I want to find the inflection points(i.e. $I_P(C,T_P C) \ge 3$) of cubic curve $x^3+y^3+z^3=0$ in $\mathbb{P}^2_k$
If $(char(k),2)=1$, I know that the smooth inflection points are equal to the intersection points with Hessian curve. Since the curve is smooth, we can find exactly all inflection points.
But how can I find them if $char(k)=2$?