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Equivariant Gröbner bases of symmetric toric ideals

It has been shown previously that a large class of monomial maps equivariant under the action of an infinite symmetric group have finitely generated kernels up to the symmetric action. We prove that these symmetric toric ideals also have finite Gröbner bases up to symmetry for certain monomial orders. An algorithm is presented for computing equivariant Gröbner bases that terminates whenever a finite basis exists, improving on previous algorithms that only guaranteed termination in rings Noetherian up to symmetry. This algorithm can be used to compute equivariant Gröbner bases of the above toric ideals, given the monomial map.