  One standard approach to compute the Hilbert function of any graded module
over a field is to come up with a free-resolution for the graded module and
another is via a Hilbert power series which serves as a generating function.
The proposed approaches enable generating the values of a Hilbert function when
the graded module is a quotient ring over a field by using combinatorics and
homological algebra. Two of these approaches named the lcm-Lattice method and
the Syzygy method, are conceptually combinatorial and work for any polynomial
quotient ring over a field. The third approach named Hilbert function table
method, also uses syzygies but the approach is better described in terms of
homological algebra.
