# Properties of Hesse derivatives of cubic curves

Sayan Dutta, Lorenz Halbeisen, Norbert Hungerbühler

Sayan Dutta, Lorenz Halbeisen, Norbert Hungerbühler

**10/9/23**Published in : arXiv:2309.05048

The Hesse curve or Hesse derivative Hess(Γ_f) of a cubic curve Γ_f given by a homogeneous polynomial f is the set of points P such that \det \left(H_f (P)\right)=0, where H_f (P) is the Hesse matrix of f evaluated at P. Also Hess(Γ_f) is again a cubic curve. We show that for a point P∈Hess(Γ_f), all the contact points of tangents from P to the curves Γ_f and Hess(Γ_f) are intersection points of two straight lines \ell_1^P and \ell_2^P (meeting on Hess(Γ_f)) with Γ_f and Hess(Γ_f), where the product of \ell_1^P and \ell_2^P is the polar conic of Γ_f at P. The operator Hess defines an iterative discrete dynamical system on the set of the cubic curves. We identify the two fixed points of this system, investigate orbits that end in the fixed points, and discuss the closed orbits of the dynamical system.