An Algorithm for Finding Symmetric Gr\"obner Bases in Infinite Dimensional Rings

A $\mathit{\text{symmetric}}\mathit{\text{ideal}}$ $I\subseteq R=K[x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.}]$ is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Gr"obner bases for symmetric ideals in the infinite dimensional polynomial ring $R$. This allows for symbolic computation in a new class of rings. In particular, we solve the ideal membership problem for symmetric ideals of $R$.