It is known that if
k
k
is a field and
F
:
k
[
X
1
,
…
,
X
n
]
→
k
[
X
1
,
…
,
X
n
]
{\mathbf {F}}:k[{X_1}, \ldots ,{X_n}] \to k[{X_1}, \ldots ,{X_n}]
is a polynomial automorphism, then
deg
(
F
−
1
)
⩽
(
deg
F
)
n
−
1
\deg ({{\mathbf {F}}^{ - 1}}) \leqslant {(\deg \,{\mathbf {F}})^{n - 1}}
. We extend this result to the case where
k
k
is a reduced ring. Furthermore, if
k
k
is not a reduced ring, we show that for any integer
n
⩾
1
n \geqslant 1
and any integer
λ
⩾
0
\lambda \geqslant 0
there exists a polynomial automorphism
F
{\mathbf {F}}
such that
deg
(
F
−
1
)
=
λ
+
(
deg
F
)
n
−
1
\deg ({{\mathbf {F}}^{ - 1}}) = \lambda + {(\deg \,{\mathbf {F}})^{n - 1}}
.