  The BRST structure of polynomial Poisson algebras is investigated. It is
shown that Poisson algebras provide non trivial models where the full BRST
recursive procedure is needed. Quadratic Poisson algebras may already be of
arbitrarily high rank. Explicit examples are provided, for which the first
terms of the BRST generator are given. The calculations are cumbersome but
purely algorithmic, and have been treated by means of the computer algebra
system REDUCE. Our analysis is classical ($=$ non quantum) throughout.
