Transforming St\"{a}ckel Hamiltonians of Benenti type to polynomial form

In this paper we discuss two canonical transformations that turn St"{a}ckel separable Hamiltonians of Benenti type into polynomial form: transformation to Vi`ete coordinates and transformation to Newton coordinates. Transformation to Newton coordinates has been applied to these systems only very recently and in this paper we present a new proof that this transformation indeed leads to polynomial form of St"{a}ckel Hamiltonians of Benenti type. Moreover we present all geometric ingredients of these Hamiltonians in both Vi`ete and Newton coordinates.