Converting subalgebra bases with the Sagbi walk

We present an algorithm which converts a given Sagbi basis of a polynomial K  -subalgebra AA with respect to one monomial ordering to the Sagbi basis of AA with respect to another monomial ordering, under the assumption that the subalgebra AA admits a finite Sagbi basis with respect to all monomial orderings. The Sagbi walk method converts a Sagbi basis by partitioning the computations following a path in the Sagbi fan.