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
{e_i}
is the number of generators of the dual module
A
^
\hat A
of
A
A
, having degree
i
i
. The variety
Z
(
E
)
Z(E)
is locally closed, irreducible, and is a bundle over
G
(
E
)
G(E)
, fibred by affine spaces
A
N
{{\mathbf {A}}^N}
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
)
(k + \cdots + k)
—for dimension reasons. For some choices of the sequence
E
,
D
E,{\text {D}}
. Buchsbaum,
D
{\text {D}}
. Eisenbud and the author have shown that the graded compressed algebras of socle-type
E
E
have almost linear minimal resolutions over
R
R
, with ranks and degrees determined by
E
E
. Other examples have given type
e
=
dim
k
(
socle
A
)
e = {\dim _k}\;({\text {socle}}\;A)
and are defined by an ideal
I
I
with certain given numbers of generators in
R
=
k
[
[
x
1
,
…
,
x
r
]
]
R = k[[{x_1},\ldots \;,{x_r}]]
. An analogous construction of thin algebras
A
=
R
/
(
f
1
,
…
,
f
s
)
A = R/({f_1},\ldots ,{f_s})
of minimal length given the initial degrees of
f
1
,
…
,
f
s
{f_1},\ldots ,{f_s}
is compared to the compressed algebras. When
r
=
2
r = 2
, the thin algebras are characterized and parametrized, but in general when
r
>
3
r > 3
, even their length is unknown. Although
k
=
C
k = {\mathbf {C}}
through most of the paper, the results extend to characteristic
p
p
.