Solvable Lie algebras, Lie groups and polynomial structures

In this paper, we study polynomial structures by starting on the Lie algebra level, then passing to Lie groups to finally arrive at the polycyclic-by-finite group level. To be more precise, we first show how a general solvable Lie algebra can be decomposed into a sum of two nilpotent subalgebras. Using this result, we construct, for any simply connected, connected solvable Lie group G of dim n, a simply transitive action on R-n which is polynomial and of degree less than or equal to n(3). Finally, we show the existence of a polynomial structure on any polycyclic-by-finite group Gamma, which is of degree less than or equal to h(Gamma)(3) on almost the entire group (h (Gamma) being the Hirsch length of Gamma).