Abstract. J. Emsalem and the author showed in [18] that a general polynomial/of degree j in the ring !%= k[yx,... ,yr\ has ('*'\X) linearly independent partial dérivâtes of order ¡, for i = 0,1,..., t = [j/2]. Here we generalize the proof to show that the various partial dérivâtes of 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) and Z(E) parametrizing the graded and nongraded compressed algebra quotients A = R/I of the power series ring R = k [[xx,...,xr]], having given socle type E. These algebras are Artin algebras having maximal length dim^ A possible, given the embedding degree r and given the socle-type sequence E = (ex,...,es), where e, is the number of generators of the dual module A of A, having degree i. The variety Z(E) is locally closed, irreducible, and is a bundle over G(E), fibred by affine spaces A" whose dimension is known.We consider the compressed algebras a new class of interesting algebras and a source of examples. Many of them are nonsmoothable-have no deformation to (k + ■ ■ ■ + k)-for dimension reasons. For some choices of the sequence E, D. Buchsbaum, D. Eisenbud and the author have shown that the graded compressed algebras of socle-type E have almost linear minimal resolutions over R, with ranks and degrees determined by E. Other examples have given type e = dim^ (socle A) and are defined by an ideal / with certain given numbers of generators in R = k [[xx,...,xr]].An analogous construction of thin algebras A = R/(fx,... ,fs) of minimal length given the initial degrees of/,,... ,fs is compared to the compressed algebras. When r = 2, the thin algebras are characterized and parametrized, but in general when r > 3, even their length is unknown. Although k = C through most of the paper, the results extend to characteristic p.
