Suppose that
h
h
and
g
g
belong to the algebra
B
\mathcal {B}
generated by the rational functions and an entire function
f
f
of finite order on
C
n
\mathbb {C}^n
and that
h
/
g
h/g
has algebraic polar variety. We show that either
h
/
g
∈
B
h/g\in \mathcal {B}
or
f
=
q
1
e
p
+
q
2
f=q_1e^p+q_2
, where
p
p
is a polynomial and
q
1
,
q
2
q_1,q_2
are rational functions. In the latter case,
h
/
g
h/g
belongs to the algebra generated by the rational functions,
e
p
e^p
and
e
−
p
e^{-p}
.
The stability property is related to the problem of algebraic dependence of entire functions over the ring of polynomials. The case of algebraic dependence over
C
\mathbb {C}
of two entire or meromorphic functions on
C
n
\mathbb {C}^n
is completely resolved in this paper.