Stable algebras of entire functions

Suppose that h and g belong to the algebra B generated by the rational functions and an entire function f of finite order on ℂⁿ and that h/g has algebraic polar variety. We show that either h/g ∈ B or f = q₁e^p + q₂, where p is a polynomial and q ₁,q₂ are rational functions. In the latter case, h/g belongs to the algebra generated by the rational functions, e^p and 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 of two entire or meromorphic functions on ℂⁿ is completely resolved in this paper.