Ergodic Theory and Dynamical Systems



Center conditions, compositions of polynomials and moments on algebraic curves


M. BRISKIN a1, J.-P. FRANCOISE a2 and Y. YOMDIN a3
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: yomdin@wisdom.weizmann.ac.il)

Abstract

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)