We attach a limit mixed Hodge structure to any polynomial map $f:{ℂ}^n\to ℂ$. The equivariant Hodge numbers of this mixed Hodge structure are invariants of $f$ which reflect its asymptotic behaviour. We compute them for a generic class of polynomials in terms of equivariant Hodge numbers attached to isolated hypersurface singularities and equivariant Hodge numbers of cyclic coverings of projective space branched along a hypersurface. We show how these invariants allow to determine topological invariants of $f$ such...