J. Emsalem and the author showed in [18] that a general polynomial f f of degree j j in the ring R = k [ y 1 , … , y r ] \mathcal {R} = k[ {{y_1},\ldots ,{y_r}} ] has ( j + r − 1 r − 1 ) \left ( {\begin {array}{*{20}{c}} {j + r - 1} \\ {r - 1} \\ \end {array} } \right ) linearly independent partial derivates of order i i , for i = 0 , 1 , … , t = [ j / 2 ] i = 0,1,\ldots ,t = [ {j/2} ] . Here we generalize the proof to show that the various partial derivates of s s polynomials of specified degrees are as independent as possible, given the room available. Using this result, we construct and describe the varieties G ( E ) G(E) and Z ( E ) Z(E) parametrizing the graded and nongraded compressed algebra quotients A = R / I A = R/I of the power series ring R = k [ [ x 1 , … , x r ] ] R = k[[{x_1},\ldots ,{x_r}]] , having given socle type E E . These algebras are Artin algebras having maximal length dim ⁡ k A \dim {_{k}}A possible, given the embedding degree r r and given the socle-type sequence E = ( e 1 , … , e s ) E = ({e_1},\ldots ,{e_s}) , where e i

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