In this paper we give a notion of polynomial type
p
(
X
)
p(X)
of a Noetherian scheme
X
X
and define the function
d
p
:
X
⟶
Z
dp:\, X\longrightarrow \mathbb {Z}
by
d
p
(
x
)
=
dim
O
X
,
x
−
p
(
O
X
,
x
)
dp(x)=\dim O_{X,x} -p(O_{X,x} )
for all
x
∈
X
.
x\in X.
Then we show that if
X
X
admits a dualizing complex and
X
X
is equidimensional,
d
p
dp
is (lower) semicontinuous; moreover, in that case, the non-Cohen-Macaulay locus nCM
(
X
)
=
{
x
∈
X
∣
O
X
,
x
(X)=\{ x\in X\mid O_{X,x}
is not Cohen-Macaulay} is biequidimensional iff
d
p
dp
is constant on nCM
(
X
)
.
(X).