On undulation invariants of plane curves (Q2346063)

Let \(X\) be a smooth plane curve of degree \(r\) defined by a ternary homogeneous polynomial \(P\). An \textit{undulation point} of \(X\) is a point on the curve \(X\) at which the tangent line has a quadruple intersection. (Such a tangent line, or the corresponding point, is often also called a \textit{hyperflex}.) A classical result by \textit{G. Salmon} [A treatise on the higher plane curves: intended as a sequel to ``A treatise of conic sections''. Second edition. Dublin: Hodges, Foster and Co (1873; JFM 05.0340.03)] shows that the locus of curves that have an undulation point is of codimension \(1\), defined by a homogeneous polynomial \(I\) of degree \(6 (r - 3) (3 r - 2)\) in the \((r + 1)(r + 2)/2\) coefficients of the polynomial \(P\). The authors give a way to obtain \(I\) explicitly for \(r \in \left\{ 4, 5 \right\}\). They achieve this by interpreting the ideal generated by \(I\) as the generator of the degree \(0\) part of a certain graded ideal in a larger ring whose variables also include the defining coefficients of the chosen tangent line. The formula for \(I\) is then obtained as a suitable determinant with respect to small terms in the grading; by uniqueness, this determinant has to equal \(I\) (up to the usual intervening scalar). The determinantal formula, while having a tendency to be large, can be kept within bounds by choosing a suitable basis; the authors have taken care to find such a basis via an elegant application of representation theory. The authors' method is likely to generalize to other tangent configurations. All the same, in the original context it is not yet known whether similar determinantal formulas exist for any \(r\); this is an interesting open problem raised by the paper.