缺二次项的三次系统中心焦点的判定

Criteria of Center and Focus for Cubic System Without Quadratic Terms

本文研究缺二次项的三次系统,直接由系统的系数给出了焦点量的算法,用muMATH软件在PC机上进行符号运算,导出了计算焦点量的最简公式,并获得了判定原点为中心的充要条件,从而彻底解决了缺二次项的三次系统的中心焦点的判定问题。此外,本文还对[4]的结果进行了核算,指出了他的疏忽之处。

This paper studies cubic system (LE_3) without quadratic terms. An algorithm for determining the focal values of (LE_3) is given directly by the coefficients of the system. Through symbol operations in IBM-PC with the software muMATH, the simplest formula describing the focal values is achieved. The necessary and sufficient condition for determining center is obtained. Therfore, the problem of how to distingush between Center and focus for the system (LE_3) is completly solved. Moreover we have checked the resulst of [4] and have found some errors in it.