REU Meeting  20240709
This following is a brief summary of our research meeting on 20240709.
What happened

Aaron outlined a solution to Exercise 1 in Section 1.4. For part (a), he used the second equation to write
, then substituted into the first equation to get . Clearing denominators then gave the equation . He then showed how to write that quartic as an linear combination of and . 
Liam presented a solution for all three parts of Exercise 3 in Section 1.4. He proved each equality by showing each ideal contained the generators for the other ideal. He noted that usually one containment was easy, while the other was a bit sneakier.

Liam then used his work in Exercise 3 to quickly solve Exercise 5.

Aaron and Nicholas both took turns analyzing Exercise 2 in Section 4.1. Aaron first showed that
is the single point , and then gave several functions that vanished at that point, such as and . Nicholas gave a geometric idea of something similar. The entire group then discussed strategies of proving that (or ) is not in the ideal . The best strategy is probably to argue by degrees, but the result was certainly less than obvious! 
We then took the last few minutes to outline the problem and strategy for solving Exercise 1 in Section 5.1. We first noted that the image of the map
is the variety given parametrically by , , . We noted that the point of the problem was to find an implicit description of this variety, i.e., equations in that would determine exactly that variety. Aaron noted that the equations certainly satisfied , which is one linear equation. We ran out of time before finding a second polynomial relation satisfied by . Can you find it?
Tasks for next meeting
 Read/skim Sections 8.1β8.3
 Work through the following exercises:
 8.1: 4, 5, 11
 8.2: 1