\documentclass[twoside]{article} \usepackage{amssymb, amsmath} \pagestyle{myheadings} \markboth{\hfil On plane polynomial vector fields \hfil EJDE--2002/37} {EJDE--2002/37\hfil M'hammed El Kahoui \hfil} \begin{document} \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)} \vspace{\bigskipamount} \\ % On plane polynomial vector fields and the Poincar\'e problem % \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{} % \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} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \newtheorem{definition}{Definition}[section] \numberwithin{equation}{section} \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