# Resolution of singularities of 2-dimensional real analytic constrained
differential systems

Otavio Henrique Perez and Paulo Ricardo da Silva  São Paulo State University
(Unesp), Institute of Biosciences, Humanities and Exact Sciences. Rua C.
Colombo, 2265, CEP 15054–000. S. J. Rio Preto, São Paulo, Brazil.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We present a theorem of resolution of singularities for real analytic
constrained differential systems $A(x)\dot{x}=F(x)$ defined on a 2-manifold
with corners having impasse set $\\{x;\det A(x)=0\\}$. This result can be seen
as a generalization of the classical one for 2-dimensional real analytic
vector fields. Our proof is based on weighted blow-ups and the Newton polygon.

###### Key words and phrases:

Constrained Differential Systems, Impasses, Resolution of Singularities, Blow
ups, Newton Polygon.

###### 2020 Mathematics Subject Classification:

34A09, 34C08.

.

##  1\. Introduction and main result

A $n$-dimensional constrained differential system (or simply constrained
system) in an open set $U\subseteq\mathbb{R}^{n}$ is given by

(1) $A(\mathbf{x})\dot{\mathbf{x}}=F(\mathbf{x}),$

where $\mathbf{x}\in U\subseteq\mathbb{R}^{n}$, $A:U\rightarrow
Mat(\mathbb{R},n)$ and $F:U\rightarrow\mathbb{R}^{n}$ are smooth functions. A
point $p$ such that $\det A(p)=0$ is called impasse point, and the so called
impasse set is the set $\Delta=\\{p\in U;\det A(p)=0\\}$.

Along this paper, we suppose that $A$ does not have constant rank. We will
further suppose that $\det A(\mathbf{x})=0$ only in a closed subset of $U$.
This implies that the adjoint matrix $A^{*}$ of $A$ (the transpose of the
cofactor matrix of $A$) is not identically zero and $AA^{*}=A^{*}A=\det(A)I$
(see Lemma 37.1, [13]).

Observe that outside the impasse set a constrained system can be written as

$\dot{\mathbf{x}}=A(\mathbf{x})^{-1}F(\mathbf{x}),$

and we have a classical ordinary differential equation.

Multiplying both sides of (1) by the adjoint matrix $A^{*}$ of $A$, it can be
shown (see Lemma 37.2, [13]) that $\gamma$ is solution of (1) if, and only if,
$\gamma$ is a solution of

(2) $\delta(\mathbf{x})\dot{\mathbf{x}}=A^{*}(\mathbf{x})F(\mathbf{x}),$

where $\delta:U\rightarrow\mathbb{R}$ is the determinant function and
$\\{\mathbf{x}\in U;\delta(\mathbf{x})=0\\}$ is the impasse set $\Delta$. The
vector field $X(\mathbf{x})=A^{*}(\mathbf{x})F(\mathbf{x})$ will be called
adjoint vector field and a constrained system in the form (2) will be called
diagonalized constrained system.

At an impasse point $p$, since the operator $A(p)$ is not invertible, the
classical results on the existence and uniqueness of the solutions break down.
However, near such point we can describe the phase portrait as follows. In the
open set where $\det A(q)>0$, the phase portrait is given by the adjoint
vector field $X$. On the other hand, in the open set where $\det A(q)<0$ the
phase portrait is given by $-X$.

######  Definition 1.

We say that $p\in U$ is a singularity of the constrained system if one of the
following conditions is satisfied:

  1. (1)

$p$ is an equilibrium point of the adjoint vector field $X$, that is,
$X(p)=0$;

  2. (2)

the impasse set $\Delta$ is not smooth at $p$;

  3. (3)

$p$ is a tangency point between $X$ and $\Delta$, that is, $d\delta(X)=0$.

Otherwise, we say that $p$ is non-singular. See Figures 1 and 2.

Figure 1. Singularities of a planar constrained system. The impasse set is
highlighted in red.
Figure 2. Non-singular points of a planar constrained system. The impasse set
is highlighted in red.

From now on we consider 2-dimensional constrained differential systems. We say
that an equilibrium point of a vector field is semi-hyperbolic if at least one
of the eigenvalues of its linearization has nonzero real part.

######  Definition 2.

A 2-dimensional constrained differential system is elementary at $p$ if one of
the following conditions holds:

  1. (1)

$p$ is a non singular point;

  2. (2)

$p$ is a semi-hyperbolic equilibrium point of $X$ and $p\not\in\Delta$;

  3. (3)

If $p$ is a semi-hyperbolic equilibrium point of $X$ and $p\in\Delta$, then
$\Delta$ coincides with a local separatrix of $X$ at $p$.

See Figure 3.

Figure 3. Elementary points of a planar constrained system. The impasse set is
highlighted in red.

In [17, 18] the authors studied singularities of constrained systems near a
smooth point of the impasse set, in dimensions $n\geq 3$ and $n=2$,
respectively. Here we also consider singular points of the impasse set. Beyond
the mathematical interest in such problem, some models from applied sciences
may present this kind of singularity (e.g. see Example 1, [16]). In [5] the
authors considered homogeneous blow ups to study singular points of the
impasse set of real polynomial constrained systems whose impasse set is
defined by a homogeneous polynomial, among others non-degeneracy conditions.
In the present work we are going to generalize these results by proving a
theorem of resolution of singularities for real analytic planar constrained
systems without such non-degeneracy assumptions.

In order to state our main theorem, we need to work in a more general category
of analytic manifolds with corners. Roughly speaking, a 2-dimensional manifold
with corners is a topological space locally modeled by open sets of
$(\mathbb{R}_{\geq 0})^{2}$. We refer to [9] for a detailed discussion on this
subject. As we will see in the discussion below, after a blow-up in a
singularity we obtain a new ambient space, which is a manifold with corners.
Throughout this paper, $\mathcal{M}$ is a 2-dimensional real analytic manifold
with corners. See Figure 4.

Figure 4. Constrained differential system in a 2-manifold with corners after a
blow-up.

A 1-dimensional analytic oriented foliation on $\mathcal{M}$ is a collection
of pairs $\mathcal{X}=\\{(U_{i},X_{i})\\}_{i\in I}$ such that
$\\{U_{i}\\}_{i\in I}$ is an open covering of $\mathcal{M}$,
$X_{i}=(P_{i},Q_{i})$ is a real analytic vector field defined in $U_{i}$ and,
for each $i,j\in I$, there is a strictly positive function $u_{i,j}$ such that
$X_{i}=u_{i,j}X_{j}$ on $U_{i}\cap U_{j}$. We further suppose that
$\partial\mathcal{M}$ is invariant by the foliation and $P_{i}$ and $Q_{i}$ do
not have common factor, which implies that equilibrium points of $X_{i}$ are
isolated.

We define the principal ideal sheaf $\mathcal{I}$ as the collection of pairs
$\mathcal{I}=\\{(U_{i},\delta_{i})\\}_{i\in I}$ such that $\\{U_{i}\\}_{i\in
I}$ is an open covering of $\mathcal{M}$, $\delta_{i}$ is an irreducible
analytic function defined in $U_{i}$ and, for each $i,j\in I$, there is a
strictly positive function $v_{i,j}$ such that $\delta_{i}=v_{i,j}\delta_{j}$
on $U_{i}\cap U_{j}$.

A constrained differential system defined in a 2-dimensional manifold with
corners is a triple $(\mathcal{M},\mathcal{X},\mathcal{I})$, where
$\mathcal{M}$ is a 2-dimensional real analytic manifold with corners,
$\mathcal{X}$ is a 1-dimensional analytic oriented foliation on $\mathcal{M}$
and $\mathcal{I}$ is the principal ideal sheaf. On each $U_{i}$ of the open
covering, we associate the diagonalized constrained differential system

$\delta_{i}(\mathbf{x})\dot{\mathbf{x}}=X_{i}(\mathbf{x}).$

For shortness we say that $\mathcal{I}$ is the impasse ideal of the
constrained system and its zero set $\Delta$ (locally given by
$\\{\delta_{i}=0\\}$) is the impasse set. A straightforward computation shows
that the phase curves of the constrained system coincide in the intersection
$U_{i}\cap U_{j}$ of two open sets of the covering.

Singularities and elementary points are defined in the same way as Definitions
1 and 2. We say that $(\mathcal{M},\mathcal{X},\mathcal{I})$ is elementary if
$(\mathcal{M},\mathcal{X},\mathcal{I})$ is elementary at every
$p\in\mathcal{M}$.

Now we are able to state the main theorem.

######  Theorem 3.

Let $(\mathcal{M},\mathcal{X},\mathcal{I})$ be a 2-dimensional real analytic
constrained differential system defined on a compact manifold with corners.
Then there is a finite sequence of weighted blow ups

$(\widetilde{\mathcal{M}},\widetilde{\mathcal{X}},\widetilde{\mathcal{I}})=(\mathcal{M}_{n},\mathcal{X}_{n},\mathcal{I}_{n})\xrightarrow{\Phi_{n}}\cdots\xrightarrow{\Phi_{0}}(\mathcal{M}_{0},\mathcal{X}_{0},\mathcal{I}_{0})=(\mathcal{M},\mathcal{X},\mathcal{I})$

such that $(\widetilde{M},\widetilde{X},\widetilde{\mathcal{I}})$ is
elementary.

We will give the proof of this theorem in Section 5.

The first theorem of resolution of singularities for 2-dimensional real
analytic vector fields was stated by Bendixson [3], and its first proof was
given by Seidenberg [15]. Such result was generalized by Dumortier [7] for
real $C^{\infty}$ planar vector fields. Brunella and Miari [4] considered
weighted blow-ups to resolve isolated singularities of 2-dimensional real
analytic vector fields under some non degeneracy conditions. Pelletier [11]
gave a proof dropping such non degeneracy conditions, and in the same paper
the author also estimates the number of weighted blow-ups needed to resolve a
singularity. The proof of resolution of singularities for 3-dimensional real
analytic vector fields was given by Panazzolo [10]. Just as in [2, 10, 11],
weighted blow-ups and the Newton polygon are our main tools.

Resolution of singularities is an important tool in the study of equilibrium
points of real planar vector fields (see [4, 7] for instance). Following this
direction, the techniques presented here will be useful in the topological
classification of phase portraits of constrained differential systems near a
singularity. Indeed, such study is the next step once the theorem of
resolution of singularities is proved (see [12]).

The paper is organized as follows. In Section 2 we present some preliminary
definitions necessary for the development of the study. In Section 3 the
Newton polygon is considered in the context of constrained systems and we find
suitable coordinate systems that will be useful in the proof of the main
theorem. In Section 4 we given a detailed description of the algorithm of
resolution of singularities of planar constrained differential systems.
Finally, in Section 5 we prove Theorem 3.

##  2\. Main tools

In this section we present the tools used in the resolution of singularities
of planar constrained differential systems. The proof of the main theorem is
based on weighted blow-ups and the weights are chosen by means of the Newton
polygon. For our knowledge, it is not defined in the literature such a polygon
for constrained systems. The idea is to associate an analytic vector field to
the constrained differential system (the so called auxiliary vector field) and
then we study the Newton polygon of such vector field. In the end of this
section, an example is presented in order to explain all the tools defined.

Firstly, we recall the notion of orbital equivalence presented in [17]. Two
constrained systems
$\delta_{i}(\mathbf{x})\dot{\mathbf{x}}=X_{i}(\mathbf{x})$, $i=1,2,$ are
orbitally equivalent at the points $p_{i}\in\Delta_{i}$, $i=1,2$, if there are
two open sets $V_{1}\ni p_{1}$ and $V_{2}\ni p_{2}$ and an analytic
diffeomorphism $\Phi:V_{1}\rightarrow V_{2}$ such that

  1. (1)

$\Phi$ maps the impasse set $\Delta_{1}$ to the impasse set $\Delta_{2}$;

  2. (2)

$\Phi$ maps the phase curves of $X_{1}$ in $V_{1}\backslash\Delta_{1}$ to the
phase curves of $X_{2}$ in $V_{2}\backslash\Delta_{2}$, not necessarily
preserving orientation.

In the case where $\Phi$ preserves orientation, we say that $\Phi$ is an
orientation preserving orbital equivalence.

###  2.1. Quasi-homogeneous blow-up

In what follows we introduce weighted blow-ups (or quasi-homogeneous blow ups)
following the definitions presented in [10] for blow-ups in $n$-dimensional
manifolds. For a good introduction of the method for 2-dimensional vector
fields, we refer to [1, 8]. Roughly speaking, blowing-up a singularity means
to replace a point by a compact set, which gives us a new ambient space. In
this new compact set, new (possibly less degenerated) singularities may
appear.

Let $(\mathcal{M},\mathcal{X},\mathcal{I})$ be a 2-dimensional real analytic
constrained system and fix local coordinates of such triple. A weight-vector
is a non-zero vector $\omega=(\omega_{1},...,\omega_{n})$ of positive
integers. A quadratic $\omega$-blow-up is a real analytic surjective map

$\Phi_{\omega}:$ | $\widetilde{M}=\mathbb{S}^{n-1}\times\mathbb{R}_{\geq 0}$ | $\rightarrow$ |  $\mathbb{R}^{n}$,
---|---|---|---
| $(\bar{x},\tau)$ | $\mapsto$ |  $\tau^{\omega}\bar{x}=(\tau^{\omega_{1}}\bar{x}_{1},...,\tau^{\omega_{n}}\bar{x}_{n})$,

where
$\mathbb{S}^{n-1}=\\{\bar{x}\in\mathbb{R}^{n};\sum_{i=1}^{n}\bar{x}_{i}^{2}=1\\}$.
The set $\Omega=\\{(0,...,0)\\}$ is called blow-up center and
$\Phi_{\omega}^{-1}(\Omega)=\mathbb{S}^{n-1}\times\\{0\\}=\mathbb{E}$ is
called exceptional divisor.

