Generalized centre conditions and multiplicities for polynomial Abel equations of small degrees
Generalized centre conditions and multiplicities for polynomial Abel equations of small degrees
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DOI：
10.1088/0951-7715/12/4/317
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发表时间：
1999-07
期刊：
影响因子：
1.7
通讯作者：
M. Blinov;Y. Yomdin
中科院分区：
文献类型：
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作者：
M. Blinov;Y. Yomdin
我们考虑一个Abel方程(*)y‘=p(X)y2+q(X)y3，其中p(X)，q(X)-多项式在x中.这个方程的一个中心条件(与平面上多项式向量场的经典中心条件密切相关)是：对任何解y(X)，y0=y(0)y(1).这个条件是由展开过程中所有泰勒系数Vk(1)的消失所给出的。在Briskin等人(中心条件，多项式的合成和代数曲线上的矩出现)之后，我们引入了方程(*)的周期，对于(*)的任何解y(X)，y(0)y()。广义中心条件是关于P，Q的条件，在该条件下，给定的A1，…，AK是(恰好都是)(*)的周期。Briskin等人(1998，The Bautin Ideas of the Abel方程非线性10)给出了理想Ik={v2，…，vk}的一个新的基，它由一个线性递推关系定义。利用这个基和多项式的一种特殊表示，我们推广了Briskin等人的结果(中心条件，多项式的合成和代数曲线上的矩出现)，证明了小次数p和q的合成猜想，如Alwash和Lloyd(1987年与多项式二维系统有关的非自治方程)所述。R.Soc.Edinburgh A 105 129-52)，Briskin等人(中心条件，多项式的合成和代数曲线上的矩)，Briskin等人(中心条件II：参数和模型中心问题待出现)。特别是，这在所考虑的情况下提供了透明的广义中心条件。我们还计算了(*)的零解的最大可能重数，推广了Alwash和Lloyd(1987)关于多项式二维系统的非自治方程的结果。R.Soc.爱丁堡A 105 129-52)。
We consider an Abel equation (*)y´ = p(x)y2+q(x)y3 with p(x), q(x) - polynomials in x. A centre condition for this equation (closely related to the classical centre condition for polynomial vector fields on the plane) is that y0 = y(0) y(1) for any solution y(x). This condition is given by the vanishing of all the Taylor coefficients vk(1) in the development . Following Briskin et al (Centre Conditions, Composition of Polynomials and Moments on Algebraic Curves to appear) we introduce periods of the equation (*) as those , for which y(0) y() for any solution y(x) of (*). The generalized centre conditions are conditions on p, q under which given a1, ... ,ak are (exactly all) the periods of (*). A new basis for the ideals Ik = {v2, ... ,vk} has been produced in Briskin et al (1998 The Bautin ideal of the Abel equation Nonlinearity 10), defined by a linear recurrence relation. Using this basis and a special representation of polynomials, we extend results of Briskin et al (Centre Conditions, Composition of Polynomials and Moments on Algebraic Curves to appear), proving for small degrees of p and q a composition conjecture, as stated in Alwash and Lloyd (1987 Non-autonomous equations related to polynomial two-dimensional systems Proc. R. Soc. Edinburgh A 105 129-52), Briskin et al (Centre Conditions, Composition of Polynomials and Moments on Algebraic Curves to appear), Briskin et al (Center Conditions II: Parametric and Model Centre Problems to appear). In particular, this provides transparent generalized centre conditions in the cases considered. We also compute maximal possible multiplicity of the zero solution of (*), extending the results of Alwash and Lloyd (1987 Non-autonomous equations related to polynomial two-dimensional systems Proc. R. Soc. Edinburgh A 105 129-52).