  We explicitly describe the defining relations for simple Lie algebra of
vector fields with polynomial coefficients and its subalgebras of divergence
free, hamiltonian and contact vector fields, and for the Poisson algebra
(realized on polynomials). We consider generators of these Lie algebras
corresponding to the systems of simple roots associated with the standard
grading of these algebras. (These systems of simple roots are distinguished in
the sense of Penkov and Serganova.
