In this text we give a decomposition result on polynomial poly-vector fields
generalizing a result on the decomposition of homogeneous Poisson structures.
We discuss consequences of this decomposition result in particular for low
dimensions and low degrees. We provide the tools to calculate simple cubic
Poisson structures in dimension three and quadratic Poisson structures in
dimension four. Our decomposition result has a nice effect on the relation
between Poisson structures and Jacobi structures.Comment: 20 pages. v4: sections rearranged and formulations clarifie
