Recall that a point on the curve $F = 0$ in $\mathbb P^2_k$ is called an inflection point if $\operatorname{Hess}(F) = 0$, where $\operatorname{Hess}(F)$ is the determinant of the $3\times 3$ matrix $\left( \frac{\partial ^2 F}{\partial x_i \partial x_j}\right)$. Find infection point on the curve $F = x^4 + y^4 + z^4=0$ in $\mathbb P^2_{\mathbb C}$.
I am having difficulty solving the problem above. I understand that I need to find the points of intersection of $\operatorname{Hess}(F) = 0$ and $F = 0$. $\operatorname{Hess}(F)$ is given by $12^3(xyz)^2$. From here, how do I deduce which points in $\mathbb P^2_{\mathbb C}$ satisfy the equation?
Setting $\operatorname{Hess}(F) = 0$ gives you either $x=0, y=0$ or $z=0$.
For $x=0$, using $F = 0$ we have $y^4 = -z^4$, which is the same as
$$ y = \pm e^{i\pi/4} z,\ \ \pm ie^{i\pi/4} z, $$
which gives rises to the four points $$ [0,1, \pm e^{i\pi/4}], [0,1,\pm ie^{i\pi/4}].$$
One works similarly for $y=0$, $z=0$ to find that there are 12 inflection points on the curve.