A commutative Banach algebra
A
\mathcal {A}
is said to have the
P
(
k
,
n
)
P(k,n)
property if the following holds: Let
M
{{M}}
be a closed subspace of finite codimension
n
n
such that, for every
x
∈
M
x\in {{M}}
, the Gelfand transform
x
^
\hat {x}
has at least
k
k
distinct zeros in
Δ
(
A
)
\Delta (\mathcal {A})
, the maximal ideal space of
A
\mathcal {A}
. Then there exists a subset
Z
Z
of
Δ
(
A
)
\Delta (\mathcal {A})
of cardinality
k
k
such that
M
^
\hat {{M}}
vanishes on
Z
Z
, the set of common zeros of
M
{{M}}
. In this paper we show that if
X
⊂
C
X\subset \mathbf {C}
is compact and nowhere dense, then
R
(
X
)
R(X)
, the uniform closure of the space of rational functions with poles off
X
X
, has the
P
(
k
,
n
)
P(k,n)
property for all
k
,
n
∈
N
k,n\in \mathbf {N}
. We also investigate the
P
(
k
,
n
)
P(k,n)
property for the algebra of real continuous functions on a compact Hausdorff space.