\documentclass[twoside]{article} \usepackage{amssymb} % used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Invariance of Poincar\'e-Lyapunov polynomials\hfil EJDE--1998/23}% {EJDE--1998/23\hfil Pierre Joyal \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.\ {\bf 1998}(1998), No.~23, pp. 1--8. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113} \vspace{\bigskipamount} \\ Invariance of Poincar\'e-Lyapunov polynomials under the group of rotations \thanks{ {\em 1991 Mathematics Subject Classifications:} 58F14, 58F21, 58F35, 34C25. \hfil\break\indent {\em Key words and phrases:} focus, invariance of Poincar\'e-Lyapunov polynomials, \hfil\break\indent weighted-homogeneity. \hfil\break\indent \copyright 1998 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted June 25, 1998. Published October 9, 1998. \hfil\break\indent This research was partially supported by NSERC and FCAR } } \date{} \author{Pierre Joyal} \maketitle \begin{abstract} We show that the Poincar\'e-Lyapunov polynomials at a focus of a family of real polynomial vector fields of degree $n$ on the plane are invariant under the group of rotations. Furthermore, we show that under the multiplicative group ${\mathbb C}^*=\{\rho {\rm e}^{i\psi}\}$, they are invariant up to a positive factor. These results follow from the weighted-homogeneity of the polynomials that we define in the text. \end{abstract} \newtheorem{prop}{Proposition} \newtheorem{lemma}[prop]{Lemma} \newtheorem{coro}[prop]{Corollary} \newcommand{\qed}{\vrule height 1.3ex width 1.0ex depth- 0.0ex} \section{Introduction} Let us consider a real analytic vector field on the plane having a non-degenerate focus at the origin, that is, the Jacobian matrix of the vector field at the focus is not singular. After a linear transformation, we can suppose that the Jacobian matrix at the focus has the form \begin{equation}\label{mat} \left(\begin{array}{rr} a & -b \\ b & a \end{array} \right),\qquad b\ne 0. \end{equation} Let $\Sigma$ be a local cross section with one end point at the origin and $U\subseteq\Sigma$, a neighborhood of the origin in $\Sigma$. Recall that the displacement function in the neighbourhood of the origin is the Poincar\'e map $P\colon U\to\Sigma$ minus the Identity. One can show that the displacement function in a neighborhood of the origin has the following form (see [1]): \begin{equation}\label{eq2} r=({\rm e}^{2\pi a/b}-1)r_0+u_3r_0^3+u_5r_0^5+u_7r_0^7+\cdots\, . \end{equation} All the coefficients of the even powers of $r_0$ are equal to zero. When all the coefficients vanish, the origin is a center. Instead of calculating these coefficients to determine if an equilibrium point is a center, Poincar\'e gave in [2] another method which resembles the search for a Lyapunov function to establish the stability of a focus. Let us recall this method. Looking at (\ref{eq2}), we see that ${dr/dr_0}\ne 0$ in a punctured neighborhood of the origin, if $a\ne 0$. Suppose that $a=0$. If the vector field is linear, the integral curves are circles around the origin: $x^2+y^2=k$ ($k$ a constant), or in polar coordinates $r^2=k$. If the vector field is not linear, it is natural to look for integral curves that are small perturbations of these circles. Using polar coordinates, one tries to find integral curves of the form \begin{equation}\label{eq3} H(r,\theta) = r^2 + H_3(\theta)r^3 + H_4(\theta)r^4 + \cdots = k\,. \end{equation} If the origin is a center and if $H=k$ is an integral curve, then $$ {dH\over dt}=\frac{\partial H}{\partial r}\dot r+ \frac{\partial H}{\partial \theta}\dot\theta = 0\,. $$ Looking at the coefficients of the powers of $r$, this equation generates an infinite system of equations with the unknows $H_j(\theta)$ (see section 2). If the origin is not a center, then the equation above cannot be solved. However, as we will see later on, one can formally solve the equation $$ {dH\over dt}= P_1r^4 + P_2r^6 + P_3r^8 + \cdots\,, $$ where $P_j$, $j=1,2,\ldots$ are constants. The sign of the first non-zero $P_j$ controls the type of stability of the focus. If $P_j>0$, the focus is unstable; it is stable otherwise. In fact, it is possible to find $H=r^2 + H_3(\theta)r^3 + \cdots + H_{2j+1}(\theta)r^{2j+1}$ such that $$ {dH/dt\over r^{2j+2}}\Big|_{r=0}= P_j\,. $$ $H$ is a Lyapunov function for the focus (see proposition 1 and corollary 2). If all the $P_j$ vanish, it is possible to solve the system and the series in (\ref{eq3}) converges in a neighborhood of the origin (see [2]). There are no standard names for the constants $P_j$. Some call them focal numbers (or quantities), others call them Lyapunov constants. These names do not match the definitions of Andronov {\it et al\/} [1]. According to [1], the $j^{\rm th}$ focal value (or quantity) is the $j^{\rm th}$ derivative of the displacement function $r$ in (\ref{eq2}). If the first non-vanishing derivative of $r$ is of order $k=2j+1\ge 3$ ($j\ge 1$), then it is called the $k^{\rm th}$ Lyapunov value. But the $P_j$ are not in general equal to the $u_j$ in (\ref{eq2}). Moreover, in the case of a family of vector fields, the $P_j$ are in fact polynomial functions of the parameters (as we will see later on). We adopt the following definition. \paragraph{Definition} $P_j$ is the $j^{\rm th}$ Poincar\'e-Lyapunov constant. In the case of a family of vector fields, $P_j$ will be called the $j^{\rm th}$ Poincar\'e-Lyapunov polynomial (associated with this family). \medskip We will study these polynomials for the family of all polynomial vector fields of degree $n$ on the plane. We will prove that they are invariant under the group of rotations $S^1=\{\,{\rm e}^{i\psi}\,\}$ and also invariant under the multiplicative group ${\mathbb C}^*=\{\,\rho {\rm e}^{i\psi}\,\}$ modulo a positive factor. Precisely, $\forall j\ge 1$ and for $g=\rho{\rm e}^{i\psi}\in {\mathbb C}^*$, \[ P_j(g(a_{rs}))=\rho^{2j}P_j(a_{rs})\,, \] where the $a_{rs}$ are the parameters of the family of all polynomial vector fields of degree $n$ on the plane. In this statement, it is important to distinguish a Poincar\'e-Lyapunov polynomial from the corresponding Poincar\'e-Lyapunov constant (the value of this polynomial for a certain vector field). Indeed, the statement says that the polynomials are also weighted-homogeneous in a certain sense that we will define in section 3. \section{ Poincar\'e's Method} We suppose that the family of all polynomial vector fields of degree $n$ has an equilibrium point at the origin with a Jacobian matrix of the form (\ref{mat}) where $a=0$. We will slightly modify Poincar\'e's procedure to obtain the main result of this article. Dividing the family by $b$, it takes the following form in the coordinates $z=x+iy$ and $\bar z$: \begin{equation}\label{eq4} \begin{array}{lll} \dot z &=& \displaystyle{ iz + \sum_{m=2}^n\sum_{j+k=m} a_{jk}z^j\bar z^k,}\\ \dot{\bar z} &=& \displaystyle{ -i\bar z + \sum_{m=2}^n\sum_{j+k=m} \bar a_{kj}z^j\bar z^k.