Notice that $\Phi_{\omega}|_{\widetilde{M}\backslash\mathbb{E}}$ is an
analytic diffeomorphism that maps its domain onto
$\mathbb{R}^{n}\backslash\Omega$. If we blow up a point outside the boundary
$\partial\mathcal{M}$, then $\mathbb{E}$ is the boundary of
$\widetilde{\mathcal{M}}$. On the other hand, if we blow up a point on
$\partial\mathcal{M}$, then we previously have a divisor $\mathcal{D}$ on
$\mathcal{M}$ and the boundary of $\widetilde{\mathcal{M}}$ is given by
$\widetilde{\mathcal{D}}=\Phi^{-1}_{\omega}(\mathcal{D})\cap\mathbb{E}$.

Given a 1-dimensional analytic oriented foliation $\mathcal{X}$ on
$\mathcal{M}$, the pull back of $\mathcal{X}$ is the 1-dimensional analytic
oriented foliation $\widetilde{\mathcal{X}}=\Phi_{\omega}^{*}\mathcal{X}$
defined in $\widetilde{\mathcal{M}}$, which is orientation preserving
orbitally equivalent to $\mathcal{X}$ outside $\mathbb{E}$. Analogously, the
pull back of $\mathcal{I}$ is the impasse ideal
$\widetilde{\mathcal{I}}=\Phi_{\omega}^{*}\mathcal{I}$ defined in
$\widetilde{\mathcal{M}}$ whose generators are diffeomorphic to that ones of
$\mathcal{I}$ outside $\mathbb{E}$. The strict transformed of
$(\mathcal{M},\mathcal{X},\mathcal{I})$ as the triple
$(\widetilde{\mathcal{M}},\widetilde{\mathcal{X}},\widetilde{\mathcal{I}})$.
It can be easily checked that outside the exceptional divisor the triples
$(\mathcal{M},\mathcal{X},\mathcal{I})$ and
$(\widetilde{\mathcal{M}},\widetilde{\mathcal{X}},\widetilde{\mathcal{I}})$
are orbitally equivalent orientation preserving.

In many situations, it is more practical to consider directional charts
instead the quadratic blow up. Take the weight vector
$\omega=(\omega_{1},...,\omega_{n})$ and $\sigma\in\\{+,-\\}$. Consider the
set $V_{i}^{\sigma}$

$V_{i}^{\sigma}=\\{(\bar{x},\tau)\in\mathbb{S}^{n-1}\times\mathbb{R}_{\geq
0};\sigma x_{i}>0\\},$

take the projection

$\Pi_{i}^{\sigma}:$ | $V_{i}^{\sigma}$ | $\rightarrow$ |  $\mathbb{R}^{n-1}\times\mathbb{R}_{\geq 0}$,
---|---|---|---
| $(\bar{x},\tau)$ | $\mapsto$ |  $(\bar{x}_{1},...,\bar{x}_{i-1},\tau,\bar{x}_{i+1},...,\bar{x}_{n})$,

and define the map

$\Phi_{i,\omega}^{\sigma}:$ | $\mathbb{R}^{n-1}\times\mathbb{R}_{\geq 0}$ | $\rightarrow$ |  $\mathbb{R}^{n}$,
---|---|---|---
| $(\tilde{x}_{1},...,\tilde{x}_{n})$ | $\mapsto$ |  $(\tilde{x}_{i}^{\omega_{1}}\tilde{x}_{1},...,\tilde{x}_{i}^{\omega_{i-1}}\tilde{x}_{i-1},\sigma\tilde{x}_{i}^{\omega_{i}},\tilde{x}_{i}^{\omega_{i+1}}\tilde{x}_{i+1},...,\tilde{x}_{i}^{\omega_{n}}\tilde{x}_{n})$.

The $x_{i}$-directional chart is given by the pairs
$(V_{i}^{\sigma},\Pi_{i}^{\sigma})$ and the blow-up in the $x_{i}$ direction
is given by $\Phi_{i,\omega}^{\sigma}$. It can be shown that Diagram 5 is
commutative.

$V_{i}^{\sigma}$$\mathbb{R}^{n}\cap\\{\sigma x_{i}\geq
0\\}$$U$$\mathbb{R}^{n}\cap\\{\sigma x_{i}\geq
0\\}$$\Phi_{\omega}$$\Pi^{\sigma}_{i}$$id$$\Phi^{\sigma}_{i,\omega}$ Figure 5.
Equivalence between directional and quadratic blow-ups.

The pull back of $\mathcal{X}$, the pull back of $\mathcal{I}$ and the strict
transformed
$(\widetilde{\mathcal{M}},\widetilde{\mathcal{X}},\widetilde{\mathcal{I}})$ of
$(\mathcal{M},\mathcal{X},\mathcal{I})$ are defined in the same way as before.

###  2.2. The Newton polygon

A question that naturally rises is how we chose the weight vector of the
weighted blow-up. In order to make a good choice for the resolution of
singularities, we utilize the Newton polygon associated to a 2-dimensional
analytic vector field. In this subsection we briefly recall the construction
of such mathematical object, which can also be found in [1, 8, 11].

We say that a polynomial $P(x,y)=\displaystyle\sum p_{r,s}x^{r}y^{s}$ is
quasi-homogeneous of degree $d$ if there is a vector of positive integers
$\omega=(\omega_{1},\omega_{2})$ such that
$P(t^{\omega_{1}}x,t^{\omega_{2}}y)=t^{d}P(x,y)$. We say that $d$ is the
degree of quasi-homogeneity.

Let $X$ be an analytic vector field. Just as in [10], we write $X$ in the so
called logarithmic basis. More precisely, consider

$X(x,y)=P(x,y)\displaystyle\frac{\partial}{\partial
x}+Q(x,y)\displaystyle\frac{\partial}{\partial y},$

where

$P(x,y)=\Big{(}\sum a_{m,n}x^{m}y^{n}\Big{)}x,\ Q(x,y)=\Big{(}\sum
b_{m,n}x^{m}y^{n}\Big{)}y,$

with $a_{m,n},b_{m,n}\in\mathbb{R}$ and $m,n\in\mathbb{Z}$ satisfying:

  1. (1)

For $m<-1$ or $n\leq-1$, $a_{m,n}=0$;

  2. (2)

For $m\leq-1$ or $n<-1$, $b_{m,n}=0$.

We recall that a diagonalized constrained system is written in the form

(3) $\delta(x,y)\dot{x}=P(x,y),\ \delta(x,y)\dot{y}=Q(x,y),$

with $\delta(x,y)$ being an irreducible real-analytic function that can be
written as

$\delta(x,y)=\sum c_{k,l}x^{k}y^{l},$

where $c_{k,l}\in\mathbb{Z}$ is such that $c_{k,l}=0$ when $k<0$ or $l<0$.

Let $\omega=(\omega_{1},\omega_{2})$ be a vector of positive integers. One can
write the planar vector field $X$ as

(4)
$X(x,y)=\displaystyle\sum_{d=-1}^{\infty}X_{d}^{(\omega_{1},\omega_{2})}(x,y),$

where

(5)
$X_{d}^{(\omega_{1},\omega_{2})}(x,y)=\displaystyle\sum_{\omega_{1}r+\omega_{2}s=d}x^{r}y^{s}\Big{(}a_{r,s}x\displaystyle\frac{\partial}{\partial
x}+b_{r,s}y\displaystyle\frac{\partial}{\partial y}\Big{)}.$

In other words, we are writing the vector field $X$ as a sum of quasi-
homogeneous components. We say that (4) is a
$(\omega_{1},\omega_{2})$-graduation of $X$. Each
$X_{d}^{(\omega_{1},\omega_{2})}$ is called $d$-level of the
$(\omega_{1},\omega_{2})$-graduation.

Given a $(\omega_{1},\omega_{2})$-graduation of $X$, we associate the
monomials $a_{r,s}x^{r}y^{s}$ and $b_{r,s}x^{r}y^{s}$ with nonzero
coefficients to a point $(r,s)$ in the plane of powers. Observe that each
point $(r,s)$ is contained in a line of the form
$\\{\omega_{1}r+\omega_{2}s=d\\}$. Furthermore, notice that the map that
relates a monomial with a point $(r,s)$ is not injective.

######  Definition 4.

The support $\mathcal{Q}$ of $X$ is the set

$\mathcal{Q}=\\{(r,s)\in\mathbb{Z}^{2};a_{r,s}^{2}+b_{r,s}^{2}\neq 0\\}.$

######  Definition 5.

The Newton polygon $\mathcal{P}$ associated to the analytic vector field $X$
is the convex envelope of the set $\mathcal{Q}+\mathbb{R}^{2}_{+}$.

The boundary of the Newton polygon $\partial\mathcal{P}$ is the union of a
finite number of segments. We enumerate them from the left to the right:
$\gamma_{0},\gamma_{1},...,\gamma_{n+1}$. Observe that $\gamma_{0}$ is
vertical and $\gamma_{n+1}$ is horizontal. Analogously, the non-regular points
of $\partial\mathcal{P}$ will be enumerated from the left to the right:
$v_{0},...,v_{n}$.

######  Definition 6.

We say that $v_{0},...,v_{n}$ are the vertices of the Newton polygon. The
vertex $v_{0}=(r_{0},s_{0})$ is called main vertex and the segment
$\gamma_{1}$ will be called main segment. The number $s_{0}$ is called height
of the Newton polygon. See Figure 6.

Figure 6. Newton polygon of a vector field. The main vertex and main segment
are highlighted in blue.

The main segment $\gamma_{1}$ is contained in an affine line of the form
$\\{r\omega_{1}+s\omega_{2}=R\\}$. Observe that the vector
$\omega=(\omega_{1},\omega_{2})$ is normal with respect to
$\\{r\omega_{1}+s\omega_{2}=R\\}$. In the resolution of singularities of
analytic vector fields, one chooses the vector $\omega$ as the weight vector
(see [1, 11]).

The Newton polygon strongly depends on the coordinate system adopted. It means
that applying a change of coordinates on a vector field, we possibly obtain a
different Newton polygon and it may be an obstacle during the resolution
process. Later on, we will see how to solve this problem.

###  2.3. The auxiliary vector field

Consider the diagonalized constrained system given by (3), that is,

$\delta(x,y)\dot{x}=P(x,y),\ \delta(x,y)\dot{y}=Q(x,y).$

######  Definition 7.

The auxiliary vector field $X_{A}$ associated to the constrained differential
equation (3) is the vector field

(6) $\scriptstyle X_{A}(x,y)=\delta(x,y)\Bigg{(}P(x,y)\frac{\partial}{\partial
x}+Q(x,y)\frac{\partial}{\partial y}\Bigg{)}=\Bigg{(}\sum
c_{k,l}x^{k}y^{l}\Bigg{)}\Bigg{(}\sum
x^{m}y^{n}\big{(}a_{m,n}x\frac{\partial}{\partial
x}+b_{m,n}y\frac{\partial}{\partial y}\big{)}\Bigg{)},$

which is real analytic.

It is easy to sketch the phase portrait of the auxiliary vector field. Outside
the impasse set, the auxiliary vector field (6) is obtained by multiplying the
constrained differential system (3) by the positive function $\delta^{2}$. On
the other hand, the impasse set $\Delta$ is a curve of equilibrium points for
(6).

Since the auxiliary vector field is analytic, all previous definitions and
remarks concerning the Newton polygon remain true for (6). However, observe
that the points on the plane of powers are of the form $(k+m,l+n)$, and
therefore the support $\mathcal{Q}$ takes form

$\mathcal{Q}=\\{(k+m,l+s)\in\mathbb{Z}^{2};c_{k,l}(a_{r,s}^{2}+b_{r,s}^{2})\neq
0\\}.$

Moreover, the levels of a $(\omega_{1},\omega_{2})$-graduation are written in
the form

$X_{A,d}^{(\omega_{1},\omega_{2})}=\Bigg{(}\displaystyle\sum_{\omega_{1}k+\omega_{2}l=d_{1}}c_{k,l}x^{k}y^{l}\Bigg{)}\Bigg{(}\displaystyle\sum_{\omega_{1}m+\omega_{2}n=d_{2}}x^{m}y^{n}\Big{(}a_{m,n}x\displaystyle\frac{\partial}{\partial
x}+b_{m,n}y\displaystyle\frac{\partial}{\partial y}\Big{)}\Bigg{)},$

where $d_{1}+d_{2}=d$. In other words, the support of the auxiliary vector
field is obtained by the Minkowski Sum [14] between the points of the support
of the adjoint vector field $X$ and the points of the support of $\delta$.
Finally, we remark that the Newton polygon of the adjoint vector field and the
auxiliary vector field are not necessarily the same.

For our knowledge, it does not exist a definition of Newton polygon for
constrained systems. In order to solve this problem, we associate a planar
constrained differential system to its auxiliary vector field $X_{A}$, and
then we utilize the Newton polygon of $X_{A}$. Furthermore, the main advantage
of writing the auxiliary vector this way is that it turns our computations
easier, and it will be easy to see what is happening with the impasse set and
the adjoint vector field during the resolution process.

###  2.4. An example

In what follows we exemplify the concepts presented. Consider the following
constrained differential equation

(7) $xy\dot{x}=y,\ xy\dot{y}=x^{2},$

whose auxiliary vector field is given by

(8) $X_{A}(x,y)=(xy)y\displaystyle\frac{\partial}{\partial
x}+(xy)x^{2}\displaystyle\frac{\partial}{\partial y}.$

The support of (8) is the set $\mathcal{Q}=\\{(0,2),(3,0)\\}$. Therefore, the
main vertex $v_{0}$ of the Newton polygon associated to (8) is $(0,2)$, the
height of the polygon is equals $2$ and the main segment $\gamma_{1}$ is
contained in the line $\\{2r+3s=6\\}$. See Figure 7.

Figure 7. Phase portrait of (7) (left) and Newton polygon of (8) (right). The
impasse set is highlighted in red. The main vertex and the main segment of the
polygon are highlighted in blue.

