For an underdetermined quadratic polynomial system like the one given below, how does one determine the number of solutions?
$a^2+4bd=0$
$-2af+4(be+cd)>0$
$f^2+4ce=0$
where $(a,b,c,d,e,f)$ are the six unknown variables.
I have been able to find a bunch of both real and complex solutions by simple trial and error, but I would like to know if there is some theorem I can apply that will tell me whether there are a finite or an infinite number of solutions.