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.