Result: Zero Sets of Generators and Differential Ideals: Zero sets of generators and differential ideals

Let \(E\) be a subring of the ring of all entire functions in \(\mathbb{C}^N\), \({\mathcal F}=\{f_1,\dots,f_m\}\) be a fixed collection of functions in \(E\), and \(E[{\mathcal F}]\) be the ideal in \(E\) generated by \({\mathcal F}\). The problem under consideration is to determine if \(E[{\mathcal F}]=E\). The authors treat the case of \[ E=E_{\mathcal P}=\bigcup_{p\in {\mathcal P}}\biggl\{f: \log\bigl|f(z) \bigr|\leq p(z)+ O(1)\biggr\}, \] \({\mathcal P}\) being a collection of plurisubharmonic functions \(p(z)=p(|z_1|, \dots,|z_N|)\), which grow faster than \(\log|z|\). Under some additional condition on \({\mathcal P}\), a complete characterization of all \({\mathcal F}\) with \(E_{\mathcal P}[{\mathcal F}]= E_{\mathcal P}\) is obtained in terms of the zero sets of the generators. As a consequence, this gives a solution to the problem for \(E\), the ring of all entire functions of finite order.