Generalized center conditions and multiplicities for polynomial Abel equations of small degrees.

M.Blinov, Y.Yomdin

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 this equation (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). This condition is given by vanishing of all the Taylor coefficients $v_k(1)$ in the development $y(x)=y_0+\sum^{\infty}_{k=2}v_k(x)y^k_0$. Following [BFY2] we introduce periods of the equation (*) as those $\omega \in \mathbb{C}$, for which $y(0)\equiv y(\omega)$ for any solution y(x) of (*). The generalized center conditions are conditions on p, q under which given $a_1, \ldots, a_k$ are (exactly all) the periods of (*).

A new basis for the ideals $I_k=\{v_2,\ldots,v_k\}$ has been produced in [BFY1], defined by a linear recurrence relation. Using this basis and a special representation of polynomials, we extend results of [BFY2], proving for small degrees of $p$ and $q$ a composition conjecture, stated in [AL], [BFY2], [BFY3]. In particular, this provides transparent generalized center conditions in the cases considered. We also compute maximal possible multiplicity of the zero solution of (*), extending results of [AL].

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