I want to show that there exists a global map of a submanifold described as the solution of the equations below. So I found one map, but I need to know if it contains all possible solutions.
How does one show that all real solutions of the system of equations:
$x^2+xy-y-z = 0$
$2x^2+3xy-2y-3z = 0$
are of the form: $(t,t^2,t^3)$ ?