= Maximum number of flexes of a smooth plane curve
{title2=$3d(d-2)$}
A nonsingular complex <projective plane curve> of degree $d\geq2$, with no common <irreducible component> with its <Hessian curve>, has at most $3d(d-2)$ distinct <flexes>. The <Hessian criterion for a flex> identifies the flexes with the intersection, and <Bézout's theorem> counts its total <intersection multiplicity>. Higher contact can make this bound on distinct points strict.
For a general nonsingular curve of degree $d\geq3$, all flexes have tangent contact exactly three, making the bound an equality. For a fixed point and line through it, contact at least four imposes four independent linear conditions on the coefficients; the space of such point-line pairs has dimension three. Thus degree-$d$ equations with contact at least four form a proper closed exceptional locus. In local graph coordinates $y=h(x)$, the Hessian determinant along the curve is a nonvanishing factor times $h''(x)$, so contact exactly three makes the intersection simple. A nonsingular <plane conic> has no flexes; a line is excluded because its tangent is a component and its Hessian determinant is identically zero.
Back to article page
