UTM Ideals Varieties and Algorithm - Chapter 1 Section 3 Exercise 11

Problem:

Solution:

Part (a) is trivial given exercise 8, we already know to parametrization of the curve $y^2 = cx^2 - x^3$, just replace $(x, y)$ by $(z ,x)$ we get what we want.

For part (b) is just as simple, we replace $y$ by $u$ and set $c = y^2 = u^2$, then we are done.

For any point on the surface, $y$ must be either positive, 0 or negative. In any case, we reduce to a particular curve $x^2 = cz^2 - z^3$, now we know by problem 8 part (c) that the parametization covers the whole curve, so the parameterization cover all points on the surface!