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Some division theorems for vector fields. (English) Zbl 0779.17022

The author deals with the following problem of division: Given vector fields \(X\), \(Z\), does there exist a vector field \(Y\) such that \([X,Y]=Z\)? The problem has been considered only for local vector fields and the full and positive answer is known whenever \(X\) has a nonvanishing germ. The author considers the case of homogeneous vector fields \(X\), \(Z\), that is \(X(0)=Z(0)=0\) and gives partial answers. The methods used are based on various estimates, in particular those concerning prolongations of dynamical systems. A generalization to polynomials of the adjoint operator \(\text{ad }X\) is given.

MSC:

17B66 Lie algebras of vector fields and related (super) algebras
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