Stable algebras of entire functions

Suppose that $h$ and $g$ belong to the algebra $\mathcal{B}$ generated by the rational functions and an entire function $f$ of finite order on $\mathbb{C}^n$ and that $h/g$ has algebraic polar variety. We show that either $h/g\in\mathcal{B}$ or $f=q_1e^p+q_2$, where $p$ is a polynomial and $q_1,q_2$ 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 $\mathbb{C}$ of two entire or meromorphic functions on $\mathbb{C}^n$ is completely resolved in this paper.