We're not ready to approach the book. First, I'm going to motivate using algebraic geometry.
Class 1/22:
Definition: the ideal generated by a set of polynomials.
Definition: an ideal. (1) should contain zero (2) should be closed under addition (3) should be closed under multiplying by any polynomial.
Definition: the vanishing set V(I) of an ideal, those points in C^n (we'll be working with the complex numbers hereafter) where all the polynomials vanish.
Definition: the ideal I(X) of all polynomials vanishing on a set X in C^n.
Homework #1, due Thursday Jan 31. If you're wondering whether "show" means "prove", the answer is... this is a real math class. Of course it does.
1. Let X be a set in C^n. Show V(I(X)) contains X.
2. Give an example of X where they're equal, and an example where they're not.
3. Let I be an ideal in C[x_1,...,x_n]. Show I(V(I)) contains I.
4. If I,J are ideals, let I+J := {i+j : i in I, j in J}. Show I+J is an ideal.
5. Show I+I = I.
6. Let I = < g_1, ..., g_m >. Show that V(I) = the set of x in C^n where every g_i vanishes. (Make sure you understand why that's different from the definition!)
[To show two sets are equal, show both containments -- if x is in the left-hand-side, then x is in the RHS, and separately, if x is in the RHS, then x is in the LHS.]
7. Assume g_1, ..., g_m are homogeneous linear polynomials, and let I be the ideal generated by them. Let p be another homogeneous linear polynomial. How would you test whether p is in I? (Describe an algorithm, perhaps, that correctly answers "yes" or "no" after finite time.)
8. What if the (g_i) and p in #7 are all homogeneous of the same degree, but that degree isn't necessarily 1?
