一类三维幂零向量场的5次超规范形

Quintic Hypernormal Form of a Class of Nilpotent Three-dimensional Vector Fields

为了研究三维幂零向量场的超规范形(最简规范形、唯一规范形),利用新次数函数和多重李括号方法,通过引入分块矩阵的新记号,研究了一类具有对称性质的三维幂零向量场的5次超规范形问题;证明了在一定条件下,此类向量场的二阶规范形是超规范形,并获得其二阶5次超规范形的唯一形式;还研究了此类向量场的退化情况,验证了与二维结论的一致性.

To research hypernormal form （ simplest normal form, unique normal form） of nilpotent three- dimensional vector fields, the problem of quintic hypernormal form of a class of nilpotent three- dimensional vector fields with symmetrical property was studied by using new grading function and the method of multiple Lie brackets and by introducing the new marks for block matrices firstly, then that the second order normal form of above vector field was hypernormal form had been proved, and the unique form of the second order quintic hypernormal form was obtained. At last, the degenerate case of above vector fields was studied, and that this could be consistent with the conclusions in dimension 2 was verified.