Eigenpoint collinearities of plane cubics

TL;DR

This paper classifies and explicitly describes configurations of eigenpoints of plane cubic polynomials, focusing on collinearities, using geometric methods and computer algebra.

Contribution

It provides a complete classification and explicit equations for collinear eigenpoint configurations of plane cubics, combining geometric analysis with computational tools.

Findings

Classification of eigenpoint collinearities for plane cubics

Explicit equations for all eigenpoint configurations

Use of geometric techniques and computer algebra in analysis

Abstract

Given a ternary homogeneous polynomial, the fixed points of the map from $\mathbb{P}^2$ to itself defined by its gradient are called its eigenpoints. We focus on cubic polynomials, and analyze configurations of eigenpoints that admit one or more alignments. We give a classification and explicit equations, in the coordinates of the points, of all configurations: this is accomplished by using both geometric techniques and by an extensive use of computer algebra.