} \end{array} \end{equation} Setting $r=\sqrt{z\bar z}$ and $\theta=(1/2i)\ln(z/\bar z)$, we obtain: \begin{equation}\label{eq5} \begin{array}{lll} \dot r &=& \displaystyle{ {1\over 2r}(\dot z\bar z + z\dot{\bar z}) = (1/2)\sum_{m=2}^n F_m({\rm e}^{i\theta})r^m }\\ [4pt] \dot\theta &=& \displaystyle{{1\over 2r^2}(-i\dot z\bar z + iz\dot{\bar z}) = 1 + (1/2)\sum_{m=2}^n G_m({\rm e}^{i\theta})r^{m-1},} \end{array} \end{equation} where \begin{eqnarray}\label{eq6} F_m({\rm e}^{i\theta}) &=& a_{0m}{\rm e}^{-(m+1)i\theta} + \sum_{j+k=m;\ j\ne 0} (a_{jk}+\bar a_{(k+1)(j-1)}){\rm e}^{(j-k-1)i\theta} \nonumber \\ &\quad& + \bar a_{0m}{\rm e}^{(m+1)i\theta} \\ G_m({\rm e}^{i\theta}) &=& -ia_{0m}{\rm e}^{-(m+1)i\theta} +\sum_{j+k=m;\ j\ne 0}(-ia_{jk}+i\bar a_{(k+1)(j-1)}){\rm e}^{(j-k-1)i\theta} \nonumber \\ &\quad& + i\bar a_{0m}{\rm e}^{(m+1)i\theta}.\nonumber \end{eqnarray} One must find a function $$ H(r,e^{i\theta}) = r^2 + H_3(e^{i\theta})r^3 + H_4(e^{i\theta})r^4 + \cdots $$ such that \begin{equation}\label{eq7} {d H\over dt} = \frac{\partial H}{\partial r}\dot r + \frac{\partial H}{\partial \theta} \dot\theta = P_1r^4 + P_2r^6 + P_3r^8 + \cdots\,. \end{equation} We will see, as Poincar\'e did, that it is in general impossible to find $H(r,e^{i\theta})$ such that ${dH/dt}=0$, except if the origin is a center. In this case, all the constants $P_j$ vanish. We have: $$\displaylines{ {d H\over dt} = (F_2 + H_3')r^3 +\left({3\over 2}H_3F_2 + F_3 + {1\over 2}H_3' G_2 + H_4'\right)r^4 + \cdots \hfill\cr +\left({n\over 2}H_nF_2 + \cdots + {3\over 2}H_3F_{n-1} + F_n + {1\over 2}H_3' G_{n-1} + \cdots + {1\over 2}H_n' G_2 + H_{n+1}'\right)r^{n+1} \hfill\cr + \left({{n+1}\over 2}H_{n+1}F_2 + \cdots + {3\over 2}H_3F_n + {1\over 2}H_3' G_n + \cdots + {1\over2}H_{n+1}' G_2 + H_{n+2}'\right)r^{n+2}\hfill\cr + \left({{n+2}\over 2}H_{n+2}F_2 + \cdots + {4\over 2}H_4F_n + {1\over 2}H_4' G_n + \cdots + {1\over2}H_{n+2}' G_2 + H_{n+3}'\right)r^{n+3}\hfill\cr + \cdots \hfill } $$ \paragraph{Notation 1} Let us denote the coefficient of $r^k$ in the previous expression by $L_k({\rm e}^{i\theta})+H'_k$. \begin{prop} Let $m$ be the smallest integer such that $P_m\ne 0$. Then the system of equations $L_k({\rm e}^{i\theta})+H'_k=0$ ($3\le k\le 2m+1$) with the unknowns $H_k$ has a solution. $H_k$ has only powers of ${\rm e}^{i\theta}$ of the same parity as $k$. There is no $H_{2m+2}$ such that $L_{2m+2}({\rm e}^{i\theta})+H'_{2m+2}=0$. \end{prop} \paragraph{Proof} In the sequel, we will say simply powers instead of powers of ${\rm e}^{i\theta}$. If we can find $H_k'$, then $H_k$ and $H_k'$ ($k\ge 3$) have the same powers. From (6) we see that $F_j$ and $G_j$ ($j\ge 2$) have (only) powers of the parity opposite to that of $j$. Since $H'_3=-F_2$, $H'_3$ and $H_3$ have odd powers. Up to constants, the terms in $L_4$ are $H_3F_2$, $F_3$ and $H'_3G_2$, where the powers in $H_3$, $F_2$, $H'_3$ and $G_2$ are odd. Then $L_4$ has even powers. The coefficient of ${\rm e}^{0i\theta}$ in $L_4$ is $P_1$. If $P_1=0$, we can find $H_4({\rm e}^{i\theta})$ such that $L_4({\rm e}^{i\theta})+H'_4=0$; in this case $H_4$ has even powers. If $P_1\ne 0$, it is impossible to solve the equation. Let $m\ge 2$. We proceed by induction. Let us suppose that it is possible to solve the equations $L_k({\rm e}^{i\theta})+H'_k=0$ up to $k=2m$ and that the powers in $H'_k$ and $H_k$ have the same parity as $k$. Up to constants, the terms in $L_k$ are of the form $H_rF_s$, $F_{k-1}$ and $H'_rG_s$, where $r+s=k+1$. If $k=2m+1$ is