This following is a brief summary of our research meeting on 2024-07-09.

What happened

• Aaron outlined a solution to Exercise 1 in Section 1.4. For part (a), he used the second equation to write $y=\frac{1}{x}$, then substituted into the first equation to get $x^2+\frac{1}{x^2}-1=0$. Clearing denominators then gave the equation $x^4-x^2+1=0$. He then showed how to write that quartic as an $\mathbf{R}[x,y]$-linear combination of $x^2+y^2-1$ and $xy-1$.

• 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 $\mathbf{V}(J)$ is the single point $(0,1)$, and then gave several functions that vanished at that point, such as $f(x,y)=x+y-1$ and $g(x,y)=x$. Nicholas gave a geometric idea of something similar. The entire group then discussed strategies of proving that $f(x,y)$ (or $g(x,y)$) is not in the ideal $J$. 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 $\varphi$ is the variety given parametrically by $w_1=t^2-t$, $w_2=t^2$, $w_3=t-3t^2$. We noted that the point of the problem was to find an implicit description of this variety, i.e., equations in $w_1,w_2,w_3$ that would determine exactly that variety. Aaron noted that the equations certainly satisfied $w_1+2w_2+w_3=0$, which is one linear equation. We ran out of time before finding a second polynomial relation satisfied by $w_1,w_2,w_3$. Can you find it?

Tasks for next meeting

• Read/skim Sections 8.1–8.3
• Work through the following exercises:
  • 8.1: 4, 5, 11
  • 8.2: 1