# Stable algebras of entire functions

## TL;DR

This paper characterizes when ratios of functions in an algebra generated by rational functions and an entire function of finite order are contained within the algebra, revealing a specific exponential form in certain cases.

## Contribution

It establishes a clear criterion for when the ratio of two functions in the algebra remains in the algebra, identifying a special exponential form involving polynomials.

## Key findings

- If h/g is in the algebra, then h/g belongs to the algebra generated by rational functions and exponentials of a polynomial.
- If h/g is not in the algebra, then the entire function f must be of the form q_1e^p + q_2, with p polynomial and q_1,q_2 rational.
- The algebra generated by rational functions, e^p, and e^{-p} contains h/g in the case where it is not originally in the algebra.

## Abstract

Suppose that $h$ and $g$ belong to the algebra $\B$ generated by the rational functions and an entire function $f$ of finite order on ${\Bbb C}^n$ and that $h/g$ has algebraic polar variety. We show that either $h/g\in\B$ 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 $e^{-p}$.