In the proof of the main theorem we consider directional blow ups. The weight
vector will be that one that is normal to the affine line $\\{2r+3s=6\\}$. In
this example, such weight vector is $\omega=(2,3)$. We will only make the
computations in the positive $x$ and $y$ charts, because in the negative
charts the computations are analogous.

Our transformations are of the form

$(x,y)=(\tilde{x}^{2},\tilde{x}^{3}\tilde{y}),\
(x,y)=(\bar{y}^{2}\bar{x},\bar{y}^{3}),$

in the $x$ and $y$ directions, respectively. In the $x$-chart, the auxiliary
vector field takes the form

$\widetilde{X}_{A}=(\tilde{x}^{5}\tilde{y})\tilde{x}\displaystyle\frac{\tilde{y}}{2}\tilde{x}\displaystyle\frac{\partial}{\partial\tilde{x}}+(\tilde{x}^{5}\tilde{y})\tilde{x}\Big{(}\tilde{y}^{-1}-\displaystyle\frac{3}{2}\tilde{x}\tilde{y}\Big{)}\tilde{y}\displaystyle\frac{\partial}{\partial\tilde{y}},$

and then multiplying such expression by $\tilde{x}^{-6}$ we have

$\widetilde{X}_{A}=(\tilde{y})\displaystyle\frac{\tilde{y}}{2}\tilde{x}\displaystyle\frac{\partial}{\partial\tilde{x}}+(\tilde{y})\Big{(}1-\displaystyle\frac{3}{2}\tilde{y}^{2}\Big{)}\displaystyle\frac{\partial}{\partial\tilde{y}},$

which implies that we have the following constrained system in the positive
$x$-direction:

$\tilde{y}\dot{\tilde{x}}=\displaystyle\frac{\tilde{x}\tilde{y}}{2},\
\tilde{y}\dot{\tilde{y}}=1-\displaystyle\frac{3\tilde{y}^{2}}{2}.$

In this chart, the points $(0,\pm\sqrt{\frac{2}{3}})$ are hyperbolic saddles
of the adjoint vector field and the impasse set intersects the exceptional
divisor $\\{x=0\\}$ at the origin. It is also interesting to observe what
happens to the Newton polygon of the auxiliary vector field (8) when we blow-
up the origin in this direction. The main segment $\gamma_{1}$ is sent to the
vertical segment $\widetilde{\gamma}_{1}$ contained in the line $\\{r=6\\}$ of
the $rs$-plane. When we multiply the auxiliary vector field by
$\tilde{x}^{-6}$, we translate the vertical segment $\widetilde{\gamma}_{1}$
to the $s$-axis. See Figure 8.

Analogously, in the positive $y$-chart the auxiliary vector field (8) takes
the form

$\overline{X}_{A}=(\bar{x}\bar{y}^{5})\bar{y}\Big{(}\bar{x}^{-1}-\displaystyle\frac{2}{3}\bar{x}^{2}\Big{)}\bar{x}\displaystyle\frac{\partial}{\partial\bar{x}}+(\bar{x}\bar{y}^{5})\bar{y}\displaystyle\frac{\bar{x}^{2}\bar{y}}{3}\bar{y}\displaystyle\frac{\partial}{\partial\bar{y}},$

and then multiplying such expression by $\bar{y}^{-6}$ we have

$\overline{X}_{A}=(\bar{x})\Big{(}\bar{x}^{-1}-\displaystyle\frac{2}{3}\bar{x}^{2}\Big{)}\bar{x}\displaystyle\frac{\partial}{\partial\bar{x}}+(\bar{x})\displaystyle\frac{\bar{x}^{2}\bar{y}}{3}\bar{y}\displaystyle\frac{\partial}{\partial\bar{y}},$

which implies that we have the following constrained system in the positive
$y$-direction:

$\bar{x}\dot{\bar{x}}=1-\displaystyle\frac{2}{3}\bar{x}^{3},\
\bar{x}\dot{\bar{y}}=\displaystyle\frac{\bar{x}^{2}\bar{y}}{3}.$

Observe that in this chart the impasse set intersects the exceptional divisor
$\\{y=0\\}$ at the origin and the point
$\big{(}(\frac{3}{2})^{\frac{1}{3}},0\big{)}$ is a hyperbolic saddle of the
adjoint vector field. The main segment $\gamma_{1}$ is sent to the horizontal
segment $\bar{\gamma}_{1}$ contained in the line $\\{s=6\\}$ of the
$rs$-plane. When we multiply the auxiliary vector field by $\bar{y}^{-6}$, we
translate the horizontal segment $\bar{\gamma}_{1}$ to the $r$-axis. See
Figure 8.

Figure 8. Main segment after the blow-up in the $y$ direction (above) and
after the blow-up in the $x$ direction (below).

Observe that all points in the exceptional divisor $\\{x=0\\}$ are elementary,
therefore the strict transformed of $\eqref{exe-cusp-fold}$ is elementary in
$\\{x=0\\}$. See Figure 9.

Figure 9. Constrained system (7) after (left) and before (right) the weighted
blow-up at the origin.

##  3\. Properties of the Newton polygon: controllability and the notion of
elementary point

As we discussed previously, the Newton polygon strongly depends on the
coordinates adopted. This means that after a blow up or a change of
coordinates, we may obtain a completely different Newton polygon. The next
goal is to defined tools in order to control the polygon during the resolution
of singularities. More Precisely, we define tools to control the height of the
Newton polygon. We also introduce a change of coordinates that will be useful
during the resolution process (the so called admissible change of
coordinates). In the end of this section, we show that non elementary points
of a constrained system can be identified by means of the Newton polygon of
its auxiliary vector field.

###  3.1. Preparation of coordinates

We say that a coordinate system is adapted to the divisor $\mathbb{E}$ if in
such coordinate system the exceptional divisor is given by
$\mathbb{E}_{x}=\\{x=0\\}$, $\mathbb{E}_{y}=\\{y=0\\}$ or
$\mathbb{E}_{x,y}=\\{xy=0\\}$ (see Definition 5.1, [7]).

######  Definition 8.

The Newton polygon $\mathcal{P}$ associated to a vector field is controllable
if the main vertex $v_{0}$ is contained in $\\{0\\}\times\mathbb{N}$ or
$\\{-1\\}\times\mathbb{N}$.

Take the diagonalized constrained system

(9) $\delta(x,y)\dot{x}=P(x,y),\ \delta(x,y)\dot{y}=Q(x,y),$

where $\delta$, $P$ and $Q$ are real analytic functions written as

$\delta(x,y)=\sum c_{k,l}x^{k}y^{l}$, $P(x,y)=\sum a_{m,n}x^{m}y^{n}x$ and
$Q(x,y)=\sum b_{m,n}x^{m}y^{n}y$.

The auxiliary vector field $X_{A}$ associated to (9) is then given by

(10) $X_{A}(x,y)=\Bigg{(}\sum c_{k,l}x^{k}y^{l}\Bigg{)}\Bigg{(}\sum
x^{m}y^{n}\big{(}a_{m,n}x\displaystyle\frac{\partial}{\partial
x}+b_{m,n}y\displaystyle\frac{\partial}{\partial y}\big{)}\Bigg{)}$

In [11] was considered vector fields with isolated equilibrium points. This
condition implies that $\partial\mathcal{P}$ intersects the $rs$-axes of the
$rs$-plane. However, this is not always true when we consider vector fields
with curves of singularities, which is the case of the auxiliary vector field.
For example, consider the constrained system

$xy\dot{x}=x^{k};\ xy\dot{y}=2y^{l},$

whose auxiliary vector field is given by

$X_{A}=xyx^{k}\displaystyle\frac{\partial}{\partial
x}+xyy^{l}\displaystyle\frac{\partial}{\partial y}.$

The support of $X_{A}$ is then given by $\mathcal{Q}=\\{(k,1),(1,l)\\}$, and
therefore $\partial\mathcal{P}$ does not intersect the $rs$-axes of the
$rs$-plane.

In the resolution of singularities, we will assume that $\partial\mathcal{P}$
intersects at least the $s$-axis. The next lemma shows that this is not a
restrictive condition.

######  Lemma 9.

Consider the constrained system (9) near a point $p\not\in\partial\mathcal{M}$
and its auxiliary vector field (10). Define the change of coordinates

(11) $\Phi(\bar{x},\bar{y})=(\bar{x}+\lambda\bar{y},\bar{y});$

with $\lambda\in\mathbb{R}$. Then the following statements are true:

  1. (1)

The diffeomorphism $\Phi$ gives us an orbital equivalence between (9) and

(12)
$\left\\{\begin{array}[]{rcl}\delta(\bar{x}+\lambda\bar{y},\bar{y})\dot{\bar{x}}&=&P(\bar{x}+\lambda\bar{y},\bar{y})-\lambda
Q(\bar{x}+\lambda\bar{y},\bar{y}),\\\
\delta(\bar{x}+\lambda\bar{y},\bar{y})\dot{\bar{y}}&=&Q(\bar{x}+\lambda\bar{y},\bar{y}).\end{array}\right.$

  2. (2)

The diffeomorphism $\Phi$ gives us a topological equivalence between (10) and
the auxiliary vector field of (12).

  3. (3)

The Newton polygon $\mathcal{P}$ associated to the auxiliary vector field of
(12) has $(0,r)$ or $(-1,r)$ as main vertex, where $r$ is the lowest degree of
its expansion.

###### Proof.

Observe that

$\left\\{\begin{array}[]{rcl}\bar{x}&=&x-\lambda y,\\\
\bar{y}&=&y,\end{array}\right.$ $\Rightarrow$
$\left\\{\begin{array}[]{rcl}\dot{\bar{x}}&=&\dot{x}-\lambda\dot{y},\\\
\dot{\bar{y}}&=&\dot{y},\end{array}\right.$

and then we obtain the system (12). Since $\Phi$ is a diffeomorphism, it maps
the impasse set of (12) onto the impasse set of (9). Moreover, $\Phi$ maps
phase curves of (12) in phase curves of (9). This implies that (12) and (9)
are orbitally equivalent. This also implies that (10) and the auxiliary vector
field of (12) are topologically equivalent, provided outside the impasse set
the phase curves of a constrained system and its auxiliary vector field are
the same. Then items 1 and 2 are true.

After this coordinate change, the auxiliary vector field takes the form

$\scriptstyle X_{A}=\Bigg{(}\sum
c_{k,l}(\bar{x}+\lambda\bar{y})^{k}\bar{y}^{l}\Bigg{)}\Bigg{(}\sum(\bar{x}+\lambda\bar{y})^{m}\bar{y}^{n}\big{(}(a_{m,n}(\bar{x}+\lambda\bar{y})-\lambda
b_{m,n}\bar{y})\frac{\partial}{\partial\bar{x}}+b_{m,n}\bar{y}\frac{\partial}{\partial\bar{y}}\big{)}\Bigg{)}.$

Without loss of generality (and in order to simplify the notation), we will
consider the first jet from now. Then we obtain

$\scriptstyle\sum_{i=0}^{k}c_{k,l}\binom{k}{i}\lambda^{i}\bar{x}^{k-i}\bar{y}^{l+i}\Bigg{(}\sum_{j=0}^{m}\binom{m}{j}\lambda^{j}\bar{x}^{m-j}\bar{y}^{n+j}\big{(}a_{m,n}\bar{x}+\lambda\bar{y}(a_{m,n}-b_{m,n})\big{)}\Bigg{)}\frac{\partial}{\partial\bar{x}}$

in the $\bar{x}$ component and

$\scriptstyle\sum_{i=0}^{k}c_{k,l}\binom{k}{i}\lambda^{i}\bar{x}^{k-i}\bar{y}^{l+i}\Bigg{(}\sum_{j=0}^{m}\binom{m}{j}\lambda^{j}\bar{x}^{m-j}\bar{y}^{n+j}b_{m,n}\Bigg{)}\bar{y}\frac{\partial}{\partial\bar{y}}$

in the $\bar{y}$ component.

If there is a pair $(m,n)$ such that $a_{m,n}-b_{m,n}\neq 0$, then the
$\bar{x}$ component gives a contribution of the form $(-1,r)$ for the new
support $\overline{\mathcal{Q}}$, where $r=(k+m)+(l+n)$. If
$a_{m,n}-b_{m,n}=0$ for all $(m,n)$, observe that at least one $b_{m,n}$ is
nonzero and then the $\bar{y}$ component gives a contribution of the form
$(0,r)$. ∎

By Lemma 9 we can always suppose that the Newton polygon of the auxiliary
vector field is controllable. Furthermore, the case where
$p\in\partial\mathcal{M}$ gives us a controllable polygon in adapted
coordinates since $\partial\mathcal{M}$ is invariant by the adjoint vector
field.

###  3.2. Admissible change of coordinates

In this section we consider another change of coordinates that will be useful
during the resolution process. Such change of coordinates was called
admissible in [11].

######  Lemma 10.

Consider a 2-dimensional constrained system near a point $p\in\mathcal{M}$,
written in local coordinates as (9). Consider also its auxiliary vector field
(10). Suppose also that its Newton polygon $\mathcal{P}_{1}$ is controllable.
Define the change of coordinates

(13) $\Psi(\bar{x},\bar{y})=(\bar{x},-\alpha\bar{x}^{\beta}+\bar{y}),$

with $\alpha\in\mathbb{R}$ and $\beta\in\mathbb{N}$. Then the following
statements are true:

  1. (1)

The diffeomorphism $\Psi$ is an orbital equivalence between (9) and

(14)
$\left\\{\begin{array}[]{rcl}\delta(\bar{x},\bar{y}-\alpha\bar{x}^{\beta})\dot{\bar{x}}&=&P(\bar{x},\bar{y}-\alpha\bar{x}^{\beta}),\\\
\delta(\bar{x},\bar{y}-\alpha\bar{x}^{\beta})\dot{\bar{y}}&=&\alpha\beta
P(\bar{x},\bar{y}-\alpha\bar{x}^{\beta})\bar{x}^{\beta-1}+Q(\bar{x},\bar{y}-\alpha\bar{x}^{\beta}).\end{array}\right.$

  2. (2)

The diffeomorphism $\Psi$ is a topological equivalence between (10) and the
auxiliary vector field of (14).

