In an earlier work, the authors prove Stillman’s conjecture in all characteristics and all degrees by showing that, independent of the algebraically closed field
K
K
or the number of variables,
n
n
forms of degree at most
d
d
in a polynomial ring
R
R
over
K
K
are contained in a polynomial subalgebra of
R
R
generated by a regular sequence consisting of at most
η
B
(
n
,
d
)
{}^\eta \!B(n,d)
forms of degree at most
d
d
; we refer to these informally as “small” subalgebras. Moreover, these forms can be chosen so that the ideal generated by any subset defines a ring satisfying the Serre condition R
η
_\eta
. A critical element in the proof is to show that there are functions
η
A
(
n
,
d
)
{}^\eta \!A(n,d)
with the following property: in a graded
n
n
-dimensional
K
K
-vector subspace
V
V
of
R
R
spanned by forms of degree at most
d
d
, if no nonzero form in
V
V
is in an ideal generated by
η
A
(
n
,
d
)
{}^\eta \!A(n,d)
forms of strictly lower degree (we call this a strength condition), then any homogeneous basis for
V
V
is an R
η
_\eta
sequence. The methods of our earlier work are not constructive. In this paper, we use related but different ideas that emphasize the notion of a key function to obtain the functions
η
A
(
n
,
d
)
{}^\eta \!A(n,d)
in degrees 2, 3, and 4 (in degree 4 we must restrict to characteristic not 2, 3). We give bounds in closed form for the key functions and the
η
A
_
{}^\eta \!{\underline {A}}
functions, and explicit recursions that determine the functions
η
B
{}^\eta \!B
from the
η
A
_
{}^\eta \!{\underline {A}}
functions. In degree 2, we obtain an explicit value for
η
B
(
n
,
2
)
{}^\eta \!B(n,2)
that gives the best known bound in Stillman’s conjecture for quadrics when there is no restriction on
n
n
. In particular, for an ideal
I
I
generated by
n
n
quadrics, the projective dimension
R
/
I
R/I
is at most
2
n
+
1
(
n
−
2
)
+
4
2^{n+1}(n - 2) + 4
.