Hostname: page-component-7c8c6479df-ph5wq Total loading time: 0 Render date: 2024-03-28T13:23:04.237Z Has data issue: false hasContentIssue false

Center conditions, compositions of polynomials and moments on algebraic curves

Published online by Cambridge University Press:  01 October 1999

M. BRISKIN
Affiliation:
Ort Academic College for Teachers in Technology, Givat Ram, Jerusalem 95435, Israel
J.-P. FRANCOISE
Affiliation:
Université de Paris VI, U.F.R. 920, 46–56, B.P. 172, Mathematiques, 4 Pl. Jussieu, 75252 Paris, France
Y. YOMDIN
Affiliation:
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 $q$.

Type
Research Article
Copyright
1999 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)