Power-closed ideals of polynomial and Laurent polynomial rings

Abstract

We investigate the structure of power-closed ideals of the complex polynomial ring $R=C[x_1,\dots ,x_d]$ and the Laurent polynomial ring $R^{\pm }=C{[x_1,\dots ,x_d]}^{\pm }=S^{-1}C[x_1,\dots ,x_d]$, where S is the multiplicatively closed semigroup $S=[x_1,\dots ,x_d]$. Here, an ideal I is 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 number i. Important examples of such ideals are provided by the ideals of relations in Minkowski rings of convex polytopes. We investigate related closure and interior operators on the set of ideals of R and $R^{\pm }$ and we give a complete description of principal power-closed ideals and of radicals of general power-closed ideals of R and $R^{\pm }$.