Math 480, Spring 2014, Assignment 14

Carefully define the following terms, then give one example and one non-example of each:

1. Krull dimension (of a variety).
2. Transcendence degree (of $\mathsf{k}[V]$).
3. Filtration (on a $\mathsf{k}$-algebra).
4. Standard filtration on $\mathsf{k}[V]$ (a.k.a. filtration by degree).
5. Affine Hilbert function (of a filtered algebra).
6. Affine Hilbert polynomial.
7. Dimension (of an affine variety).

Carefully state the following theorems (you need not prove them):

1. Theorem relating the affine Hilbert function of $\mathsf{k}[V]$ to monomials lying outside $\mathrm{LT}(I)$ (taken with respect to a graded term order).
2. Formula for the affine Hilbert function of $\mathsf{k}[x_1,\dots,x_n]$.
3. Algorithm to compute the affine Hilbert polynomial of any affine variety (this should involve a Groebner basis calculation, the previous formula, and the Inclusion-Exclusion Principle).

Solve the following problems:

1. Section 9.3, problem 12.