Formulas for the eigendiscriminants of ternary and quaternary forms

TL;DR

This paper derives explicit formulas for eigendiscriminants of ternary and quaternary forms, enabling the detection of singular eigenschemes in tensors and symmetric tensors such as plane curves and surfaces.

Contribution

It provides the first explicit formulas for eigendiscriminants in these cases, including closed-form expressions for symmetric tensors in projective spaces.

Findings

Formulas for eigendiscriminants when n=3 and n=4.

Closed formulas for symmetric tensors representing plane curves and surfaces.

Determinant-based expressions for eigendiscriminants.

Abstract

A $d$-dimensional tensor $A$ of format $n\times n\times \cdots \times n$ defines naturally a rational map $\Psi$ from the projective space $\mathbb{P}^{n-1}$ to itself and its eigenscheme is then the subscheme of $\mathbb{P}^{n-1}$ of fixed points of $\Psi$. The eigendiscriminant is an irreducible polynomial in the coefficients of $A$ that vanishes for a given tensor if and only if its eigenscheme is singular. In this paper we contribute two formulas for the computation of eigendiscriminants in the cases $n=3$ and $n=4$. In particular, by restriction to symmetric tensors, we obtain closed formulas for the eigendiscriminants of plane curves and surfaces in $\mathbb{P}^3$ as the ratio of some determinants of resultant matrices.