Center conditions, compositions of polynomials and moments on algebraic curves
Center conditions, compositions of polynomials and moments on algebraic curves
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DOI：
10.1017/s0143385799141737
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发表时间：
1999-10
影响因子：
0.9
通讯作者：
M. Briskin;J. Françoise;Y. Yomdin
中科院分区：
文献类型：
--
作者：
M. Briskin;J. Françoise;Y. Yomdin
考虑阿贝尔方程y ^{\prime}=p（x）y^2+q（x）y^3，其中p（x），q（x）是x中的多项式.（$*$）的中心条件（与平面上多项式向量场的经典中心条件密切相关）是对于（$*$）的任何解$y（x）$，$y_0=y（0）\equiv y（1）$。这个条件是由展开式$y（x）=y_0+\sum^{\infty}_{k=2}v_k（x）y^k_0$中所有泰勒系数$v_k（1）$为零给出的。最近产生了理想$I_k=\{v_2，\dots，v_k\}$的一个新的基，由线性递归关系定义.研究这个递归关系，我们将中心条件与$P=\int p$和$Q=\int q$的可表示性以某种复合形式连接起来（进一步发展了Alwash和Lloyd的一些结果），并与矩$\int P^kq$的行为连接起来。在此基础上，得到了小次数p和q的显式中心方程.
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 $q$.