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Article Dans Une Revue Ergodic Theory and Dynamical Systems Année : 1999

Center conditions, compositions of polynomials and moments on algebraic curves

Myriam Briskin
  • Fonction : Auteur
Yosi Yomdin
  • Fonction : Auteur

Résumé

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 $q$.
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Dates et versions

hal-01408118 , version 1 (03-12-2016)

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  • HAL Id : hal-01408118 , version 1

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Myriam Briskin, Jean-Pierre Françoise, Yosi Yomdin. Center conditions, compositions of polynomials and moments on algebraic curves. Ergodic Theory and Dynamical Systems, 1999. ⟨hal-01408118⟩
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