= Solution
An <inflection point> is a smooth point where the tangent line has <intersection multiplicity> at least three with the curve. For a nonsingular plane curve of degree $n\geq3$, these are the intersections with its <Hessian curve>, whose degree is $3(n-2)$. The original curve is not a component of its Hessian: otherwise in each graph chart the tangent direction would have zero derivative identically, making the curve locally a line and hence a line globally. Bézout therefore gives the upper bound \b[$3n(n-2)$], counting multiplicity. A nonsingular <conic> has no inflection points; lines are excluded from this finite-contact definition.
Take $F=y^2z-x^3+xz^2$. Its Hessian determinant is
$$
\det D^2F=8(3xy^2-3x^2z-z^3).
$$
At infinity the curve has only $O=[0:1:0]$, which is an <inflection point>: its tangent is $z=0$ and the restricted equation is $-x^3=0$. In the chart $z=1$, substitute $y^2=x^3-x$ into the Hessian equation to obtain
$$
3x^4-6x^2-1=0,\qquad x^2=1\pm\frac2{\sqrt3}.
$$
Thus the full answer is
$$
\boxed{O=[0:1:0],\qquad [x:y:1]\text{ with }x^2=1\pm2/\sqrt3,\quad y=\pm\sqrt{x^3-x}.}
$$
The four complex values of $x$ are distinct and none is $0,1,-1$, so each gives two distinct nonzero values of $y$. Together with $O$ these are all nine inflection points, exhausting the Bézout bound.
Back to article page
