Lattice isomorphic Banach lattices of polynomials

We study Díaz-Dineen’s problem for regular homogeneous vector-valued polynomials. In particular, we prove that if $E^{\ast }$ and $F^{\ast }$ are lattice isomorphic with at least one having order continuous norm, then ${\mathcal{P}}^r{(}^{n}E;G^{\ast })$ and ${\mathcal{P}}^r{(}^{n}F;G^{\ast })$ are lattice isomorphic for every $n\in \mathbb{N}$ and every Banach lattice $G$. We also study the analogous problem for the classes of regular compact, regular weakly compact, orthogonally additive and regular nuclear polynomials.