  A form in a polynomial ring over a field is said to be homaloidal if its
polar map is a Cremona map, i.e., if the rational map defined by the partial
derivatives of the form has an inverse rational map. The object of this work is
the search for homaloidal polynomials that are the determinants of sufficiently
structured matrices. We focus on generic catatalecticants, with special
emphasis on the Hankel matrix. An additional focus is on certain degenerations
or specializations thereof. In addition to studying the homaloidal nature of
these determinants, one establishes several results on the ideal theoretic
invariants of the respective gradient ideals, such as primary components,
multiplicity, reductions and free resolutions.
