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

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