Prolongation of holomorphic vector fields on a tube domain

In this paper, we discuss some algebraic criterion on the completeness of holomorphic vector fields on a tube domain TΩ. Our objects of consideration are polynomial vector fields on TΩ. We give a method of determining the higher degree complete polynomial vector fields from the data on the lower degree complete polynomial vector fields, which we call prolongation. Furthermore, we give its applications to the holomorphic equivalence problem for tube domains.