  3. (3)

The Newton polygons $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ associated to the
auxiliary vector field of (14) and (10), respectively, have the same height.

  4. (4)

If the Newton polygon of (10) has a non horizontal main segment, then the
slope of the main segment increases after this change of coordinates.

  5. (5)

If before this change of coordinates the divisor is locally given by
$\mathbb{E}_{x}$, then in this new coordinate system the divisor is still
invariant by the adjoint vector field.

###### Proof.

Items 1 and 2 are proved applying the same ideas of the proof of the Lemma 9,
so we will prove items 3, 4 and 5.

After a change of coordinates of the form (13), the auxiliary vector field
associated to (14) takes the form

$\scriptstyle\Bigg{(}\sum
c_{k,l}\bar{x}^{k}(\bar{y}-\alpha\bar{x}^{\beta})^{l}\Bigg{)}\Bigg{(}\sum\bar{x}^{m}(\bar{y}-\alpha\bar{x}^{\beta})^{n}a_{m,n}\Bigg{)}\bar{x}\frac{\partial}{\partial\bar{x}}$

in the $\bar{x}$ component and

$\scriptstyle\Bigg{(}\sum
c_{k,l}\bar{x}^{k}(\bar{y}-\alpha\bar{x}^{\beta})^{l}\Bigg{)}\Bigg{(}\sum\bar{x}^{m}(\bar{y}-\alpha\bar{x}^{\beta})^{n}\big{(}(\alpha\beta
a_{m,n}-\alpha
b_{m,n})\bar{x}^{\beta}+b_{m,n}\bar{y}\big{)}\Bigg{)}\frac{\partial}{\partial\bar{y}}$

in the $\bar{y}$ component. Once again, without loss of generality we will
consider the first jet (in order to simplify the notation). Then we obtain

$\scriptstyle\Bigg{(}\sum_{i=0}^{l}c_{k,l}\binom{l}{i}(-\alpha)^{l-i}\bar{x}^{k+\beta(l-i)}\bar{y}^{i}\Bigg{)}\Bigg{(}\sum_{j=0}^{n}\binom{n}{j}(-\alpha)^{n-j}a_{m,n}\bar{x}^{m+\beta(n-j)}\bar{y}^{j}\Bigg{)}\bar{x}\frac{\partial}{\partial\bar{x}}$

in the $\bar{x}$ component and

$\scriptstyle\Bigg{(}\sum_{i=0}^{l}c_{k,l}\binom{l}{i}(-\alpha)^{l-i}\bar{x}^{k+\beta(l-i)}\bar{y}^{i}\Bigg{)}\Bigg{(}\sum_{j=0}^{n}\binom{n}{j}(-\alpha)^{n-j}\bar{x}^{m+\beta(n-j)}\bar{y}^{j}\big{(}(\alpha\beta
a_{m,n}-\alpha
b_{m,n})\bar{x}^{\beta}+b_{m,n}\bar{y}\big{)}\Bigg{)}\frac{\partial}{\partial\bar{y}}$

in the $\bar{y}$ component.

By assumption, the auxiliary vector field (10) has $(-1,r)$ or $(0,r)$ as main
vertex, where $r$ is the degree of the lowest graduation. This implies that
there are $(k,l)$ and $(m,n)$ such that $(k+m)=-1$ (respectively $0$) and
$(l+n)=r$. By these two expressions obtained previously, this property is
still true for $\beta\in\mathbb{N}$, then we still have a point of the form
$(-1,r)$ or $(0,r)$ as main vertex.

Since the main vertex was preserved, observe now that this operation increases
the powers of the variable $x$ (because $\beta>0$), then the $r$ entries of
the points of the new support $\overline{\mathcal{Q}}$ are increased.
Therefore, the points of the support (except the main vertex) are translated
to the right and thus the slope of the main segment increases. See Figure 10.

Figure 10. Newton polygon of the auxiliary vector field after the change of
coordinates (13).

In the case where the divisor is locally given by $\mathbb{E}_{x}$, we have
that the main vertex is of the form $(0,r)$ provided that the divisor is
invariant by the adjoint vector field. After this coordinate change, the
divisor is still locally given by $\mathbb{E}_{x}$. Moreover, by these two
expressions of the auxiliar vector field obtained previously it follows that
the divisor will be once again invariant by the adjoint vector field. ∎

######  Definition 11.

The finite composition of change of coordinates of the form (13) is called
admissible change of coordinates.

After an admissible change of coordinates, we obtain a coordinate system
adapted to the divisor, the exceptional divisor is still invariant by the
adjoint vector and the Newton polygon of the auxiliary vector field is still
controllable.

###  3.3. The relation between an elementary 2-dimensional constrained system
and its Newton polygon

During the resolution of singularities, we must apply a finite number of
operations (blow-ups) in order to obtain a “simpler” constrained system. This
notion of “simple” comes from the notion of elementary constrained system
(Definition 2). In what follows we show how to identify elementary points of
the constrained system by means of the Newton polygon of the auxiliary vector
field.

######  Definition 12.

A planar constrained differential system is Newton elementary at $p$ if the
Newton polygon $\mathcal{P}$ associated to the auxiliary vector field $X_{A}$
satisfies one of the following:

  * •

The main vertex of $\mathcal{P}$ is $(0,0)$, $(0,-1)$ or $(-1,0)$ (that is, if
the height is less or equal to zero);

  * •

The main segment $\gamma_{1}$ is horizontal.

Example: We present an example to justify why we consider the case where
$\gamma_{1}$ is horizontal as a final situation. Consider the constrained
system

$y\dot{x}=2x,\ y\dot{y}=-y.$

The support of the auxiliary vector field only contains the point $(0,1)$, and
it is straightforward to see that the origin is an hyperbolic saddle of the
adjoint vector field, where $\\{x=0\\}$ and $\\{y=0\\}$ are the separatrices.
This means that $\\{y=0\\}$ is at the same time a separatrix and the impasse
set. It is not possible to change this configuration with a finite sequence of
blowing-ups at the origin.

######  Theorem 13.

A planar constrained system is elementary at $p$ if, and only if, it is Newton
elementary at $p$.

###### Proof.

Fix a controllable coordinate system near $p$ and consider the auxiliary
vector field

(15) $\scriptstyle X_{A}(x,y)=\Bigg{(}\sum
c_{k,l}x^{k}y^{l}\Bigg{)}\Bigg{(}\sum
x^{m}y^{n}\big{(}a_{m,n}x\frac{\partial}{\partial
x}+b_{m,n}y\frac{\partial}{\partial y}\big{)}\Bigg{)}.$

Firstly, suppose that the constrained system is Newton elementary at $p$.
Observe that we do not have $m=-1$ and $n=-1$ at the same time in (15). Take
without loss of generality $m\geq 0$ and $n\geq-1$.

If $(0,0)\in\mathcal{P}$, by the expression (15) we have $(k+m)=0$ and
$(l+n)=0$. It follows that $k=m=0$ because $m\geq 0$. Since $(l+n)=0$,
$n\geq-1$ and $l\geq 0$, we have two cases to consider:

  * •

If $l=1$ and $n=-1$, then $\delta$ is regular at $p$ and $p$ is not an
equilibrium point of $X$. If $p\in\mathbb{E}_{x}$, then the impasse set
$\Delta$ intersects the divisor transversally at $p$.

  * •

If $l=n=0$, then $p\not\in\Delta$. Moreover, since $a_{-1,1}=0$ and
$a_{00}^{2}+b^{2}_{00}\neq 0$, $p$ is an elementary equilibrium point of $X$.

It follows that if $(0,0)\in\mathcal{P}$, then
$(\mathcal{M},\mathcal{X},\mathcal{I})$ is elementary at $p$.

Suppose now that $(0,-1)\in\mathcal{P}$. We have that $(k+m)=0$ and
$(l+n)=-1$, and this implies that $k=m=l=0$ and $n=-1$. Therefore,
$p\not\in\Delta$ and $p$ is not an equilibrium point of $X$. The proof for the
case where $(-1,0)$ is the main vertex is completely analogous.

Finally, suppose that the main segment $\gamma_{1}$ is horizontal. Since the
Newton polygon of the auxiliary vector field is obtained by means of the
Minkowski sum between the support of the adjoint vector field and the impasse
curve, then the Newton polygon of the impasse curve and the Newton polygon of
the adjoint vector field have horizontal main segment. This implies that in
this system of coordinates the impasse curve is given by

$\delta=y\Bigg{(}c_{0,1}+\sum c_{k,l}x^{k}y^{l}\Bigg{)},$

and we have the following cases, since $p$ is either an isolated equilibrium
point or a regular point of the adjoint vector field $X$:

  * •

$a_{-1,0}\neq 0$, which implies that $b_{n,-1}=0$ for all $n$ and then the
impasse curve coincides with a phase curve of $X$.

  * •

$b_{0,-1}\neq 0$, which implies that $X$ is transversal to $\Delta$ near the
origin.

  * •

$a_{00}^{2}+b_{00}^{2}\neq 0$, and then a separatrix of the equilibrium point
(eventually the central manifold) coincides with the impasse curve.

In these three cases, the constrained system is elementary at $p$.

Now suppose that $(\mathcal{M},\mathcal{X},\mathcal{I})$ is elementary at $p$.
Without loss of generality we can choose coordinates such that
$\delta(x,y)=y$.

When $p$ is not an equilibrium point of $X$ and $p\not\in\Delta$, then
$a^{2}_{-1,0}+b^{2}_{0,-1}\neq 0\neq c_{00}$ and we have
$(0,-1)\in\mathcal{P}$ or $(-1,0)\in\mathcal{P}$. If $p\in\Delta$ but $p$ is
not an equilibrium point of $X$, then we have two possibilities:

  * •

The adjoint vector field is transversal to the impasse set $\Delta$ and
therefore $c_{0,1}\neq 0$ and $b_{0,-1}\neq 0$, which implies that
$(0,0)\in\partial\mathcal{P}$.

  * •

The phase curve passing through $p$ coincides with the impasse set $\Delta$,
and therefore $a_{-1,0}\neq 0$ and $b_{n,-1}=0$ for all $n$. Then
$a_{-1,0}\neq 0\neq c_{0,1}$ gives the contribution $(-1,1)\in\mathcal{P}$ and
the main segment of the polygon is horizontal.

If $p$ is a semi-hyperbolic equilibrium point of $X$ and $p\not\in\Delta$, we
are in the classical case studied in [11] and therefore $(0,0)\in\mathcal{P}$.

Finally, suppose that $p$ is a semi hyperbolic equilibrium point of $X$ and
$p\in\Delta$. Remember that we are considering adapted coordinates and
$\delta(x,y)=y$. By hypothesis, $\Delta$ must be contained in a separatrix (or
the central manifold) of $p$. This means that such phase curve is not
transversal to $\Delta$ and all coefficients of the form $b_{n,-1}$ are zero.
On the other hand, $a_{0,0}^{2}+b_{0,0}^{2}\neq 0$, thus the main segment is
horizontal and the statement is true. ∎

To end this section, we remark that when the divisor is locally given by
$\mathbb{E}_{xy}$ we cannot have an elementary constrained system with
$\Delta$ intersecting the origin. Indeed, since $\Delta$ is not contained in
$\mathbb{E}_{x}\cup\mathbb{E}_{y}$ and $\Delta$ is transversal to
$\mathbb{E}_{x}$ and $\mathbb{E}_{y}$ simultaneously, it follows that
$c_{1,0}^{2}+c_{0,1}^{2}\neq 0$. Regarding that the origin in
$\mathbb{E}_{xy}$ is always a hyperbolic equilibrium point of the adjoint
vector field, we have that $a_{00}^{2}+b_{00}^{2}\neq 0$, hence
$(1,0),(0,1)\in\mathcal{P}$ and then we must continue the resolution process.

##  4\. Resolution of singularities: local strategy and proof of the main
Theorem

In this section we study the local theorem of the resolution of singularities.
Firstly, we give the proof of the local theorem (which is basically the
description of the algorithm of resolution of singularities) and later we give
the proof of the lemmas and propositions used in the proof of Theorem 14. We
will focus only in the positives $x$ and $y$ charts because the computations
on the negative charts are completely analogous. Remember that all singular
points of the constrained differential system are isolated.

######  Theorem 14.

(Local Theorem) Let $(\mathcal{M},\mathcal{X},\mathcal{I})$ be a 2-dimensional
constrained system and let $p\in\mathcal{M}$. Fix local coordinates such that
the Newton polygon of the auxiliary vector field $X_{A}$ is controllable. Then
there is a finite sequence of weighted blow ups such that the strict
transformed is elementary in all points of the excepcional divisor.

###### Proof.

As usual, denote by $X$ the adjoint vector field in such local coordinates and
by $\Delta$ the impasse set. Let $X_{A}$ be the auxiliary vector field.
Firstly, we find the expression of the affine line
$\\{r\omega_{1}+s\omega_{2}=R\\}$ that contains the main segment $\gamma_{1}$.
If $\gamma_{1}$ is horizontal, we are done by Theorem 13. If this is not the
case, denote the height of the Newton polygon by $h$ and define the weight
vector as $\omega=(\omega_{1},\omega_{2})$. After a weighted blow-up in the
the $y$ direction, it follows by Lemma 16 that the origin of the excepcional
divisor $\mathbb{E}_{y}$ is elementary. In other words, after just one
weighted blow up to in the $y$ direction the origin of $\mathbb{E}_{y}$ is
elementary.

