Given a field k, the dimension of a vector space V over k completely describes V. Is there a similar way to describe other algebraic objects? We will consider a finite set M of monomials in n+1 variables, and try to count all monomials of degree d that are multiples of elements of M; this will give the Hilbert function of an ideal. We will show that the Hilbert function becomes a polynomial in large enough degree (depending on the ideal), called the Hilbert polynomial; use the Hilbert polynomial to give the degree and dimension of an ideal; and compute the Hilbert polynomial of the intersection of two curves to prove Bezout's theorem.
* Pizza and soda will be served.