一类三次多项式微分系统的代数分类及其应用

利用Llibre的代数不变式理论,首先由二元二次和二元四次多项式的分类结果,对一次和三次齐次多项式微分系统进行代数分类,同时补充了已有结果中出现的漏洞;其次,由称共变张量空间的性质,对缺二次项的三次微分系统在保证轨线走向不变的前提下进行代数分类,使分类后同类系统的示性多项式有相同零点;最后通过讨论一类简单系统的有界性说明了分类的方便方处.

Algebraic Classification of a Kind of Cubic Polynomial System and Its Applications

By Llibre's algebraic invariant theory, at first on the basis of classification for quadric and quatic homogeneous binary polynomial, the plane linear and cubic homogeneous polynomial differential systems are classified. Some deficiency in the references is complemented. Secondly, according to the space of symmetric covariant tensors, on algebraic classification to cubic polynomial differential systems without quadratic terms is obtained, keeping the trajectory direction, which gives the same determinant polynomial for the same class. At last, as an application, the boundedness of a class of cubic systems is disscussed.