Power-closed ideals of polynomial and Laurent polynomial rings

We investigate the structure of power-closed ideals of the complex polynomial ring $R=\mathbb{C}[x_1,\dots ,x_d]$ and the Laurent polynomial ring $R^{\pm }=\mathbb{C}[x_1,\dots ,x_d{]}^{\pm }=M^{-1}\mathbb{C}[x_1,\dots ,x_d]$, where $M$ is the multiplicative sub-monoid $M=[x_1,\dots ,x_d]$ of $R$. Here, an ideal $I$ is {\em power-closed} if $f(x_1,\dots ,x_d)\in I$ implies $f(x_1^i,\dots ,x_d^i)\in I$ for each natural $i$. In particular, we investigate related closure and interior operators on the set of ideals of $R$ and $R^{\pm }$. Finally, we give a complete description of principal power-closed ideals and of the radicals of general power-closed ideals of $R$ and $R^{\pm }$.