Entire functions polynomially bounded in several variables

Abstract

In this paper we show that if an entire function $f(z_1,z_2)$ of two (or more) complex variables verifies $\norm{f(z_1,z_2)}\leq K(\norm{P(z_1,z_2)})$, where $P(z_1,z_2)$ is a polynomial that is not a power in $\CC[[z_1,z_2]]$, and $K$ is any positive-valued real function, then $f(z_1,z_2)$ can be written as a holomorphic function of $P$.