A polynomial bracket for the Dubrovin-Zhang hierarchies

We define a hierarchy of Hamiltonian PDEs associated to an arbitrary
tau-function in the semi-simple orbit of the Givental group
action on genus expansions of Frobenius manifolds. We prove that
the equations, the Hamiltonians, and the bracket are weightedhomogeneous
polynomials in the derivatives of the dependent variables
with respect to the space variable.
In the particular case of a conformal (homogeneous) Frobenius
structure, our hierarchy coincides with the Dubrovin-Zhang hierarchy
that is canonically associated to the underlying Frobenius
structure. Therefore, our approach allows to prove the polynomiality
of the equations, Hamiltonians, and one of the Poisson brackets
of these hierarchies, as conjectured by Dubrovin and Zhang.