On the realization problem of plane real algebraic curves as Hessian curves

TL;DR

This paper investigates which real plane algebraic curves, specifically those with only outer or inner ovals, can be realized as Hessian curves of smooth functions, establishing some realizability results.

Contribution

It provides new results on the realization problem of certain plane algebraic curves as Hessian curves, expanding understanding in differential geometry and singularity theory.

Findings

Some curves with only outer ovals are realizable as Hessian curves.

Some curves with only inner ovals are realizable as Hessian curves.

The study links Hessian topology with algebraic curve properties.

Abstract

The Hessian Topology is a subject having interesting relations with several areas, for instance, differential geometry, implicit differential equations, analysis and singularity theory. In this article we study the problem of realization of a real plane curve as the Hessian curve of a smooth function. The plane curves we consider are constituted either by only outer ovals or inner ovals. We prove that some of such curves are realizable as Hessian curves.