= Hessian criterion for a flex
{c}
{title2=$\det\operatorname{Hess}F(p)=0$}
For a <plane curve> defined by a <homogeneous polynomial> $F$ of degree $d\ge2$, assume the <field characteristic> is different from two and $d-1$ is nonzero in the <field>. A <smooth point of a variety> is a <flex> exactly when the <determinant> of the homogeneous <Hessian matrix> vanishes there, provided its <tangent line> is not a component. Put the point at $[0:0:1]$ with tangent $y=0$ and $F_y(p)\ne0$. The homogeneous identities give $F_{xz}(p)=F_{zz}(p)=0$ and $F_{yz}(p)=(d-1)F_y(p)$, so the determinant is $-(d-1)^2F_y(p)^2F_{xx}(p)$. In the completed <local ring>, write the curve as a <formal power series> $y=h(x)$. Its linear coefficient is zero and its quadratic coefficient is $-F_{xx}(p)/(2F_y(p))$. Vanishing of that coefficient is exactly <intersection multiplicity> at least three with the tangent, proving the criterion in these positive characteristics as well as in characteristic zero.
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