  The analogue of the Poisson bracket for the De Donder-Weyl (DW) Hamiltonian
formulation of field theory is proposed. We start from the Hamilton-
Poincar\'{e}-Cartan (HPC) form of the multidimensional variational calculus and
define the bracket on the differential forms over the space-time (=horizontal
forms). This bracket is related to the Schouten-Nijenhuis bracket of the
multivector fields which are associated with the horizontal forms by means of
the "polysymplectic form". The latter is given by the HPC form and generalizes
the symplectic form to field theory. We point out that the algebra of forms
with respect to our Poisson bracket and the exterior product has the structure
of the Gerstenhaber graded algebra. It is shown that the Poisson bracket with
the DW Hamiltonian function generates the exterior differential thus leading to
the bracket representation of the DW Hamiltonian field equations. Few
illustrative examples are also presented.