Blowing up the point $p$ in the $x$ direction, by Lemma 17 the new height
$h_{1}$ of the Newton polygon in the origin of the divisor $\mathbb{E}_{x}$ is
less than $h$, that is, the height decreased. Now, we need to study the other
points of $\mathbb{E}_{x}$. Due to Proposition 19, there are two
possibilities:

  1. (1)

There is a finite number $N$ of points $p_{i}$, $1\leq i\leq N$ in the
exceptional divisor such that the height $h_{p_{i}}$ of the Newton polygon at
$p_{i}$ is positive. Moreover, $h_{p_{i}}\leq h$ for all $i$.

  2. (2)

If there is a point in $\mathbb{E}_{x}$ such that $h_{p_{i}}=h$, this is the
only point in the divisor that has positive height.

In the first case, there is a finite number of points $p_{i}$ in the divisor
such that or $p_{i}$ is not a semi-hyperbolic equilibrium point of the adjoint
vector field or the impasse set intersects the divisor at $p_{i}$. Eventually,
a point on the divisor can be an equilibrium point and an impasse point at the
same time. We must continue blowing up the points in which the height is
positive and the main segment is not horizontal. The weights are chosen in the
same way as before.

For the second case, we remark that this point cannot be the origin. Indeed,
we show that in this case the first graduation of the auxiliary vector field
is very particular (see Proposition 20). So we apply an admissible change of
coordinates, such as in Lemma 10. Such change of coordinates does not increase
the height of the polygon, but it increases the slope of the main segment. Now
we obtain a new weight vector to continue the process, due to Proposition 21.

At each step, we have the same two possibilities. Since there is a finite
number of non-elementary points on the divisor, the height is finite and at
each step the height decreases (eventually applying admissible change of
coordinates), after a finite number $n_{0}$ of blowing ups we have that all
points on the divisor do not have positive height or the main segment is
horizontal, that is, all points are elementary.

The worst scenario is when a finite composition of admissible change of
coordinates is not sufficient to obtain a weight vector such that the blow-up
such vector decreases the height. In this case, there is an analytic change of
coordinates such that, in this new system of coordinates, the main segment is
horizontal and then we finish the resolution process (see Proposition 21). ∎

In the next subsection we show the lemmas and propositions used in the
previous proof. Before such study, we discuss some similarities and
differences between our approach and that one in [11].

Just as in the work of Pelletier [11], our proof is based on weighted blow ups
and the Newton polygon. Therefore, many ideas presented here can be found in
such work, for example, the behavior of the Newton polygon during the
resolution process. Nevertheless, some definitions were inspired in [10].

We recall that in [11] the author considered analytic vector fields such that
the origin in an isolated equilibrium point. Furthermore, in such paper the
author proved that applying a finite sequence of blowing-ups in a singularity,
we decrease the so called total multiplicity of a singularity (eventually
applying some change of coordinates).

However, in this work we consider the Newton polygon of the auxiliary vector
field, which is an analytic vector field that has a curve of equilibrium
points. Here, we show that applying a finite sequence of blowing-ups in a
singularity, we decrease the height of the Newton polygon (also applying some
change of coordinates when necessary). In this sense, some ideas of our
approach were inspired in [2].

###  4.1. The local strategy

From now on, fix local coordinates for the constrained differential system
$(\mathcal{M},\mathcal{X},\mathcal{I})$, denote the local vector field by $X$
and the impasse set by $\Delta$. Let $X_{A}$ be the auxiliary vector field
(10) and assume that the Newton polygon is controllable and the main segment
$\gamma_{1}$ is not horizontal. Recall that the weight vector
$\omega=(\omega_{1},\omega_{2})$ is chosen by the expression of the affine
line $\\{r\omega_{1}+s\omega_{2}=R\\}$ that contains the main segment
$\gamma_{1}$.

######  Lemma 15.

After a blow-up with weight $\omega=(\omega_{1},\omega_{2})$, the auxiliary
vector field (10) takes the form

(16) $\scriptstyle\widetilde{X}_{A}(\tilde{x},\tilde{y})=\Bigg{(}\sum
c_{k,l}\tilde{x}^{k\omega_{1}+l\omega_{2}}\tilde{y}^{l}\Bigg{)}\Bigg{(}\sum\tilde{x}^{m\omega_{1}+n\omega_{2}}\tilde{y}^{n}\big{(}\frac{a_{m,n}}{\omega_{1}}\tilde{x}\frac{\partial}{\partial\tilde{x}}+(b_{m,n}-\frac{\omega_{2}}{\omega_{1}}a_{m,n})\tilde{y}\frac{\partial}{\partial\tilde{y}}\big{)}\Bigg{)}$

in the $x$ direction and

(17)
$\displaystyle\scriptstyle\widetilde{X}_{A}(\tilde{x},\tilde{y})=\Bigg{(}\sum
c_{k,l}\tilde{x}^{k}\tilde{y}^{k\omega_{1}+l\omega_{2}}\Bigg{)}\Bigg{(}\sum\tilde{x}^{m}\tilde{y}^{m\omega_{1}+n\omega_{2}}\big{(}(a_{m,n}-\frac{\omega_{1}}{\omega_{2}}b_{m,n})\tilde{x}\frac{\partial}{\partial\tilde{x}}+\frac{b_{m,n}}{\omega_{2}}\tilde{y}\frac{\partial}{\partial\tilde{y}}\big{)}\Bigg{)}$

in the $y$ direction.

###### Proof.

For the $x$ direction, observe that the blow-up

$x=\tilde{x}^{\omega_{1}}$, $y=\tilde{x}^{\omega_{2}}\tilde{y}$

gives us the relation

$\left(\begin{array}[]{c}\dot{x}\\\ \dot{y}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\omega_{1}\tilde{x}^{\omega_{1}-1}&0\\\
\omega_{2}\tilde{x}^{\omega_{2}-1}\tilde{y}&\tilde{x}^{\omega_{2}}\\\
\end{array}\right)\left(\begin{array}[]{c}\dot{\tilde{x}}\\\
\dot{\tilde{y}}\\\ \end{array}\right),$

and (9) becomes the constrained system

$\left\\{\begin{array}[]{rcl}\delta(\tilde{x}^{\omega_{1}},\tilde{x}^{\omega_{2}}\tilde{y})\dot{\tilde{x}}&=&\displaystyle\sum\tilde{x}^{m\omega_{1}+n\omega_{2}}\tilde{y}^{n}\frac{a_{m,n}}{\omega_{1}}\tilde{x},\\\
\delta(\tilde{x}^{\omega_{1}},\tilde{x}^{\omega_{2}}\tilde{y})\dot{\tilde{y}}&=&\displaystyle\sum\tilde{x}^{m\omega_{1}+n\omega_{2}}\tilde{y}^{n}\big{(}b_{m,n}-\frac{\omega_{2}}{\omega_{1}}a_{m,n}\big{)}\tilde{y},\end{array}\right.$

whose auxiliary vector field is of the form (16). Analogously we obtain the
expression (17) in the $y$ direction. ∎

######  Lemma 16.

After a blow-up in the $y$ direction, the Newton polygon is still
controllable, the divisor $\mathbb{E}_{y}$ is invariant by the adjoint vector
field $X$ and the origin is an elementary point of the constrained system.

###### Proof.

Take the first level of the $(\omega_{1},\omega_{2})$-graduation

$\scriptstyle
X_{A,d}^{(\omega_{1},\omega_{2})}(x,y)=\Bigg{(}\sum_{\omega_{1}k+\omega_{2}l=d_{1}}c_{k,l}x^{k}y^{l}\Bigg{)}\Bigg{(}\sum_{\omega_{1}m+\omega_{2}n=d_{2}}x^{m}y^{n}\Big{(}a_{m,n}x\frac{\partial}{\partial
x}+b_{m,n}y\frac{\partial}{\partial y}\Big{)}\Bigg{)},$

where $d_{1}+d_{2}=d$. After a blow up in the $y$ direction the auxiliary
vector field takes the from

$\displaystyle\widetilde{X}_{A}(\tilde{x},\tilde{y})={}$
$\displaystyle\Bigg{(}\sum
c_{k,l}\tilde{x}^{k}\tilde{y}^{k\omega_{1}+l\omega_{2}}\Bigg{)}\Bigg{(}\sum\tilde{x}^{m}\tilde{y}^{m\omega_{1}+n\omega_{2}}\big{(}(a_{m,n}-\frac{\omega_{1}}{\omega_{2}}b_{m,n})\tilde{x}\frac{\partial}{\partial\tilde{x}}+\frac{b_{m,n}}{\omega_{2}}\tilde{y}\frac{\partial}{\partial\tilde{y}}\big{)}\Bigg{)}=$
$\displaystyle\Bigg{(}\sum
c_{k,l}\tilde{x}^{k}\tilde{y}^{d_{1}}\Bigg{)}\Bigg{(}\sum\tilde{x}^{m}\tilde{y}^{d_{2}}\big{(}(a_{m,n}-\frac{\omega_{1}}{\omega_{2}}b_{m,n})\tilde{x}\frac{\partial}{\partial\tilde{x}}+\frac{b_{m,n}}{\omega_{2}}\tilde{y}\frac{\partial}{\partial\tilde{y}}\big{)}\Bigg{)}.$

In the $(r,s)$-plane, this operation sends lines of the form
$\\{r\omega_{1}+s\omega_{2}=R\\}$ into horizonal lines. In other words, the
main segment $\gamma_{1}$ is sent to a horizontal segment. Dividing the
expression above by $\tilde{y}^{d}$, we move this horizontal segment to the
$r$-axis (see Figure 11). Observe that the first entries of the points of the
polygon do not change. In these new coordinates, the main vertex of the Newton
polygon ate the origin is $(-1,0)$ or $(0,0)$. By Theorem 13, the origin is
now an elementary point. ∎

Figure 11. Main segment of the Newton polygon after a blow up at the $y$
direction.

######  Lemma 17.

After a blow-up in the $x$ direction, the main vertex is contained in
$\\{0\\}\times\mathbb{N}$ (and therefore the Newton polygon is still
controllable), the divisor $\mathbb{E}_{x}$ is invariant by the adjoint vector
field $X$. Moreover, the height of the Newton polygon decreases.

###### Proof.

We recall that the choice of the weight vector depends on the equation of the
main segment $\gamma_{1}$. Once the choice is made, we take the first level of
the $(\omega_{1},\omega_{2})$-graduation

$\scriptstyle
X_{A,d}^{(\omega_{1},\omega_{2})}(x,y)=\Bigg{(}\sum_{\omega_{1}k+\omega_{2}l=d_{1}}c_{k,l}x^{k}y^{l}\Bigg{)}\Bigg{(}\sum_{\omega_{1}m+\omega_{2}n=d_{2}}x^{m}y^{n}\Big{(}a_{m,n}x\frac{\partial}{\partial
x}+b_{m,n}y\frac{\partial}{\partial y}\Big{)}\Bigg{)},$

where $d_{1}+d_{2}=d$.

Firstly, we simplify the notation. It is sufficient to study the first level
of the graduation. For each $l$, there is only one $k$ such that
$\omega_{1}k+\omega_{2}l=d_{1}$. Analogously, for each $n$, there is only one
$m$ such that $\omega_{1}m+\omega_{2}n=d_{2}$. So we rewrite the last
expression as

$\scriptstyle
X_{A,d}^{(\omega_{1},\omega_{2})}(x,y)=\Bigg{(}\sum_{l=0}^{r_{1}}c_{k,l}x^{k}y^{l}\Bigg{)}\Bigg{(}\sum_{n=-1}^{r_{2}}x^{m}y^{n}\Big{(}a_{m,n}x\frac{\partial}{\partial
x}+b_{m,n}y\frac{\partial}{\partial y}\Big{)}\Bigg{)},$

where $\omega_{2}r_{1}=d_{1}$ and $\omega_{2}r_{2}=d_{2}$ (if the main vertex
is contained in $\\{0\\}\times\mathbb{N}$) or
$\omega_{2}r_{2}=d_{2}+\omega_{1}$ (if the main vertex is contained in
$\\{-1\\}\times\mathbb{N}$). From (16), after a blow-up in the $x$ direction,
we have

$\scriptstyle\Bigg{(}\sum_{l=0}^{r_{1}}c_{k,l}\tilde{x}^{\omega_{1}k+\omega_{2}l}\tilde{y}^{l}\Bigg{)}\Bigg{(}\sum_{n=-1}^{r_{2}}\tilde{x}^{\omega_{1}m+\omega_{2}n}\tilde{y}^{n}\Big{(}\frac{a_{m,n}}{\omega_{1}}\tilde{x}\frac{\partial}{\partial\tilde{x}}+(b_{m,n}-\frac{\omega_{2}}{\omega_{1}}a_{m,n})\tilde{y}\frac{\partial}{\partial\tilde{y}}\Big{)}\Bigg{)}.$

Concerning the Newton polygon, this operation sends lines of the form
$\\{r\omega_{1}+s\omega_{2}=R\\}$ into vertical lines. Moreover, the second
entries of the points of the polygon do not change. Therefore, $\gamma_{1}$ is
sent into a vertical segment. Now, multiplying the auxiliary vector field by
$\tilde{x}^{-d}$, we get

(18)
$\scriptstyle\Bigg{(}\sum_{l=0}^{r_{1}}c_{k,l}\tilde{y}^{l}\Bigg{)}\Bigg{(}\sum_{n=-1}^{r_{2}}\tilde{y}^{n}\Big{(}\frac{a_{m,n}}{\omega_{1}}\tilde{x}\frac{\partial}{\partial\tilde{x}}+(b_{m,n}-\frac{\omega_{2}}{\omega_{1}}a_{m,n})\tilde{y}\frac{\partial}{\partial\tilde{y}}\Big{)}\Bigg{)}.$

This operation moves the vertical lines to the left. In particular, the main
segment $\gamma_{1}$ is now contained in $\\{0\\}\times\mathbb{N}$ (see Figure
12). Hence in this new coordinate system the main vertex has the form $(0,r)$
and the Newton polygon is controllable. Moreover, the divisor $\mathbb{E}_{x}$
is invariant by the adjoint vector field. Finally, since there is a vertex
$v_{1}=(i_{1},j_{1})\in\gamma_{1}$ where $j_{1}$ is smaller than the ordinate
the old main vertex $v_{0}$, the height of the Newton polygon decreased. ∎

Figure 12. Main segment of the Newton polygon after a blow up at the $x$
direction.

