Entire functions polynomially bounded in several variables

TL;DR

This paper proves that entire functions of multiple complex variables, which are polynomially bounded by a non-power polynomial, can be expressed as holomorphic functions of that polynomial.

Contribution

It establishes a new characterization of entire functions bounded by non-power polynomials in several variables, linking them to holomorphic functions of those polynomials.

Findings

Entire functions bounded by non-power polynomials are functions of those polynomials.

The result generalizes classical one-variable polynomial bounds to several variables.

Provides a structural understanding of polynomially bounded entire functions.

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 $P$.