Polynomial poly-vector fields

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.