Indecomposable Matrices Defining Plane Cubics

TL;DR

This paper classifies all 6x6 linear determinantal representations of smooth Weierstrass cubics, including indecomposable cases, and confirms the Kippenhahn conjecture for matrices of size 6.

Contribution

It provides a complete classification of 6x6 determinantal representations of plane cubics and verifies a significant conjecture in matrix analysis.

Findings

All 6x6 determinantal representations of smooth Weierstrass cubics are characterized.

The Kippenhahn conjecture is verified for 6x6 matrices.

Abstract

In this article we find all (decomposable and indecomposable) $6\times 6$ linear determinantal representations of smooth Weierstrass cubics. As a corollary we verify the Kippenhahn conjecture for $M_6$.