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Stable algebras of entire functions

Authors:	Dan Coman Evgeny A Poletsky

Institution:	Department of Mathematics, 215 Carnegie Hall, Syracuse University, Syracuse, New York 13244-1150 ; Department of Mathematics, 215 Carnegie Hall, Syracuse University, Syracuse, New York 13244-1150

Abstract:	Suppose that and belong to the algebra $\mathcal{B}$ generated by the rational functions and an entire function of finite order on $\mathbb{C}^n$ and that has algebraic polar variety. We show that either $h/g\in\mathcal{B}$ or , where is a polynomial and are rational functions. In the latter case, belongs to the algebra generated by the rational functions, 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.

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