Now, we must study what happens to the other points of the exceptional
divisor.

######  Lemma 18.

Consider the auxiliary vector field (18), which is the auxiliary vector field
after a blow-up in the $x$ direction. After a translation of the form

(19) $\Omega(\bar{x},\bar{y})=(\bar{x},\bar{y}+\xi),$

where $\xi\in\mathbb{R}$, the auxiliary vector field takes the form (dropping
the tildes and bars in order to simplify the notation)

(20) $\scriptstyle
X_{A}(x,y)=\Bigg{(}\sum_{i=0}^{r_{1}}C_{i}(\xi)y^{i}\Bigg{)}\Bigg{(}\sum_{j=0}^{r_{2}}\alpha_{j}(\xi)y^{j}x\frac{\partial}{\partial
x}+\sum_{j=-1}^{r_{2}}\beta_{j}(\xi)y^{j}y\frac{\partial}{\partial
y}\Bigg{)},$

where

$\scriptstyle C_{i}(\xi)=\sum_{l=i}^{r_{1}}c_{l}\xi^{l-i}\binom{l}{i}$,
$\scriptstyle\alpha_{j}(\xi)=\sum_{n=j}^{r_{2}}\frac{a_{n}}{\omega_{1}}\xi^{n-j}\binom{n}{j}$
and
$\scriptstyle\beta_{j}(\xi)=\sum_{n=j}^{r_{2}}(b_{n}-\frac{\omega_{2}}{\omega_{1}}a_{n})\xi^{n-j}\binom{n+1}{j+1}$,

and such constants satisfy:

$\scriptstyle\frac{\partial}{\partial\xi}C_{i}(\xi)=(i+1)C_{i+1}(\xi)$,
$\scriptstyle\frac{\partial}{\partial\xi}\alpha_{j}(\xi)=(j+1)\alpha_{j+1}(\xi)$
and
$\scriptstyle\frac{\partial}{\partial\xi}\beta_{j}(\xi)=(j+2)\alpha_{j+1}(\xi)$.

Moreover, after this change of coordinates, the height of the Newton polygon
does not increase, the main vertex is contained in $\\{0\\}\times\mathbb{N}$
and the exceptional divisor is still invariant by the adjoint vector field.

###### Proof.

Take the equation (18) and the translation (19). First, we recall the
following property of sum:

$\scriptstyle\sum_{i=k}^{n}\sum_{j=k}^{i}z_{i,j}=\sum_{j=k}^{n}\sum_{i=j}^{n}z_{i,j},$

where, $k\leq j\leq i\leq n$ are integers. In the $x$ component of the
auxiliary vector field, we have (dropping the tildes):

$\displaystyle\Bigg{(}{}$
$\displaystyle\sum_{l=0}^{r_{1}}c_{l}(y+\xi)^{l}\Bigg{)}\Bigg{(}\sum_{n=0}^{r_{2}}(y+\xi)^{n}\Big{(}\frac{a_{n}}{\omega_{1}}\Big{)}\Bigg{)}x=\Bigg{(}\sum_{l=0}^{r_{1}}c_{l}\sum_{i=0}^{l}y^{i}\xi^{l-i}\binom{l}{i}\Bigg{)}\Bigg{(}\sum_{n=0}^{r_{2}}\frac{a_{n}}{\omega_{1}}\sum_{j=0}^{n}y^{j}\xi^{n-j}\binom{n}{j}\Bigg{)}x=$
$\displaystyle=\Bigg{(}\sum_{i=0}^{r_{1}}y^{i}\sum_{l=i}^{r_{1}}c_{l}\xi^{l-i}\binom{l}{i}\Bigg{)}\Bigg{(}\sum_{j=0}^{r_{2}}y^{j}\sum_{n=j}^{r_{2}}\frac{a_{n}}{\omega_{1}}\xi^{n-j}\binom{n}{j}\Bigg{)}x=\Bigg{(}\sum_{i=0}^{r_{1}}y^{i}C_{i}(\xi)\Bigg{)}\Bigg{(}\sum_{j=0}^{r_{2}}y^{j}\alpha_{j}(\xi)\Bigg{)}x.$

In the $y$ component, we have (again dropping the tildes):

$\displaystyle\Bigg{(}\sum_{i=0}^{r_{1}}y^{i}C_{i}(\xi)\Bigg{)}$
$\displaystyle\Bigg{(}\sum_{n=-1}^{r_{2}}\big{(}b_{n}-\frac{\omega_{2}}{\omega_{1}}a_{n}\big{)}(y+\xi)^{n+1}\Bigg{)}=$
$\displaystyle=\Bigg{(}\sum_{i=0}^{r_{1}}y^{i}C_{i}(\xi)\Bigg{)}\Bigg{(}\sum_{n=-1}^{r_{2}}\big{(}b_{n}-\frac{\omega_{2}}{\omega_{1}}a_{n}\big{)}(y+\xi)^{n+1}\frac{1}{y}\Bigg{)}y=$
$\displaystyle=\Bigg{(}\sum_{i=0}^{r_{1}}y^{i}C_{i}(\xi)\Bigg{)}\Bigg{(}\sum_{n=-1}^{r_{2}}\big{(}b_{n}-\frac{\omega_{2}}{\omega_{1}}a_{n}\big{)}\sum_{j=0}^{n+1}y^{j-1}\xi^{n+1-j}\binom{n+1}{j}\Bigg{)}y=$
$\displaystyle=\Bigg{(}\sum_{i=0}^{r_{1}}y^{i}C_{i}(\xi)\Bigg{)}\Bigg{(}\sum_{N=0}^{r_{2}}\big{(}b_{N-1}-\frac{\omega_{2}}{\omega_{1}}a_{N-1}\big{)}\sum_{j=0}^{N}y^{j-1}\xi^{N-j}\binom{N}{j}\Bigg{)}y=$
$\displaystyle=\Bigg{(}\sum_{i=0}^{r_{1}}y^{i}C_{i}(\xi)\Bigg{)}\Bigg{(}\sum_{j=0}^{r_{2}}y^{j-1}\sum_{N=j}^{r_{2}}\big{(}b_{N-1}-\frac{\omega_{2}}{\omega_{1}}a_{N-1}\big{)}\xi^{N-j}\binom{N}{j}\Bigg{)}y=$
$\displaystyle=\Bigg{(}\sum_{i=0}^{r_{1}}y^{i}C_{i}(\xi)\Bigg{)}\Bigg{(}\sum_{j=-1}^{r_{2}}y^{j}\sum_{n=j}^{r_{2}}\big{(}b_{n}-\frac{\omega_{2}}{\omega_{1}}a_{n}\big{)}\xi^{n-j}\binom{n+1}{j+1}\Bigg{)}y=$
$\displaystyle=\Bigg{(}\sum_{i=0}^{r_{1}}y^{i}C_{i}(\xi)\Bigg{)}\Bigg{(}\sum_{j=-1}^{r_{2}}y^{j}\beta_{j}(\xi)\Bigg{)}y.$

Note that

$\displaystyle\frac{\partial}{\partial\xi}C_{i}(\xi)=\frac{\partial}{\partial\xi}\Bigg{(}c_{i}+\sum_{l=i+1}^{r_{1}}c_{l}\xi^{l-i}\binom{l}{i}\Bigg{)}=\sum_{l=i+1}^{r_{1}}c_{l}(l-i)\xi^{l-(i+1)}\binom{l}{i}=$
$\displaystyle=\sum_{l=i+1}^{r_{1}}c_{l}\xi^{l-(i+1)}\frac{l!}{i!(l-(i+1))!}=\sum_{l=i+1}^{r_{1}}c_{l}\xi^{l-(i+1)}\frac{(i+1)l!}{(i+1)!(l-(i+1))!}=(i+1)C_{i+1}(\xi).$

The other expressions involving the derivative with respect to $\xi$ can be
shown analogously. Observe that $C_{r_{1}}=c_{r_{1}}\neq 0$, and

$\alpha_{r_{2}}=\displaystyle\frac{a_{r_{2}}}{\omega_{1}}$ and
$\beta_{r_{2}}=\big{(}b_{r_{2}}-\displaystyle\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}\big{)}$

does not vanish simultaneously. Therefore, the height of the Newton polygon
does not increases. We remark that this operation does not change the first
entries of the points of the polygon. It is also clear that the exceptional
divisor is invariant by the adjoint vector field. ∎

######  Proposition 19.

After a blow-up at the $x$ direction, the following statements are true:

  1. (1)

The impasse set $\Delta$ intersects the exceptional divisor only in a finite
number of points;

  2. (2)

There is a finite number of semi-hyperbolic equilibrium points of the adjoint
vector field $X$ contained in the exceptional divisor;

  3. (3)

If there is a point such that the height has not decreased, this is the only
point on the divisor that has positive height.

###### Proof.

From Lemma 18, thus the impasse set $\Delta$ is locally given by

$\scriptstyle\sum_{i=0}^{r_{1}}y^{i}C_{i}(\xi)=C_{0}(\xi)+C_{1}(\xi)y+...+C_{r_{1}}(\xi)y^{r_{1}}.$

Observe that there is a finite number of points $(0,\xi_{k})$ on the
exceptional divisor such that $C_{0}(\xi_{k})=0$. In other words, since there
is a finite number of roots for $C_{0}(\xi)$, the impasse set $\Delta$
intersects the exceptional divisor in a finite number of points.

Concerning the adjoint vector field, let us notice that the polynomials

$\scriptsize\alpha_{0}(\xi)=\sum_{n=0}^{r_{2}}\frac{a_{n}}{\omega_{1}}\xi^{n},\
\beta_{-1}(\xi)=\sum_{n=-1}^{r_{2}}(b_{n}-\frac{\omega_{2}}{\omega_{1}}a_{n})\xi^{n+1}$

are not identically zero simultaneously. Indeed, if this was the case, we
would have $a_{n}=b_{n}=0$ for all $-1\leq n\leq r_{2}$ and it would
contradict the fact that the translation does not increase the height of the
Newton polygon. Hence $\beta_{-1}\equiv 0$ or $\alpha_{0}\equiv 0$.

If $\beta_{-1}\not\equiv 0$, this polynomial has a finite number of roots,
thus by the expression of the auxiliary vector field (20) there is a finite
number of equilibrium points $(0,\xi_{k})$ of the adjoint vector field in the
exceptional divisor (eventually some of them are not elementary). If
$\beta_{-1}\equiv 0$, then $\alpha_{0}\not\equiv 0$. In this case, all points
of the exceptional divisor are equilibrium points of the adjoint vector field.
The polynomial $\alpha_{0}$ has a finite number of roots, and then by (20)
there is a finite number of points $(0,\xi_{k})$ that have positive height. In
other words, there is a finite number of non elementary equilibrium points in
the divisor.

Finally, we show that if there is a point in the excepcional divisor such that
the height of the polygon has not decreased, this is the only point in the
divisor with positive height. Let $(0,\xi_{0})$ be the point such that the
height did not decrease. We recall that $\xi_{0}\neq 0$, since by Lemma 17 the
height at the origin decreases. Then we must have

$\scriptsize\left\\{\begin{array}[]{rcccccccccl}C_{0}(\xi_{0})&=&...&=&C_{r_{1}-1}(\xi_{0})&=&0&,&C_{r_{1}}(\xi_{0})&\neq&0;\\\
\alpha_{0}(\xi_{0})&=&...&=&\alpha_{r_{2}-1}(\xi_{0})&=&0&,&\alpha_{r_{2}}(\xi_{0})&\neq&0;\\\
\beta_{-1}(\xi_{0})&=&...&=&\beta_{r_{2}-1}(\xi_{0})&=&0&,&\beta_{r_{2}}(\xi_{0})&\neq&0.\end{array}\right.$

From Lemma 18 we know that

$\scriptstyle\frac{\partial}{\partial\xi}C_{i}(\xi)=(i+1)C_{i+1}(\xi)$,
$\scriptstyle\frac{\partial}{\partial\xi}\alpha_{j}(\xi)=(j+1)\alpha_{j+1}(\xi)$
and
$\scriptstyle\frac{\partial}{\partial\xi}\beta_{j}(\xi)=(j+2)\alpha_{j+1}(\xi)$.

Then $\xi_{0}$ must be, simultaneously, root of multiplicity $r_{1}$ of
$C_{0}$, root of multiplicity $r_{2}$ of $\alpha_{0}$ and root of multiplicity
$r_{2}+1$ of $\beta_{-1}$. By the degree of such polynomials, such root is
unique, then $(0,\xi_{0})$ is the only point in the exceptional divisor with
positive height. ∎

In the case where the height does not decrease, geometrically we have a point
$p$ such that it is an equilibrium point of the adjoint vector field and an
impasse point simultaneously.

######  Proposition 20.

Under the conditions of the Proposition 19, suppose that after a weighted blow
up at the $x$ direction the height did not decrease. Then, before the blow up
in the $x$ direction, the first graduation of the auxiliary vector field (10)
is of the form

(21) $\displaystyle\scriptstyle
X_{A}=\scriptstyle\Bigg{(}c_{r_{1}}\big{(}y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}}\big{)}^{r_{1}}\Bigg{)}\scriptstyle\Bigg{(}\Big{(}a_{r_{2}}\big{(}y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}}\big{)}^{r_{2}}\Big{)}x\frac{\partial}{\partial
x}{}$
$\displaystyle\scriptstyle+\Big{(}y^{-1}\big{(}b_{r_{2}}-\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}\big{)}(y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}})^{r_{2}+1}+$
$\displaystyle\scriptstyle+\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}(y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}})^{r_{2}}\Big{)}y\frac{\partial}{\partial
y}\Bigg{)}$

if the main vertex is contained in $\\{0\\}\times\mathbb{N}$ or

(22) $\scriptstyle
X_{A}=\Bigg{(}c_{r_{1}}\big{(}y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}}\big{)}^{r_{1}}\Bigg{)}\Bigg{(}\Big{(}a_{r_{2}}\big{(}y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}}\big{)}^{r_{2}}\Big{)}\frac{\partial}{\partial
x}+\Big{(}-\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}(-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}-1})(y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}})^{r_{2}}{)}\frac{\partial}{\partial
y}\Bigg{)}$

if the main vertex is contained in $\\{-1\\}\times\mathbb{N}$. In both cases,
$\frac{\omega_{2}}{\omega_{1}}\in\mathbb{N}$.

