Is it possible to define a line with a single polynomial equation?

Question

Is it possible to come up with a polynomial $F(x,y,z)$ such that the solutions to the equation $F(x,y,z)=0$ are all points that lie on a given line?

Answer 1

Yes.

For example $(a_1x+b_1y+c_1z−d_1)^2+(a_2x+b_2y+c_2z−d_2)^2=0$ gives the intersection of two planes.

Proof sketch: If $u,v$ are real numbers, then $u^2+v^2=0$ if and only if $u=0$ and $v=0$

(See Thomas Andrews' comment, which gives the correct answer)

Answer 2

If the solution set is a line, then the distance from any member of the solution set to the line is zero.

In other words, find the equation of a cylinder of radius $r$ centered on the line, then set $r=0$.

So, for instance, the solution set $y^2 + z^2 = 0$ will be the line parameterized by the lines $x = t, y= z = 0.$