The cardinality of a minimal basis of an ideal I is denoted
ν
(
I
)
\nu (I)
. Let A be a polynomial ring in
n
>
0
n > 0
variables with coefficients in a noetherian (commutative with
1
≠
0
1 \ne 0
) ring R, and let M be a maximal ideal of A. In general
ν
(
M
A
M
)
+
1
⩾
ν
(
M
)
⩾
ν
(
M
A
M
)
\nu (M{A_M}) + 1 \geqslant \nu (M) \geqslant \nu (M{A_M})
. This paper is concerned with the attaining of equality with the lower bound. It is shown that equality is attained in each of the following cases: (1)
A
M
{A_M}
is not regular (valid even if A is not a polynomial ring), (2)
M
∩
R
M \cap R
is maximal in R and (3)
n
>
1
n > 1
. Equality may fail for
n
=
1
n = 1
, even for R of dimension 1 (but not regular), and it is an open question whether equality holds for R regular of dimension
>
1
> 1
. In case
n
=
1
n = 1
and
dim
(
R
)
=
2
\dim (R) = 2
the attaining of equality is related to questions in the K-theory of projective modules. Corollary to (1) and (2) is the confirmation, for the case of maximal ideals, of one of the Eisenbud-Evans conjectures; namely,
ν
(
M
)
⩽
max
{
ν
(
M
A
M
)
,
dim
(
A
)
}
\nu (M) \leqslant \max \{ \nu (M{A_M}),\dim (A)\}
. Corollary to (3) is that for R regular and
n
>
1
n > 1
, every maximal ideal of A is generated by a regular sequence—a result well known (for all
n
⩾
1
n \geqslant 1
) if R is a field (and somewhat less well known for R a Dedekind domain).