An exponential formula for polynomial vector fields - II. Lie series, Exponential Substituition, and rooted trees

摘要

Grobner's Lie Series and the Exponential Formula provide different explicit formulas for the flow generated by a finite-dimensional polynomial vector Field. The present paper gives (1) a generalization of the Lie series in case of non-commuting variables called Exponential Substituition, (2) a structural understanding of the three Formulas and their mutual relationships in terms of rooted trees. and (3) as a byproduct new results on the enumeration, coding, and statistics of different kinds Of rooted trees. (C) 1999 Academic Press. [References: 38]