Abstract Using the Euler-Jacobi formula there is a relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the polynomial differential systems x ˙ = P ( x , y , z ) , y ˙ = Q ( x , y , z ) , z ˙ = R ( x , y , z ) with degrees of P, Q and R equal to two when these systems have the maximum number of isolated singular points, i.e., 8 singular points. In other words we extend the well-known Berlinskii's Theorem for quadratic polynomial differential systems in the plane to the space.

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空间二次多项式微分系统奇异点的构型及其拓扑指数

摘要 利用欧拉-雅可比公式，多项式向量场的奇异点与其拓扑指数之间存在关系。使用这个公式，我们获得多项式微分系统的奇异点的配置及其拓扑指数 x ˙ = P ( x , y , z ) , y ˙ = Q ( x , y , z ) , z = x , y , z ) 且 P、Q 和 R 的次数等于 2，当这些系统具有最大数量的孤立奇异点时，即 8 个奇异点。换句话说，我们将平面二次多项式微分系统的著名柏林斯基定理扩展到空间。