There are two valid inputs for FrobeniusRootStrategy, namely Substitution and MonomialBasis. In the end, for each generator f of an ideal I, we are simply writing f = ∑aipe mi where m is a monomial all of whose exponents are < pe, then all the possible ai generate the frobeniusRoot. Substitution computes this by doing a Grobner basis computation in a ring with twice as many variables. MonomialBasis does this more directly and naively. There does not appear to be a single case where one is much faster than the other.