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\markboth{\hfil On plane polynomial vector fields \hfil EJDE--2002/37}
{EJDE--2002/37\hfil M'hammed El Kahoui \hfil}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 37, pp. 1--23. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
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On plane polynomial vector fields and the Poincar\'e problem
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\thanks{ {\em Mathematics Subject Classifications:} 34C05, 34A34, 34C14.
\hfil\break\indent
{\em Key words:} Polynomial vector fields, Invariant
algebraic curves, Intersection numbers, \hfil\break\indent
Tjurina number, B\'ezout theorem.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted November 20, 2001. Published May 6, 2002.} }
\date{}
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\author{M'hammed El Kahoui}
\maketitle
\begin{abstract}
In this paper we address the Poincar\'e problem, on plane
polynomial vector fields, under some conditions on the nature
of the singularities of invariant curves. Our main idea
consists in transforming a given vector field of degree $m$
into another one of degree at most $m+1$ having its invariant
curves in projective quasi-generic position. This allows us to
give bounds on degree for some well known classes of curves
such as the nonsingular ones and curves with ordinary nodes.
We also give a bound on degree for any invariant curve in terms
of the maximum Tjurina number of its singularities and the
degree of the vector field.
\end{abstract}
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\section{Introduction}
The study of algebraic invariant curves and integrating factors of
plane polynomial vector fields goes back at least to Darboux
\cite{darboux78} and Poincar\'e \cite{poincare28}. We refer the reader
to \cite{schlomiuk93a,schlomiuk93b,christopherllibre99} for an
interesting survey and historical remarks on the problem. For a
given polynomial vector field the question of finding invariant
algebraic curves reduces mainly to the so-called {\it Poincar\'e
problem} which consists in finding an upper bound on the degree of
such curves. Indeed, any time such bound is found for a given
vector field the question of finding its invariant curves can be
algorithmically solved by using linear algebra (see e.g.
\cite{christopher94a,maccallum97a,christopher96a}).
Solving the Poincar\'e problem, and hence finding invariant
curves, yields great advances in the algorithmic study of plane
polynomial vector fields. Darboux \cite{darboux78} showed that the
abundance of invariant algebraic curves of a plane polynomial
vector field ensures its integrability. More precisely, he proved
that a vector field of degree $m$ with a least
$\frac{m(m+1)}{2}+1$ invariant curves has a first integral. Later
on Jouanolou \cite{jouanolou79} showed that any degree $m$ plane
polynomial vector field with at least $\frac{m(m+1)}{2}+2$
invariant curves has a rational first integral. In this direction
Prelle and Singer studied in \cite{prellesinger82} another kind of
first integrals, namely {\it elementary first integrals}. They
proved that the existence of algebraic integrating factors is
necessary for the existence of elementary first integrals, and
that deciding about the existence of algebraic integrating factors
is the main question to be solved in order to decide about the
existence of elementary first integrals. Ten years later, Singer
\cite{singer92} used differential algebra techniques to study a
wider class of first integrals, namely the {\it Liouvillian first
integrals}, and he proved that they have elementary functions as
integrating factors.
It is well known from the work of Jouanolou that a plane
polynomial vector field has either a rational first integral or
finitely many invariant curves. This gives an indirect proof for
the existence of an upper bound for the degree of irreducible
invariant curves of a given vector field. As far as we know there
is actually no effective method to compute such bound for any
given vector field, even more the question promises to be hard.
For example, the related question of deciding whether the closure
of the set of vector fields with given degree and having invariant
curves is an algebraic set is still open (see e.g.
\cite{christopherllibre2000, linsneto2000} for more details on the
question). On the other hand, it is well known that the degree of
the given vector field is not enough in order to get control on
the maximal degree of its irreducible invariant curves. It is for
instance easy to find linear vector fields having rational first
integrals of arbitrarily high degree. Even more, its is
established in \cite{moulin2000} and \cite{christopherllibre2000a}
the existence of quadratic plane vector fields without rational
first integral and having invariant algebraic curves of any given
degree.
Partial answers to the Poincar\'e problem have been given in
recent years. All of them follow the same strategy, which consists
in finding an upper bound in terms of the degree of the vector
field under some additional conditions on its fixed points or on
the nature of the singularities the invariant curves have (see
e.g. \cite{llibrechavarriga2000,
carnicer94,cerveaulinsneto91,tsygvintsev2001,walcher2000a,campillocarnicer97}).
\subsection*{Outline of the paper}
In this paper we study the Poincar\'e problem from ``algebraic
geometry" point of view. For this purpose it is natural to state
the problem in the general setting of a commutative field of
characteristic zero. Our main idea consists in reducing the
problem , by means of projective transformations, to a situation
where invariant curves have no critical points at infinity. This
reduction has a double advantage: first it keeps the geometric
properties of the invariant curves. Secondly, it allows to use
some basic results of projective algebraic geometry such as
B\'ezout theorem.
The paper is structured as follows: in section \ref{sec:1}
we define the concept of curves in projective quasi-generic
position and we show explicitly how to transform projectively any
curve to a curve in such position. Section \ref{sec:2} is devoted
to show how to transform vector fields, without loss of control
on their degree, into vector fields having their invariant curves
in projective quasi-generic position. In section \ref{sec:3} we
apply the techniques developed in sections \ref{sec:1} and
\ref{sec:2} to the Poincar\'e problem. We recover the classical
bound given in the case of nonsingular curves and we give better
bounds than the known ones in the case of curves with ordinary
nodes. A bound in terms of the maximum Tjurina number of the
singularities of an invariant curve is also given in this section.
\subsection*{The setting of a commutative field of characteristic
zero}
Polynomial vector fields and invariant algebraic curves are
objects of algebraic nature. It is hence natural to study their
properties in the general setting of a commutative field of
characteristic zero. This gives as well some flexibility to our
study of vector fields; we shall for example see that this allows
to treat in the same way invariant curves and rational first
integrals (lemma \ref{firstintegral}). Another practical reason
lies in the study of parameterized vector fields, since any given
vector fields ${\cal X}\in \mathbb{R}[u,x,y]$, where
$u=(u_1,\ldots, u_r)$ is a list of parameters, can be viewed as
vector field over $\mathbb{R}(u)[x,y]$.
\subsection*{Notation}
Let $\mathbb{K}$ be a commutative field of characteristic zero and
$\overline{\mathbb{K}}$ its algebraic closure. Let $f$ be a squarefree
polynomial in $\mathbb{K}[x,y]$ and ${\cal C}(f)$ be the affine plane algebraic
curve, over the field $\overline{\mathbb{K}}$, defined by the equation
$f(x,y)=0$.
The zeros in $\overline{\mathbb{K}}^{2}$ of the ideal ${\cal I}(f,
{\partial}_yf)$ are called the {\it critical points} of the curve
${\cal C}(f)$ with respect to the projection on the $x$-axis. In the same
way, the zeros of the ideal ${\cal I}(f, {\partial}_xf)$ are
called the critical points of the curve ${\cal C}(f)$ with respect to the
projection on the $y$-axis. A critical point of the curve ${\cal C}(f)$
with respect to one of the projections is simply called a
critical point of the curve.
The multiplicity of a point $(\alpha,\beta)$ of the curve
${\cal C}(f)$ is defined as the smallest integer $s$ such that
${\partial}^{i+j}_{x^{i}y^{j}}f(\alpha,\beta)=0$ for any $i+j