Compressed algebras: Artin algebras having given socle degrees and maximal length

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 ] R = k[ {{y_1},… ,{y_r}} ] has ( j + r − 1 r − 1 ) ( { {array}{*{20}{c}} {j + r - 1} \\ {r - 1} \\ {array} } ) linearly independent partial derivates of order i i , for i = 0 , 1 , … , t = [ j / 2 ] i = 0,1,… ,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