Computing Equivariant Gröbner Bases

A polynomial ring over a countably infinite number of 
variables presents some obstacles to computation 
because it is not Noetherian. However, often ideals 
of interest in this setting are endowed with certain 
symmetry. Given an action of a monoid G on the set of 
variables, we consider G-equivariant ideals finitely 
generated up to the action of G. I will describe a 
known algorithm that can compute equivariant Gröbner 
bases. We implemented it as a package for Macaulay2, 
and applied this software to computing the kernels of 
certain toric maps on infinite dimensional rings. In 
particular we answer the next case of an open question 
about their finite-generation.