Ergodic Theory and Dynamical Systems

Center conditions, compositions of polynomials and moments on algebraic curves

a1 Ort Academic College for Teachers in Technology, Givat Ram, Jerusalem 95435, Israel
a2 Université de Paris VI, U.F.R. 920, 46–56, B.P. 172, Mathematiques, 4 Pl. Jussieu, 75252 Paris, France
a3 Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel (e-mail:


We consider an Abel equation $(*)$ $y^{\prime}=p(x)y^2+q(x)y^3$ with $p(x)$, $q(x)$ polynomials in $x$. A center condition for ($*$) (closely related to the classical center condition for polynomial vector fields on the plane) is that $y_0=y(0)\equiv y(1)$ for any solution $y(x)$ of ($*$). This condition is given by the vanishing of all the Taylor coefficients $v_k(1)$ in the development $y(x)=y_0+\sum^{\infty}_{k=2}v_k(x)y^k_0$. A new basis for the ideals $I_k=\{v_2,\dots,v_k\}$ has recently been produced, defined by a linear recurrence relation. Studying this recurrence relation, we connect center conditions with a representability of $P=\int p$ and $Q=\int q$ in a certain composition form (developing further some results of Alwash and Lloyd), and with a behavior of the moments $\int P^kq$. On this base, explicit center equations are obtained for small degrees of $p$ and $q$.

(Received March 27 1997)
(Revised September 25 1998)