This is an exercise (exercise 3.3) question from the text "Ideals, Varieties, and Algorithms" by Cox, Little, and O'Shea. I have question about part c.
Using Macaulay2 (web) I got the second elimination ideal $I_2=I\cap\mathbb{R}[x,y,z]=\langle x^2-yz\rangle$. I get this from the reduced Groebner basis of $$I=\langle x-uv,y-u^2,z-v^2\rangle$$ which is $GB=\{u^2-y,uv-x,ux-vy,uz-vx,v^2-z,x^2-yz\}$.
Now solving $x^2-yz=0$ we need $y,z$ to have the same sign. But from $v^2-z=0$, that sign needs to be plus to be over $\mathbb{R}$. What is the answer to the last part of the question: as to which parametrization would cover the other "half."