It was known that the various higher partial derivatives of a set of c general enough polynomials b1, ..., b(c) in P = k[Y1, ..., Y(r)] of specified degrees are as independent as possible. We generalize this result to a set of c general enough elements of the direct sum P(s). The power series ring A = k[[x1, ..., x(r)]] acts as partial differential operators on P(s); the Matlis duality relates A-submodules A (f1, ..., fc) of P(s) and quotients MBAR of A(s). The degrees of f1, ..., fc determine the socle type of MBAR . A quotient MBAR of As is termed compressed if it has maximal length given (r,s, socle type of MBAR}. A compressed module MBAR is the Matlis dual to a set of elements b1, ..., b(c) of specified degrees in P(s), whose partial derivatives of all orders are as independent as possible. We term a quotient MBAR of A(s) Gorenstein of socle type t(j) if its Matlis dual is generated by a single polynomial b of degree j. We use the Matlis duality to construct and describe the varieties G(j) and Z(j) parametrizing the compressed graded and nongraded Gorenstein Artin modules MBAR of socle type t(j), that are quotients of M = A(s). We determine the Hilbert series of such modules and show that they are termwise external among the Hilbert series of quotients of M having the same socle type. We also determine the dimensions of G(j) and Z(j); and we show that they are locally affine spaces. We apply the dimension result to show, when r = 3 and s = 2, that there is a length-526 type-one (Gorenatein) quotient MBAR of A+A having no deformation to k+ ... +k. Thus, MBAR is not smoothable. This complements the result of H. Kleppe that when r = 3 and s = 1, Gorenstein Artin algebra quotients of A are smoothable.
