标题： 关于三元多项式代数上的局部幂零导子 显示英文标题

标题： On locally nilpotent derivations of polynomial algebra in three variables

摘要： 在本文中，我们研究特征为零的域上的三个变量多项式代数上的局部幂零导子。 我们引入一种迭代构造，给出所有秩为$2$的局部幂零导子。 这种构造可以得到秩为$2$的不可三角化的局部幂零导子的例子。 我们还证明了由Freudenburg给出的秩为$3$的局部幂零导子的著名例子是秩为$3$的新的局部幂零导子一大类例子中的一个成员。 我们的方法基于考虑与给定导子交换的所有局部幂零导子。 我们根据交换的局部幂零导子集合，对具有给定秩的局部幂零导子进行了表征。 显示英文摘要

摘要： In this paper we investigate locally nilpotent derivations on the polynomial algebra in three variables over a field of characteristic zero. We introduce an iterating construction giving all locally nilpotent derivations of rank $2$. This construction allows to get examples of non-triangularizable locally nilpotent derivations of rank $2$. We also show that the well-known example of a locally nilpotent derivation of rank $3$, given by Freudenburg, is a member of a large family of new examples of rank $3$ locally nilpotent derivations. Our approach is based on considering all locally nilpotent derivations commuting with a given one. We obtain a characterization of locally nilpotent derivations with a given rank in terms of sets of commuting locally nilpotent derivations.