odd, then $F_{k-1}$ has odd powers. Since $r+s$ is even, $s$ and $r$ have the same parity and the powers in $H_rF_s$ and $H'_rG_s$ are odd. We conclude that $L_{2m+1}({\rm e}^{i\theta})+H'_{2m+1}=0$ has a solution and that $H'_{2m+1}$ and $H_{2m+1}$ have odd powers. Similar arguments show that, when $k=2m+2$, $F_{k-1}$, $H_rF_s$ and $H'_rG_s$ have even powers; then $L_{2m+2}({\rm e}^{i\theta})+H'_{2m+2}=0$ has a solution if and only if $P_m$, the coefficient of ${\rm e}^{0i\theta}$ in $L_{2m+2}$, is zero. If $P_m=0$, then $H'_{2m+2}$ and $H_k$ have even powers. \hfill\qed \begin{coro} Let $m$ be the smallest integer such that $P_m\ne 0$. Then the function $r^2 + H_3(\theta)r^3 + \cdots + H_{2m+1}(\theta)r^{2m+1}$, i.e., the solution of the system of equations $L_k({\rm e}^{i\theta})+H'_k=0$ ($3\le k\le 2m+1$), is a Lyapunov function for the focus. If $P_m<0$, the focus is stable. Otherwise it is unstable. \end{coro} To find the Poincar\'e-Lyapunov polynomials we proceed as follows. Equating ${dH/dt}$ with the right hand side of (\ref{eq7}), we get an infinite set of differential equations with the unknowns $H_j$ ($j\ge 3$) and $P_k$ ($k\ge 1$), where $P_k$ is the coefficient of ${\rm e}^{0i\theta}$ in $L_{2k+2}$. If $n=2k$ is even, the system is: \begin{eqnarray}\label{eq8} H_3' &=& -F_2 \nonumber\\ H_4' &=& P_1- {3\over2}H_3F_2 + F_3 - {1\over2}H_3' G_2 \\ & &\hskip -20pt\cdots \nonumber \\ H_{2k+1}' &=&\hskip -1pt -{2k\over2}H_{2k}F_2 -\cdots-{3\over2}H_3F_{2k-1} - F_{2k} - {1\over2}H_3' G_{2k-1} - \cdots - {1\over2}H_{2k}' G_2 \nonumber \\ H_{2k+2}'&=& P_k-{{2k+1}\over2}H_{2k+1}F_2 - \cdots -{3\over2}H_3F_{2k} \nonumber \\ & &\qquad\qquad\qquad\qquad\qquad\qquad - {1\over2}H_3' G_{2k} - \cdots - {1\over2}H_{2k+1}' G_2 \nonumber \\ & &\hskip -20pt\cdots \nonumber \end{eqnarray} If $n=2k-1$ is odd, the last lines become: \begin{eqnarray}\label{eq9} H_{2k+1}' &=& - {2k\over2}H_{2k}F_2 - \cdots - {3\over2}H_3F_{2k-1} - {1\over2}H_3' G_{2k-1} - \cdots - {1\over2}H_{2k}' G_2 \nonumber \\ H_{2k+2}'&=& P_k-{{2k+1}\over2}H_{2k+1}F_2 - \cdots -{4\over2}H_4F_{2k-1} \\ & &\qquad\qquad\qquad\qquad\qquad\qquad - {1\over2}H_4' G_{2k-1}-\cdots - {1\over2}H_{2k+1}' G_2 \nonumber \\ & &\hskip -20pt\cdots \nonumber \end{eqnarray} Poincar\'e used the sine and the cosine functions instead of ${\rm e}^{i\theta}$. \section{ The Main Result} Letting $z=\alpha w$ ($\alpha = \rho {\rm e}^{i\psi}$), the vector field (\ref{eq4}) becomes (writing just one equation): $$ \dot w = iw + \sum_{m=2}^n\sum_{j+k=m} a_{jk}\alpha^{j-1}\bar a^kw^j\bar w^k. $$ Then we obtain: \begin{lemma} Under the action of the element $\rho e^{i\psi}$ of the group ${\mathbb C}^*$, $a_{rs}$ and $\bar a_{rs}$, where $r+s=m$, are respectively changed to $a_{rs}\rho^{m-1}{\rm e}^{(r-s-1)i\psi}$ and\newline $\bar a_{rs}\rho^{m-1}{\rm e}^{(s-r+1)i\psi}$. \end{lemma} \paragraph{Definition} Let $c\in{\mathbb C}$ be a constant. If $r+s=m$, the weight of $ca_{rs}$ or $c\bar a_{rs}$ with respect to $\rho$ is $m-1$ . The respective weights of $ca_{rs}$ and $c\bar a_{rs}$ with respect to $\psi$ are $r-s-1$ and $s-r+1$. \begin{lemma}\label{lem4} Let $c\in{\mathbb C}$ be a constant. Each $ca_{rs}$ or $c\bar a_{rs}$ in $F_m$ and $G_m$ (see (\ref{eq6})) have a weight with respect of $\rho$ equal to $m-1$. The weight with respect to $\psi$ of each monomial in the coefficient of ${\rm e}^{ti\theta}$ is $t$. \end{lemma} \paragraph{Proof} Because $j+k=m$ ($j,k\ge 0$), $(k+1)+(j-1)=m$ ($j\ne 0$) and $0+m=m$, equation (6) implies that the weights with respect to $\rho$ of $ca_{jk}$, $c\bar a_{(k+1)(j-1)}$ and $c\bar a_{m0}$ in $F_m$ and $G_m$ are indeed equal to $m-1$. The weight with respect to $\psi$ of $ca_{jk}$ is $j-k-1$, that of $c\bar a_{(k+1)(j-1)}$ ($j\ne 0$), $(j-1)-(k+1)+1=j-k-1$ and that of $c\bar a_{0m}$, $m-0+1=m+1$. \hfill\qed \medskip Since each monomial in the coefficient of ${\rm e}^{si\theta}$ has the same weights, we can, without ambiguity, talk about of the {\it weights of this coefficient\/}. The following notation will help to easily determine the weights of the coefficient of ${\rm e}^{si\theta}$ in $F_m$ and $G_m$. \paragraph{Notation} Let us denote the coefficient of ${\rm e}^{si\theta}$ in $F_m$ by $c{\scriptstyle [m-1,s]}$. The coefficients of the ${\rm e}^{si\theta}$'s in $G_m$ will be denoted in order by $$ -ic{\scriptstyle [m-1,-m-1]}, d{\scriptstyle [m-1,-m+1]},\ldots, d{\scriptstyle [m-1,m-1]}, ic{\scriptstyle [m-1,m+1]}\,. $$ In the particular case of the family of polynomial vector fields of degree 3, one gets: \begin{eqnarray*} \dot r &=& {1\over2}\left(c{\scriptstyle [1,-3]}{\rm e}^{-3i\theta} + c{\scriptstyle [1,-1]}{\rm e}^{-i\theta} + c{\scriptstyle [1,1]}{\rm e}^{i\theta} + c{\scriptstyle [1,3]}{\rm e}^{3i\theta}\right)r^2 \\ & & + {1\over2}\left(c{\scriptstyle [2,-4]}{\rm e}^{-4i\theta} + c{\scriptstyle [2,-2]}{\rm e}^{-2i\theta} + c{\scriptstyle [2,0]} + c{\scriptstyle [2,2]}{\rm e}^{2i\theta} + c{\scriptstyle [2,4]}{\rm e}^{4i\theta}\right)r^3\\ \dot\theta &=& 1+ {1\over2}\left(-ic{\scriptstyle [1,-3]}{\rm e}^{-3i\theta} +d{\scriptstyle [1,-1]}{\rm e}^{-i\theta}+d{\scriptstyle [1,1]}{\rm e}^{i\theta} +ic{\scriptstyle [1,3]}{\rm e}^{3i\theta}\right)r \\ & & +{1\over2}\left(-ic{\scriptstyle [2,-4]}{\rm e}^{-4i\theta} +d{\scriptstyle [2,-2]}{\rm e}^{-2i\theta} +d{\scriptstyle [2,0]} + d{\scriptstyle [2,2]}{\rm e}^{2i\theta} + ic{\scriptstyle [2,4]}{\rm e}^{4i\theta}\right)r^2. \end{eqnarray*} \begin{lemma} The following relations are satisfied: $$ \bar c{\scriptstyle [m-1,s]}= c{\scriptstyle [m-1,-s]}\ \hbox{and}\ \bar d{\scriptstyle [m-1,s]} = d{\scriptstyle [m-1,-s]}. $$ Moreover, $c{\scriptstyle [m-1,0]}$ and $d{\scriptstyle [m-1,0]}$ are real. \end{lemma} \paragraph{Proof} $F_m$ and $G_{m}$ are real expressions, since the original family of vector fields is real. Because in (\ref{eq4}), $\dot z\bar z+z\dot{\bar z}$ and $-i\dot z\bar z +iz\dot{\bar z}$ are sums of conjugate terms, $F_m$ and $G_m$ are are also sums of conjugate terms. Precisely, the conjugate of the coefficient of ${\rm e}^{si\theta}$ is the coefficient of the conjugate of ${\rm e}^{si\theta}$. Then $\bar c{\scriptstyle [m-1,s]}=c{\scriptstyle [m-1,-s]}$ and $\bar d{\scriptstyle [m-1,s]}=d{\scriptstyle [m-1,-s]}$. When $s=0$, the terms $c{\scriptstyle [m-1,0]}$ and $d{\scriptstyle [m-1,0]}$ are self-conjugate, and therefore real. \hfill\qed \paragraph{Definition} Let $h$ be a monomial in the unknowns $c{\scriptstyle [j,s]}$ and $d{\scriptstyle [k,t]}$. The weights of $h$ with respect to $\rho$ and $\psi$ are the sums of the respective weights of its unknowns. We will say that a polynomial is weighted-homogeneous of degree $(k,r)$ if all its monomials have the same weights $k$ and $r$ with respect to $\rho$ and $\psi$ respectively. \begin{prop} Let $Q_t$ be the coefficient of ${\rm e}^{ti\theta}$ in $H_s$. Then $Q_t$ is weighted-homogeneous of degree $(s-2,t)$. $P_k$ is weighted-homogeneous of degree $(2k,0)$. \end{prop} \paragraph{Proof} Let