for the Poincaré compactification in the Poincaré disc of polynomial systems.

**Theorem 5** (Markus-Neumann-Peixoto). *Let
*

Hence, in order to qualitatively describe the phase portrait on the Poincaré disc of system (1) it is enough to qualitatively describe its separatrix configuration. This was done in Fig. 1, where we have drawn the separatrices other than singular points with bold lines. The other lines are orbits contained in its respective canonical regions. We observe from Fig. 1 that system (1) has

Below, we prove Theorem 3 by proving that Fig. 1 is a separatrix configuration of system (1).

From the previous sections we conclude that close enough to the singular points, the phase portrait of system (1) is qualitatively the one presented in Fig. 6.

For further references we label the hyperbolic, parabolic and elliptic sectors presenting in the origins of the charts

From the definition of system (1), each of its orbits is a connected component of a level set of
*asymptotic flower* of

Since for each

and this polynomial of degree
*starts* or *finishes* at

We call

The point

Now we analyze the parabolic sectors

Close to the two points of
*claim that the correct possibility is the first one*. Indeed, we can increase the radii of these circles containing the images of the orbits of

So, the image of the orbits of the parabolic sectors

Now, since the image of this last orbit cuts the curves
*big orbits*. A big orbit whose image is contained in a circle close enough to

The orbits whose images are arcs of the circles with radii smaller than the radius of

The orbits whose images are arcs starting and finishing at

In particular, this means that the big orbits must finish at the origin of the chart

The big orbits also produce a strip canonical region.

The orbits of the strip canonical region placed above the hyperbolic sector

The elliptic sectors

Hence we have