number of solutions for an underdetermined quadratic polynomial system

Question

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.

What is the source of this system? Knowing that might give some geometric insight into the problem. — Travis Willse, May 08 '23 at 23:52
This systems arises from trying to fulfill some conditions placed on the determinants of some matrices. — user6006085, May 09 '23 at 00:03
Just as a simple example, $,(a,b,c,d,e,f) = (2t, -t, t, t, 0, 0),$ is a solution for any $,t \ne 0,$. What you have is two independent equations in three unknowns each, which will have infinitely many solutions. Then the inequality will reject some of those solutions, but you'll still have infinitely many left. — dxiv, May 09 '23 at 00:29
thanks @dxiv. Appreciate your input. I have found it difficult to find a reference with an explicit theorem that can be cited. Are there no established general results for this kind of thing. — user6006085, May 09 '23 at 00:47
@user6006085 $a^2+4bd=0$ defines a surface in the 3D space $(a,b,d)$, which normally has infinitely many points. See also When does a system of polynomial equations have infinitely many solutions? — dxiv, May 09 '23 at 01:36

number of solutions for an underdetermined quadratic polynomial system

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