A standard monomial for an ideal I and term order < is a monomial not in init(I).
Theorem: the images in C[x_1..x_n]/I of the standard monomials are a basis of that space.
Definition of graded ideal and Hilbert function h_I(d).
Many examples of Hilbert functions.
h_I(d) = the number of standard monomials of degree d.
Basic case: I=0. Then the Hilbert function is (d+n-1 choose n-1), by a stars and bars argument.
