The option PreviousGens is a list of polynomials of total degree at most $d-1$ which can be used with componentOfKernel to trim the monomial basis for that multidegree.
Functions with optional argument named PreviousGens:
componentOfKernel(...,PreviousGens=>...) -- see componentOfKernel -- Finds all minimal generators of a given degree in the kernel of a ring map