us look at the system of equations (\ref{eq8}) or (\ref{eq9}). According to lemma~\ref{lem4} and the paragraph following it, the statement is true for all the coefficients $Q_t$ in $H_3$, since $H_3' = - F_2$. Since $H_s'=-L_s$ (see notation 1), the result follows by induction. \begin{coro} $P_k$ is invariant under the group of rotations $S^1$ and is invariant under the group $C^*$ modulo a positive constant. \end{coro} \section{ Conclusion} We have proved not only that $\forall j\ge 1$ and for $g=\rho{\rm e}^{i\psi}\in {\mathbb C}^*$, $P_j(g(a_{rs}))=\rho^{2j}P_j(a_{rs})$, where $P_j$ is a Poincar\'e-Lyapunov polynomial, but also that $P_j$ is weighted-homogeneous of degree $(2j,0)$ (according to definition 3). This result has at least two goals. New directions of research related to Hilbert's $16^{\rm th}$ problem which look promising have been given by H. Zoladek in [3] and [4]. One of the questions raised by the Hilbert's $16^{\rm th}$ problem is about the maximum number of limit cycles that exist in the family of polynomial vector fields of degree less or equal to $n$. A minor question, but closely related to, is to determine the maximum number of limit cycles near a center-focus. Zoladek proved in [3] that the family of polynomial vector fields of degree less or equal to two has at most 3 limit cycles near a center-focus. In [4], he proved that a family of degree less or equal to three, but without its quadratic part, has at most 5 limit cycles near a center-focus. The proofs follow from his main result that says the ideal generated by the Poincar\'e-Lyapunov polynomials is a linear combination, with polynomial coefficients in the $a_{rs}$, of the first Poincar\'e-Lyapunov polynomials. He utilizes for it the invariance of the Poincar\'e-Lyapunov polynomials under the group of rotations, but the arguments for proving the invariance, though correct, are rather elliptic. The present article gives a detailed proof. One knows the importance of the Poincar\'e-Lyapunov polynomials to determine the stability of an equilibrium point. One could hope to find the Poincar\'e-Lyapunov polynomials for certain low degree polynomial vector fields. Indeed, using a computer, one could list all the monomials of $P_j$, since they must satisfy the (two) homogeneity condition(s). Using the explicit system (8) or (9), one could find the coefficients of the monomials. \paragraph{\bf Remark} The author has received from J.P. Fran\-\c coi\-se, C. Rousseau and R. Roussarie the main arguments of another proof of the invariance of the Poincar\'e-Lyapunov polynomials under the group of rotations. They do not have a result on the homogeneity with respect to the weights. \begin{thebibliography}{0} \bibitem{[1]} A. A. Andronov {\it et al}, Theory of Bifurcations of Dynamic Systems on a plane, {\it John Wiley \& Sons\/}, 1973. \bibitem{[2]} H. Poincar\'e, Oeuvres de Poincar\'e, Chapitre 11 (Th\'eorie des centres), pp 95-114. \bibitem{[3]} H. Zoladek, Quadratic systems with center and their perturbations, {\it J. of Diff. Eqns.}, {\bf 109}, 1994, pp 223-273. \bibitem{[4]} H. Zoladek, On a certain generalization of Bautin's theorem, {\it Non-linearity}, {\bf 7}, 1994, pp 273-279. \end{thebibliography} \bigskip {\sc Pierre Joyal}\\\ D\'epartement d'informatique et de math\'ematique\\ Universit\'e du Qu\'ebec \`a Chicoutimi\\ 555 boul. de l'Universit\'e, Chicoutimi, G7H 2B1, Canada\\ E-mail address: Pierre\_Joyal@uqac.uquebec.ca \end{document}