Polynomial first integrals for quasi-homogeneous polynomial differential systems
Polynomial first integrals for quasi-homogeneous polynomial differential systems
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DOI：
10.1088/0951-7715/15/4/313
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发表时间：
2002-07
期刊：
影响因子：
1.7
通讯作者：
J. Llibre;Xiang Zhang
中科院分区：
文献类型：
--
作者：
J. Llibre;Xiang Zhang
1996 年，Furta (Furta S D 1996 Z. Angew Math. Phys. 47 112-31) 和 Goriely (Goriely A 1996 J. Math. Phys. 37 1871-93) 独立证明了拟齐次多项式微分系统的 Kowalevskaya 指数与其阶数之间存在联系。拟齐次多项式优先 积分。在这里，我们提供了一个新链接。在与二次齐次多项式微分系统相关的 Kowalevskaya 矩阵可对角化的特殊情况下，Tsygvintsev (Tsygvintsev A 2001 J. Phys. A: Math. Gen. 34 2185-93) 在 2001 年对 Furta 和 Goriely 发现的链接进行了改进，他还为这些系统证明了任意 给定次数的齐次多项式第一积分是一组固定多项式的线性组合。我们证明，当该系统没有可对角化的 Kowalevskaya 矩阵时，Tsygvintsev 的结果也是正确的。最后，我们用 Kowalevskaya 指数来描述具有拟齐次多项式一阶积分的权重 2 次的二维拟齐次多项式微分系统。
In 1996, Furta (Furta S D 1996 Z. Angew Math. Phys. 47 112-31) and Goriely (Goriely A 1996 J. Math. Phys. 37 1871-93) proved, independently, the existence of a link between the Kowalevskaya exponents of quasi-homogeneous polynomial differential systems and the degree of their quasi-homogeneous polynomial first integrals. Here, we provide a new link. In the particular case that a Kowalevskaya matrix associated with a quadratic homogeneous polynomial differential system is diagonalizable, an improvement of the link found by Furta and Goriely has been obtained by Tsygvintsev (Tsygvintsev A 2001 J. Phys. A: Math. Gen. 34 2185-93) in 2001, who additionally proved for these systems that an arbitrary homogeneous polynomial first integral of a given degree is a linear combination of a fixed set of polynomials. We show that Tsygvintsev's results are also true when this system has no diagonalizable Kowalevskaya matrices. Finally, we characterize in terms of the Kowalevskaya exponents the two-dimensional quasi-homogeneous polynomial differential systems of weight degree 2 which have a quasi-homogeneous polynomial first integral.