On Undulation Invariants of Plane Curves

TL;DR

This paper provides explicit formulas for the undulation invariant of plane curves of degree 4 and 5, expressed as determinants of large matrices, aiding in the algorithmic detection of undulation points.

Contribution

It introduces explicit determinant formulas for the undulation invariant of quartic and quintic plane curves, expanding on classical algebraic geometry results.

Findings

Explicit 21x21 matrix formula for quartic curves

Explicit 36x36 matrix formula for quintic curves

Facilitates algorithmic detection of undulation points

Abstract

One of the general problems in algebraic geometry is to determine algorithmically whether or not a given geometric object, defined by explicit polynomial equations (e.g. a curve or a surface), satisfies a given property (e.g. has singularities or other distinctive features of interest). A classical example of such a problem, described by A.Cayley and G.Salmon in 1852, is to determine whether or not a given plane curve of degree r > 3 has undulation points -- the points where the tangent line meets the curve with multiplicity four. They proved that there exists an invariant of degree 6(r - 3)(3 r - 2) that vanishes if and only if the curve has undulation points. In this paper we give explicit formulae for this invariant in the case of quartics (r=4) and quintics (r=5), expressing it as the determinant of a matrix with polynomial entries, of sizes 21 times 21 and 36 times 36 respectively.