A short proof of Cartan's Nullstellensatz for entire functions in $\mathbb C^n$

TL;DR

The paper provides a concise analytic proof of Cartan's Nullstellensatz for entire functions in several complex variables, utilizing properties of maximal ideals in the polydisk algebra.

Contribution

It introduces a simple formula for solving the Bézout equation in the space of entire functions, offering an accessible proof of the Nullstellensatz.

Findings

Derived a solution formula for the Bézout equation in entire functions

Provided a short, elementary proof of Cartan's Nullstellensatz

Connected maximal ideals in the polydisk algebra to point evaluations

Abstract

Using the fact that the maximal ideals in the polydisk algebra are given by the kernels of point evaluations, we derive a simple formula that gives a solution to the B\'ezout equation in the space of all entire functions of several complex variables. Thus a short and easy analytic proof of Cartan's Nullstellensatz is obtained.