###### Proof.

Now we characterize the expression of the auxiliary vector field (and hence
the constrained system) before the blow up.

Case 1: Before the blow-up, the main vertex of the Newton polygon is of the
form $(0,d_{1}+d_{2})$. We recall that $\xi_{0}\neq 0$ because at the origin
the height decreases. Remember also that $\omega_{1}r_{1}=d_{1}$ and
$\omega_{2}r_{2}=d_{2}$. Moreover, $a_{-1,r_{2}}=0$. Since at $(0,\xi_{0})$
the height does not decrease, it follows from what we have shown that

$\scriptsize\left\\{\begin{array}[]{rccccccl}\displaystyle\sum_{l=0}^{r_{1}}c_{l}\xi^{l}&=&C_{0}(\xi)&=&c_{r_{1}}(\xi-\xi_{0})^{r_{1}}&=&c_{r_{1}}\displaystyle\sum_{j=0}^{r_{1}}\xi^{j}(-\xi_{0})^{r_{1}-j}\displaystyle\binom{r_{1}}{j};\\\
\displaystyle\sum_{n=0}^{r_{2}}\frac{a_{n}}{\omega_{1}}\xi^{n}&=&\alpha_{0}(\xi)&=&\frac{a_{r_{2}}}{\omega_{1}}(\xi-\xi_{0})^{r_{2}}&=&\frac{a_{r_{2}}}{\omega_{1}}\displaystyle\sum_{j=0}^{r_{2}}\xi^{j}(-\xi_{0})^{r_{2}-j}\displaystyle\binom{r_{2}}{j};\\\
\displaystyle\sum_{n=-1}^{r_{2}}\lambda_{n}\xi^{n+1}&=&\beta_{-1}(\xi)&=&\lambda_{r_{2}}(\xi-\xi_{0})^{r_{2}+1}&=&\lambda_{r_{2}}\displaystyle\sum_{j=0}^{r_{2}+1}\xi^{j}(-\xi_{0})^{r_{2}+1-j}\displaystyle\binom{r_{2}+1}{j};\\\
\end{array}\right.$

where $\lambda_{n}=\big{(}b_{n}-\frac{\omega_{2}}{\omega_{1}}a_{n}\big{)}$.
Thus we have the following equalities, for each $l$ and $n$:

$\scriptsize\left\\{\begin{array}[]{rcl}c_{l}&=&c_{r_{1}}(-\xi_{0})^{r_{1}-l}\displaystyle\binom{r_{1}}{l};\\\
a_{n}&=&a_{r_{2}}(-\xi_{0})^{r_{2}-n}\displaystyle\binom{r_{2}}{n};\\\
b_{n}&=&\big{(}b_{r_{2}}-\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}\big{)}(-\xi_{0})^{r_{2}-n}\displaystyle\binom{r_{2}+1}{n+1}+\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}(-\xi_{0})^{r_{2}-n}\displaystyle\binom{r_{2}}{n}.\end{array}\right.$

We remark that for each $k$ and each $m$ we have
$k=\frac{\omega_{2}}{\omega_{1}}(r_{1}-l)$ and
$m=\frac{\omega_{2}}{\omega_{1}}(r_{2}-n)$. Then, before the blow up at the
$x$ direction, the first graduation the auxiliary vector field is of the form

$\displaystyle\scriptstyle\Bigg{(}$
$\displaystyle\sum_{l=0}^{r_{1}}\scriptstyle
c_{l}x^{k}y^{l}\Bigg{)}\Bigg{(}\sum_{n=0}^{r_{2}}a_{n}x^{m}y^{n}\Bigg{)}x\frac{\partial}{\partial
x}=$
$\displaystyle\scriptstyle=\Bigg{(}\sum_{l=0}^{r_{1}}c_{r_{1}}(-\xi_{0})^{r_{1}-l}\binom{r_{1}}{l}(x^{\frac{\omega_{2}}{\omega_{1}}})^{r_{1}-l}y^{l}\Bigg{)}\Bigg{(}\sum_{n=0}^{r_{2}}a_{r_{2}}(-\xi_{0})^{r_{2}-n}\binom{r_{2}}{n}(x^{\frac{\omega_{2}}{\omega_{1}}})^{r_{2}-n}y^{n}\Bigg{)}x\frac{\partial}{\partial
x}=$
$\displaystyle\scriptstyle=\Bigg{(}\sum_{l=0}^{r_{1}}c_{r_{1}}\binom{r_{1}}{l}(-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}})^{r_{1}-l}y^{l}\Bigg{)}\Bigg{(}\sum_{n=0}^{r_{2}}a_{r_{2}}\binom{r_{2}}{n}(-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}})^{r_{2}-n}y^{n}\Bigg{)}x\frac{\partial}{\partial
x}=$
$\displaystyle\scriptstyle=\Bigg{(}c_{r_{1}}\big{(}y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}}\big{)}^{r_{1}}\Bigg{)}\Bigg{(}a_{r_{2}}\big{(}y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}}\big{)}^{r_{2}}\Bigg{)}x\frac{\partial}{\partial
x}$

in the $x$ component, and in the $y$ component we have (dropping the
expression of the impasse set in order to simplify the notation)

$\displaystyle\Bigg{(}\scriptstyle\sum_{n=-1}^{r_{2}}\Big{(}\big{(}b_{r_{2}}-\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}\big{)}(-\xi_{0})^{r_{2}-n}\binom{r_{2}+1}{n+1}+\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}(-\xi_{0})^{r_{2}-n}\binom{r_{2}}{n}\Big{)}x^{\frac{\omega_{2}}{\omega_{1}}(r_{2}-n)}y^{n}\Bigg{)}y\frac{\partial}{\partial
y}=$
$\displaystyle\scriptstyle=\Bigg{(}\sum_{n=-1}^{r_{2}}\big{(}b_{r_{2}}-\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}\big{)}y^{n}(-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}})^{r_{2}-n}\binom{r_{2}+1}{n+1}+\sum_{n=0}^{r_{2}}\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}y^{n}(-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}})^{r_{2}-n}\binom{r_{2}}{n}\Bigg{)}y\frac{\partial}{\partial
y}=$
$\displaystyle\scriptstyle=\Bigg{(}\sum_{N=0}^{r_{2}}\big{(}b_{r_{2}}-\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}\big{)}y^{N-1}(-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}})^{r_{2}+1-N}\binom{r_{2}+1}{N}+\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}(y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}})^{r_{2}}\Bigg{)}y\frac{\partial}{\partial
y}=$
$\displaystyle=\scriptstyle\Bigg{(}y^{-1}\big{(}b_{r_{2}}-\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}\big{)}(y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}})^{r_{2}+1}+\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}(y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}})^{r_{2}}\Bigg{)}y\frac{\partial}{\partial
y}.$

To end the Case 1, let us remark that $c_{r_{1}}\neq 0$. Then

  * •

If $a_{r_{2}}\neq 0$, then the $x$ component gives the contributions
$(0,r_{1}+r_{2})$ and $(\frac{\omega_{2}}{\omega_{1}},r_{1}+r_{2}-1)$ to the
support $\mathcal{Q}$.

  * •

If $a_{r_{2}}=0$, then $b_{r_{2}}\neq 0$ and the $y$ component gives
contributions $(0,r_{1}+r_{2})$ and
$(\frac{\omega_{2}}{\omega_{1}},r_{1}+r_{2}-1)$ to the support $\mathcal{Q}$.

In both cases, we have $\frac{\omega_{2}}{\omega_{1}}\in\mathbb{N}$.

Case 2: Suppose that the main vertex is of the form $(-1,d_{1}+d_{2})$.
Remember that $\omega_{1}r_{1}=d_{1}$ and $\omega_{2}r_{2}=d_{2}+\omega_{1}$.
We also remark that $a_{r_{2}}=a_{-1,r_{2}}\neq 0$ and
$b_{r_{2}}=b_{-1,r_{2}}=0$. This case is almost analogous to Case 1, with one
more term in the expansion of the polynomials $\alpha_{0}(\xi)$ and
$\beta_{-1}(\xi)$. The expression of $C_{0}$ remains exactly the same, so we
focus in the other expressions. For each $k$ and each $m$ we have
$k=\frac{\omega_{2}}{\omega_{1}}(r_{1}-l)$ and
$m=\frac{\omega_{2}}{\omega_{1}}(r_{2}-n)-1$. So in the $x$ component we
obtain

$\displaystyle\scriptstyle\Bigg{(}\sum_{l=0}^{r_{1}}c_{l}x^{k}y^{l}\Bigg{)}$
$\displaystyle\scriptstyle\Bigg{(}\sum_{n=0}^{r_{2}}a_{n}x^{m}y^{n}\Bigg{)}x\frac{\partial}{\partial
x}=$
$\displaystyle\scriptstyle=\Bigg{(}c_{r_{1}}\big{(}y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}}\big{)}^{r_{1}}\Bigg{)}\Bigg{(}\sum_{n=0}^{r_{2}}a_{r_{2}}(-\xi_{0})^{r_{2}-n}\binom{r_{2}}{n}(x^{\frac{\omega_{2}}{\omega_{1}}})^{r_{2}-n}x^{-1}y^{n}\Bigg{)}x\frac{\partial}{\partial
x}=$
$\displaystyle\scriptstyle=\Bigg{(}c_{r_{1}}\big{(}y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}}\big{)}^{r_{1}}\Bigg{)}\Bigg{(}\sum_{n=0}^{r_{2}}a_{r_{2}}\binom{r_{2}}{n}(-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}})^{r_{2}-n}y^{n}\Bigg{)}\frac{\partial}{\partial
x}=$
$\displaystyle\scriptstyle=\Bigg{(}c_{r_{1}}\big{(}y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}}\big{)}^{r_{1}}\Bigg{)}\Bigg{(}a_{r_{2}}\big{(}y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}}\big{)}^{r_{2}}\Bigg{)}\frac{\partial}{\partial
x}.$

Before we compute the expression in the $y$ component, we must give the
expression of the coefficients $b_{n}$. Since $b_{r_{2}}=0$, then for each $n$
we have

$\displaystyle\scriptstyle b_{n}$
$\displaystyle\scriptstyle=-\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}(-\xi_{0})^{r_{2}-n}\binom{r_{2}+1}{n+1}+\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}(-\xi_{0})^{r_{2}-n}\binom{r_{2}}{n}\scriptstyle=-\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}(-\xi_{0})^{r_{2}-n}\Bigg{(}\binom{r_{2}+1}{n+1}-\binom{r_{2}}{n}\Bigg{)}=$
$\displaystyle\scriptstyle=-\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}(-\xi_{0})^{r_{2}-n}\binom{r_{2}}{n}\frac{r_{2}-n}{n+1}\scriptstyle=-\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}(-\xi_{0})^{r_{2}-n}\binom{r_{2}}{n+1},$

and finally we obtain

$\displaystyle\scriptstyle\Bigg{(}$
$\displaystyle\sum_{l=0}^{r_{1}}\scriptstyle
c_{l}x^{k}y^{l}\Bigg{)}\scriptstyle\Bigg{(}\sum_{n=-1}^{r_{2}}b_{n}x^{m}y^{n}\Bigg{)}y\frac{\partial}{\partial
y}=$
$\displaystyle\scriptstyle=\Bigg{(}c_{r_{1}}\big{(}y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}}\big{)}^{r_{1}}\Bigg{)}\Bigg{(}\sum_{n=-1}^{r_{2}}\Big{(}-\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}(-\xi_{0})^{r_{2}-n}\binom{r_{2}}{n+1}\Big{)}x^{\frac{\omega_{2}}{\omega_{1}}(r_{2}-n)}x^{-1}y^{n}\Bigg{)}y\frac{\partial}{\partial
y}=$
$\displaystyle\scriptstyle=\Bigg{(}c_{r_{1}}\big{(}y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}}\big{)}^{r_{1}}\Bigg{)}\Bigg{(}-\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}x^{-1}\sum_{n=-1}^{r_{2}}(-\xi_{0})^{r_{2}-n}\binom{r_{2}}{n+1}(x^{\frac{\omega_{2}}{\omega_{1}}})^{(r_{2}-n)}y^{n}\Bigg{)}y\frac{\partial}{\partial
y}=$
$\displaystyle\scriptstyle=\Bigg{(}c_{r_{1}}\big{(}y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}}\big{)}^{r_{1}}\Bigg{)}\Bigg{(}-\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}x^{-1}\sum_{N=0}^{r_{2}}\binom{r_{2}}{N}(-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}})^{(r_{2}-N+1)}y^{N-1}\Bigg{)}y\frac{\partial}{\partial
y}=$
$\displaystyle\scriptstyle=\Bigg{(}c_{r_{1}}\big{(}y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}}\big{)}^{r_{1}}\Bigg{)}\Bigg{(}-\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}(-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}-1})\sum_{N=0}^{r_{2}}\binom{r_{2}}{N}(-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}})^{(r_{2}-N)}y^{N}\Bigg{)}\frac{\partial}{\partial
y}=$
$\displaystyle\scriptstyle=\Bigg{(}c_{r_{1}}\big{(}y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}}\big{)}^{r_{1}}\Bigg{)}\Bigg{(}-\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}(-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}-1})(y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}})^{r_{2}}\Bigg{)}\frac{\partial}{\partial
y}.$

From the expression of the $x$ component, the Newton polygon contains the
point $(-1,r_{1}+r_{2})$. Moreover, the $y$ component gives the contribution
$(\frac{\omega_{2}}{\omega_{1}}-1,r_{1}+r_{2}-1)$. This implies
$\frac{\omega_{2}}{\omega_{1}}\in\mathbb{N}$. ∎

######  Proposition 21.

Suppose that the main segment $\gamma_{1}$ is not horizontal and after a blow-
up at the $x$ direction there is a point in the excepcional divisor such that
the height did not decrease. Then there is an analytic change of coordinates
such that one of the following holds:

  * •

We obtain a new weight vector $\omega=(\omega_{1},\omega_{2})$ such that,
after a blow up in the $x$ direction, the main vertex is contained in
$\\{0\\}\times\mathbb{N}$, the height decreases and there is only a finite
number of points in the exceptional divisor $\mathbb{E}_{x}$ such that they
may not be elementary.

