Let $F(x, y, z)$ - homogeneous polynomial degree six, such that the set of real solutions equation $F(x, y, z)=0$ in $\mathbb{R}P^2$ is finite. How many solutions can be?
My attempt:
Obviously, $F=\prod_{i=1}^6 (a_ix+b_iy+c_iz)$. Some of line can be equal, so answer is from 0 to 6. (Example for zero is $F=x^6+y^6+z^6$.)
Am I true?