  * •

The main segment is horizontal.

###### Proof.

We know that there is only one non-elementary point $p$ in the exceptional
divisor and the expression of the auxiliary vector field before the blow up is
given by Proposition 20. We have two cases to consider:

Case 1: The point $p$ is an equilibrium point of the adjoint vector field. In
this case, take a change of coordinates of the form

$\scriptsize\Psi_{1}(x,y)=(x,y-\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}}).$

We recall that by Proposition 20 we have
$\frac{\omega_{2}}{\omega_{1}}\in\mathbb{N}$. By Lemma 10, such change of
coordinates does not change the height of the polygon and increases the slope
of the main segment. Applying such change of coordinates in the expressions
obtained from Proposition 20, the first graduation of the auxiliary vector
field becomes

$\scriptstyle
X_{A}=c_{r_{1}}y^{r_{1}}\Bigg{(}a_{r_{2}}y^{r_{2}}x\frac{\partial}{\partial
x}+b_{r_{2}}y^{r_{2}}y\frac{\partial}{\partial y}\Bigg{)}+...$

if the main vertex is contained in $\\{0\\}\times\mathbb{N}$ or

$\scriptstyle
X_{A}=c_{r_{1}}y^{r_{1}}\Bigg{(}a_{r_{2}}y^{r_{2}}\frac{\partial}{\partial
x}+\frac{\omega_{2}}{\omega_{1}}a_{r_{2}}(\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}-1})y^{r_{2}}\frac{\partial}{\partial
y}\Bigg{)}+...$

if the main vertex is contained in $\\{-1\\}\times\mathbb{N}$. In both cases,
the main segment of the new Newton polygon is not horizontal, and then we
obtain new weights to continue the resolution of singularities, since the
equilibrium point of the adjoint vector field is isolated.

Now we must blow up the origin with these new weights. After this new blow-up,
once again we have two possibilities: the new blow up decreases the height of
the Newton polygon or it does not decrease the height. In this last case, we
apply again a change of coordinates $\Psi_{2}$ given by Lemma 10. Observe that
this operation does not change the height and it increases the slope of the
main segment. And then we continue the reasoning.

If it is necessary, we apply $n$ change of coordinates $\Psi_{n}$ given by
Lemma 10. The main segment will not be horizontal and we obtain new weights to
continue the resolution of singularities. These new weights decreases the
height of the Newton polygon and then the Proposition is true.

In this case where $p$ is an equilibrium point of the adjoint vector field, we
affirm that this iteration of admissible change of coordinates ends. Indeed,
if this was not the case, since at each step the slope of the main segment
increases, after an infinite composition of admissible coordinates

$\scriptsize\Psi_{\infty}(x,y)=\Big{(}x,y-\sum_{i}\xi_{i}x^{\tilde{\omega}_{i}}\Big{)}$

we should have a Newton polygon with horizontal main segment. Moreover, after
the change of coordinates $\Psi_{\infty}$ we would have that the adjoint
vector field is $\infty$-flat along the impasse set $\Delta$, which is a
contradiction (see Lemmas 7 and 9 of [11]). Then the iteration of admissible
change of coordinates does not continue indefinitely.

Case 2: The point $p$ is not an equilibrium point of the adjoint vector field.
This implies that $r_{2}=0$ and the main vertex is not contained in
$\\{0\\}\times\mathbb{N}$. Applying an admissible change of coordinates
$\Psi_{1}$ in the expressions obtained from Proposition 20, the first
graduation of the auxiliary vector field becomes

$\scriptstyle X_{A}=c_{r_{1}}y^{r_{1}}\Bigg{(}a_{-1,0}\frac{\partial}{\partial
x}+\frac{\omega_{2}}{\omega_{1}}a_{-1,0}(\xi_{0}x^{\frac{\omega_{2}}{\omega_{1}}-1})\frac{\partial}{\partial
y}\Bigg{)}+...$

We have now two possibilities:

  * •

The slope of the main segment increases and we obtain new weights to continue
the resolution of singularities;

  * •

The main segment is horizontal.

In the first case we must blow up the origin with these new weights. After
this new blow-up, once again we have the same two possibilities as before. If
it is necessary, we apply $n$ change of coordinates $\Psi_{n}$ given by Lemma
10. Then or the main segment will be horizontal or we obtain new weights to
continue the resolution of singularities. This leads us to a case where the
process may never ends and we must apply a change of coordinates of the form

$\Psi_{\infty}(x,y)=\Big{(}x,y-\sum_{i}\xi_{i}x^{\tilde{\omega}_{i}}\Big{)}.$

After a change of coordinates $\Psi_{\infty}$, the main segment is horizontal,
since at each step the slope increases. Now we must show that such change of
coordinates is analytic. Indeed, we remark that the impasse curve $\Delta$
always contains this non elementary point $p$. From the way we defined
$\Psi_{\infty}$, it follows that the curve
$\\{y-\sum_{i}\xi_{i}x^{\tilde{\omega}_{i}}=0\\}$ is contained in the curve
$\Delta$. In other words, the curve
$\\{y-\sum_{i}\xi_{i}x^{\tilde{\omega}_{i}}=0\\}$ must be a component of the
analytic set $\Delta$. Since the impasse set is an analytic curve, so is the
curve $\\{y-\sum_{i}\xi_{i}x^{\tilde{\omega}_{i}}=0\\}$ and therefore the
formal sum converges. ∎

Example: Consider the vector field

(23) $\scriptstyle(y-x^{s})\dot{x}=(y-x^{s})^{r}x;\
(y-x^{s})\dot{y}=-s(y-x^{s})^{r+1}+sy(y-x^{s})^{r}+y^{k+1},$

whose auxiliary vector field is

(24) $\scriptstyle X_{A}=(y-x^{s})(y-x^{s})^{r}x\frac{\partial}{\partial
x}+(y-x^{s})\Big{(}-sy^{-1}(y-x^{s})^{r+1}+s(y-x^{s})^{r}+y^{k}\Big{)}y\frac{\partial}{\partial
y};$

where $k\geq r+1$. Observe that the origin is an isolated equilibrium point of
the adjoint vector field. The main vertex is of the form $(0,r)$ and therefore
the height is $r$. Blowing up the auxiliary vector field in origin with weight
$\omega=(1,s)$ in the $x$ direction, we obtain

$\scriptstyle\widetilde{X}_{A}=(\tilde{y}-1)(\tilde{y}-1)^{r}x\frac{\partial}{\partial
x}+(\tilde{y}-1)\Big{(}-s\tilde{y}^{-1}(\tilde{y}-1)^{r+1}+\tilde{x}^{(k-r)s}\tilde{y}^{k}\Big{)}\tilde{y}\frac{\partial}{\partial
y}.$

Observe that at the origin the height has decreased, but when we translate
origin to $(0,1)\in\mathbb{E}_{x}$ we see that the height is still $r$. So,
before the blow-up we must apply a change of coordinates of the form

$\bar{x}=x;\ \bar{y}=y-x^{s},$

and then (24) takes form

(25)
$\scriptstyle\overline{X}_{A}=\bar{y}\bar{y}^{r}\bar{x}\frac{\partial}{\partial\bar{x}}+\bar{y}(\bar{y}+\bar{x}^{s})^{k}\frac{\partial}{\partial\bar{y}},$

and now we have a new weight vector to continue the process.

##  5\. Global resolution of singularities

We have already studied the local strategy of resolution of singularities and
now we are able to give the proof of Theorem 3. Suppose that
$(\mathcal{M},\mathcal{X},\mathcal{I})$ is a real analytic constrained
differential system defined on a 2-dimensional compact manifold $\mathcal{M}$
with corners, which means that $\mathcal{X}$ is a 1-dimensional analytic
foliation and $\Delta$ is an 1-dimensional analytic subset of $\mathcal{M}$.

Recall that, in our firsts assumptions, in the local representations of
$\mathcal{X}$, the components $P$ and $Q$ of $X$ do not have common factor,
which implies that all equilibrium points of the adjoint vector field are
isolated. Analogously, the impasse ideal $\mathcal{I}$ is locally generated by
an irreducible analytic function $\delta$, and then $\delta$ has isolated
singular points. Since all of our objects are analytic, there is a finite
number of tangency points between the foliation and the impasse set. Due to
the compactness of $\mathcal{M}$, there is finite set $\\{p_{i}\\}_{i=1}^{k}$
of non elementary points.

In each point $p_{i}$ we fix local coordinates $(\phi_{i},U_{i})$ such that
the Newton polygon of the auxiliary vector field is controllable. Applying
Theorem 14 in each system of coordinates, we have
$(\tilde{\phi}_{i},\widetilde{U}_{i})$ such that in each $\widetilde{U}_{i}$
there are only elementary points. Since the blow ups are analytic
diffeomorphisms that preserve orientation outside the exceptional divisor (as
well as the admissible change of coordinates), then the constrained system
remains well defined in the intersection between two charts.

Then we obtain a finite sequence of weighted blow-ups

$(\widetilde{\mathcal{M}},\widetilde{\mathcal{X}},\widetilde{\mathcal{I}})=(\mathcal{M}_{n},\mathcal{X}_{n},\mathcal{I}_{n})\xrightarrow{\Phi_{n}}\cdots\xrightarrow{\Phi_{0}}(\mathcal{M}_{0},\mathcal{X}_{0},\mathcal{I}_{0})=(\mathcal{M},\mathcal{X},\mathcal{I})$

such that the strict transformed
$(\widetilde{\mathcal{M}},\widetilde{\mathcal{X}},\widetilde{\mathcal{I}})$
has only elementary points.

##  6\. Acknowledgments

The first author is supported by Sao Paulo Research Foundation (FAPESP)
(grants 2016/22310-0 and 2018/24692-2), and by Coordenação de Aperfeiçoamento
de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. The second
author was financed by CAPES-Print and FAPESP. The authors are grateful for
Daniel Cantergiani Panazzolo for many discussions and suggestions, and for
LMIA-Université de Haute-Alsace for the hospitality during the preparation of
this work.

## References

  * [1] M. J. Álvarez, A. Ferragut and X. Jarque. _A survey on the blow up technique_. Int. J. Bifurcation and Chaos 21 (2011), 3103-3118.
  * [2] A. Belotto, D. Panazzolo. _Global desingularization of a real-analytic planar vector field_. Université de Haute-Alsace, 2011, unpublished.
  * [3] I. Bendixson. _Sur les courbes définies par des équations différentielles_. Acta Math. 24 (1901), 1-88.
  * [4] M. Brunella, M. Miari. _Topological equivalence of a plane vector field with its principal part defined through Newton Polyhedra_. J. Differ. Equations 85 (1990), 338-366.
  * [5] P. T. Cardin, P. R. da Silva, M. A. Teixeira. _Implicit differential equations with impasse singularities and singular perturbation problems_. Isr. J. Math. 189 (2012), 307-322.
  * [6] L. O. Chua, H. Oka. _Normal forms for constrained nonlinear differential equations. I. Theory_. IEEE Transactions on Circuits and Systems, vol. 35, no. 7 (1988), 881-901.
  * [7] F. Dumortier. _Singularities of vector fields on the plane_. J. Differ. Equations 23 (1977), 53-106.
  * [8] F. Dumortier, J. Llibre, J. C. Artés. _Qualitative Theory of Planar Differential Systems_. Universitext, Springer-Verlag Berlin Heidelberg (2006).
  * [9] D. Joyce. _A generalization of manifold with corners_. Adv. Math. 299 (2016), 760-862.
  * [10] D. Panazzolo. _Resolution of singularities of real-analytic vector fields in dimension three_. Acta Math. 197 (2006), no. 2, 167-289.
  * [11] M. Pelletier. _Éclatements quasi homogènes_. Ann. Fac. Sci. Toulouse Math., vol 4 (1995), 879-937.
  * [12] O. H. Perez, P. R. da Silva. _On the classification of phase portraits of constrained differential systems under topological equivalence_. In preparation.
  * [13] P. J. Rabier, W. C. Rheinboldt. _Theoretical and numerical analysis of differential-algebraic equations_ , in P. G. Ciarlet et al. (eds.), _Handbook of Numerical Analysis_ , North Holland/Elsevier, Vol. VIII (2002), 183-540.
  * [14] R. Schneider. _Convex Bodies: The Brunn-Minkowski Theory_. Encyclopedia of mathematics and its applications. 44. Cambridge University Press (1993).
  * [15] A. Seidenberg. _Reduction of singularities of the differential equation $Ady=Bdx$_. Amer. J. Math. 90 (1968), 248-269.
  * [16] S. Smale. _On the mathematical foundations of electrical circuit theory_. J. Differ. Geom. 7 (1972), 193-210.
  * [17] J. Sotomayor, M. Zhitomirskii. _Impasse singularities of differential systems of the form $A(x)x^{\prime}=F(x)$_. J. Differ. Equations 169 (2001), 567-587.
  * [18] M. Zhitomirskii. _Local normal forms for constrained systems on 2-manifolds_. Bol. Soc. Brasil. Mat. 24 (1993), 211-232.